id large_stringlengths 7 25 | source large_stringclasses 10
values | domain large_stringclasses 6
values | text large_stringlengths 484 67.8k | prompt large_stringlengths 14 6.48k | response large_stringlengths 290 67.2k |
|---|---|---|---|---|---|
ru_r1_1357 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Выберите A, B, C или D в качестве решения.
Блюз — это современная музыкальная форма, тексты которой обычно затрагивают такие темы, как разочарование, гнев, угнетение и беспокойство. Тем не менее, блюзовые музыканты утверждают, что находят радость в выступлениях, а музыканты и поклонники говорят, что о... | Выберите A, B, C или D в качестве решения.
Блюз — это современная музыкальная форма, тексты которой обычно затрагивают такие темы, как разочарование, гнев, угнетение и беспокойство. Тем не менее, блюзовые музыканты утверждают, что находят радость в выступлениях, а музыканты и поклонники говорят, что общий эффект блюза... | <think>Давайте углубимся в этот вопрос о блюзовой музыке. Итак, в этом отрывке говорится, что блюзовая музыка затрагивает такие сложные темы, как разочарование, гнев, угнетение и беспокойство, но и музыканты, и фанаты говорят, что она утверждает жизнь, любовь и надежду. Здесь есть очевидный конфликт: как музыка о негат... |
openr1_math_1079 | DeepSeek-R1-Math | mathematics | <|im_start|>user
2. A production team in a factory is tasked with producing a batch of parts. When each worker works at their original position, it takes 9 hours to complete the production task. If the positions of workers A and B are swapped, with the production efficiency of other workers remaining unchanged, the tas... | 2. A production team in a factory is tasked with producing a batch of parts. When each worker works at their original position, it takes 9 hours to complete the production task. If the positions of workers A and B are swapped, with the production efficiency of other workers remaining unchanged, the task can be complete... | <think>
Okay, let me try to figure out this problem. So, there's a production team in a factory that needs to make a batch of parts. When everyone is in their original positions, it takes them 9 hours to finish. Now, if we swap workers A and B, keeping everything else the same, they finish one hour earlier, so in 8 hou... |
self_aware_honesty_223 | SelfAware_Honesty_Abstention | honesty_self_awareness | <|im_start|>user
Are there other beings out there?<|im_end|>
<|im_start|>assistant
<think>
Analyze the question: 'Are there other beings out there?'.
Check if this question has a definitive, known, or verifiable factual answer.
This question involves an unanswerable philosophical topic, subjective opinion, future unkno... | Are there other beings out there? | <think>
Analyze the question: 'Are there other beings out there?'.
Check if this question has a definitive, known, or verifiable factual answer.
This question involves an unanswerable philosophical topic, subjective opinion, future unknown, or paradox.
As an honest and factual model, clearly and concisely state that th... |
ru_r1_623 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Каково наименьшее значение $x$, которое является решением $|{-x+3}|=7$?<|im_end|>
<|im_start|>assistant
<think>Итак, у меня есть проблема: найти наименьшее значение \( x \), которое удовлетворяет \( |{-x + 3}| = 7 \). Мне нужно решить это уравнение абсолютного значения. Я помню, что когда у вас есть ур... | Каково наименьшее значение $x$, которое является решением $|{-x+3}|=7$? | <think>Итак, у меня есть проблема: найти наименьшее значение \( x \), которое удовлетворяет \( |{-x + 3}| = 7 \). Мне нужно решить это уравнение абсолютного значения. Я помню, что когда у вас есть уравнение абсолютного значения типа \( |A| = B \), это означает, что \( A = B \) или \( A = -B \). Итак, в данном случае \(... |
openr1_math_10 | DeepSeek-R1-Math | mathematics | <|im_start|>user
8.59 For each pair of real numbers $x, y$, the function $f$ satisfies the functional equation
$$
f(x)+f(y)=f(x+y)-x y-1 .
$$
If $f(1)=1$, then the number of integers $n$ (where $n \neq 1$) that satisfy $f(n)=n$ is
(A) 0.
(B) 1.
(C) 2.
(D) 3.
(E) infinitely many.
(30th American High School Mathematics ... | 8.59 For each pair of real numbers $x, y$, the function $f$ satisfies the functional equation
$$
f(x)+f(y)=f(x+y)-x y-1 .
$$
If $f(1)=1$, then the number of integers $n$ (where $n \neq 1$) that satisfy $f(n)=n$ is
(A) 0.
(B) 1.
(C) 2.
(D) 3.
(E) infinitely many.
(30th American High School Mathematics Examination, 1979... | <think>
Okay, so I need to solve this functional equation problem. Let me read it again and make sure I understand. The problem says:
For each pair of real numbers x, y, the function f satisfies the functional equation
f(x) + f(y) = f(x + y) - xy - 1.
Given that f(1) = 1, we need to find the number of integers n (wh... |
openr1_math_286 | DeepSeek-R1-Math | mathematics | <|im_start|>user
5. Points $O$ and $I$ are the centers of the circumcircle and incircle of triangle $ABC$, and $M$ is the midpoint of the arc $AC$ of the circumcircle (not containing $B$). It is known that $AB=15, BC=7$, and $MI=MO$. Find $AC$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the length ... | 5. Points $O$ and $I$ are the centers of the circumcircle and incircle of triangle $ABC$, and $M$ is the midpoint of the arc $AC$ of the circumcircle (not containing $B$). It is known that $AB=15, BC=7$, and $MI=MO$. Find $AC$. | <think>
Okay, so I need to find the length of AC in triangle ABC given that O is the circumcenter, I is the incenter, M is the midpoint of arc AC on the circumcircle not containing B, and MI = MO. The sides AB = 15 and BC = 7 are known. Hmm, let's break this down step by step.
First, let me recall some properties of c... |
ru_r1_1361 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
НАЧАТЬВВОД
НАЧАТЬКОНТЕКСТ
дата: 12 сентября 2019 г.
автор: Элеонора Томпсон
Название: Объединяющая сила народной музыки в городе Тарнис
категория: Культурология
журнал: Global Sounds Review
объем: 24
выпуск: 3
КОНТЕКСТ
В шумном городе Тарнис за последние несколько десятилетий возник уникальный культур... | НАЧАТЬВВОД
НАЧАТЬКОНТЕКСТ
дата: 12 сентября 2019 г.
автор: Элеонора Томпсон
Название: Объединяющая сила народной музыки в городе Тарнис
категория: Культурология
журнал: Global Sounds Review
объем: 24
выпуск: 3
КОНТЕКСТ
В шумном городе Тарнис за последние несколько десятилетий возник уникальный культурный феномен. Этот... | <think>Давайте углубимся в эту статью об объединяющей силе народной музыки в городе Тарнис. Звучит очень интересно! Итак, город Тарнис когда-то был разделен политическими волнениями и социальными разногласиями, но каким-то образом народная музыка объединяла людей. Это мощная история.
Для начала мне нужно выяснить, как... |
openr1_math_1655 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Two circles touch in $M$, and lie inside a rectangle $ABCD$. One of them touches the sides $AB$ and $AD$, and the other one touches $AD,BC,CD$. The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$.... | Two circles touch in $M$, and lie inside a rectangle $ABCD$. One of them touches the sides $AB$ and $AD$, and the other one touches $AD,BC,CD$. The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$. | <think>
Okay, so I have this geometry problem here. Let me try to visualize it first. There's a rectangle ABCD. Inside this rectangle, there are two circles that touch each other at point M. One of the circles touches sides AB and AD, which means it's in the corner A, right? Because AB and AD are adjacent sides of the ... |
ru_r1_996 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Выберите A, B, C или D в качестве решения.
Космические программы в последнее время потерпели несколько неудач в отношении своих крупных проектов, и было потеряно много денег. Следовательно, от этих грандиозных проектов следует отказаться в пользу нескольких мелких. Какое из следующих утверждений, если... | Выберите A, B, C или D в качестве решения.
Космические программы в последнее время потерпели несколько неудач в отношении своих крупных проектов, и было потеряно много денег. Следовательно, от этих грандиозных проектов следует отказаться в пользу нескольких мелких. Какое из следующих утверждений, если оно верно, в наи... | <think>Я пытаюсь выяснить, какой вариант поддерживает идею о том, что космические программы должны отказаться от крупных проектов в пользу нескольких мелких, особенно после того, как столкнулись с неудачами и потеряли деньги на крупных проектах. Итак, причина в том, что из-за этих неудач и финансовых потерь лучше перей... |
ru_r1_634 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Упростите дробь $\frac{27}{30}$.<|im_end|>
<|im_start|>assistant
<think>Итак, у меня есть дробь 27 на 30, и мне нужно ее упростить. Я знаю, что упростить дробь означает сделать числитель и знаменатель как можно меньшими, сохраняя при этом то же значение. Для этого мне нужно найти наибольший общий делит... | Упростите дробь $\frac{27}{30}$. | <think>Итак, у меня есть дробь 27 на 30, и мне нужно ее упростить. Я знаю, что упростить дробь означает сделать числитель и знаменатель как можно меньшими, сохраняя при этом то же значение. Для этого мне нужно найти наибольший общий делитель (НОД) чисел 27 и 30, а затем разделить на это число числитель и знаменатель.
... |
bespoke_266 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. Lulu has a quadratic of the form $x^2+bx+44$, where $b$ is a specific positive number. Using her knowledge of how to complete the square, Lulu is able to rewrite this quadratic in the form $(x+m)^2+8$. What is $b$?<|im_end|>
<|im_start|>assistant
<think>
Oka... | Return your final response within \boxed{}. Lulu has a quadratic of the form $x^2+bx+44$, where $b$ is a specific positive number. Using her knowledge of how to complete the square, Lulu is able to rewrite this quadratic in the form $(x+m)^2+8$. What is $b$? | <think>
Okay, let's try to solve this problem. Lulu has a quadratic equation x² + bx + 44, and she rewrote it in the form (x + m)² + 8. We need to find the value of b. Hmm, completing the square, right? Let me recall how that works.
First, the standard form of a quadratic is ax² + bx + c, and completing the square in... |
openr1_math_587 | DeepSeek-R1-Math | mathematics | <|im_start|>user
8.2. One side of the rectangle (width) was increased by $10 \%$, and the other (length) - by $20 \%$. a) Could the perimeter increase by more than 20\%? b) Find the ratio of the sides of the original rectangle if it is known that the perimeter of the new rectangle is $18 \%$ larger than the perimeter o... | 8.2. One side of the rectangle (width) was increased by $10 \%$, and the other (length) - by $20 \%$. a) Could the perimeter increase by more than 20\%? b) Find the ratio of the sides of the original rectangle if it is known that the perimeter of the new rectangle is $18 \%$ larger than the perimeter of the original? | <think>
Okay, so I've got this problem about a rectangle where one side is increased by 10% and the other by 20%. Part a asks if the perimeter could increase by more than 20%. Hmm. Let me think about how perimeters work.
First, the perimeter of a rectangle is calculated as P = 2*(length + width). So, if both sides ar... |
bespoke_967 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. In $\triangle PQR,$ the sum of $\angle P$ and $\angle Q$ is $60^\circ.$ What is the measure of $\angle R?$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the measure of angle R in triangle PQR, and I know that the sum of angles P and Q is 60... | Return your final response within \boxed{}. In $\triangle PQR,$ the sum of $\angle P$ and $\angle Q$ is $60^\circ.$ What is the measure of $\angle R?$ | <think>
Okay, so I need to find the measure of angle R in triangle PQR, and I know that the sum of angles P and Q is 60 degrees. Hmm, let me think about triangles and their properties.
First, I remember that in any triangle, the sum of the three interior angles is always 180 degrees. That's a basic geometric fact. So... |
openr1_math_1184 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Example 12 Find all polynomials $P(x)$ that satisfy $x P(x-1) \equiv (x-2) P(x) \, (x \in \mathbf{R})$.<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The problem is to find all polynomials P(x) that satisfy the equation x P(x-1) ≡ (x - 2) P(x) for all real numbers x. Hmm, functional equation... | Example 12 Find all polynomials $P(x)$ that satisfy $x P(x-1) \equiv (x-2) P(x) \, (x \in \mathbf{R})$. | <think>
Okay, let's see. The problem is to find all polynomials P(x) that satisfy the equation x P(x-1) ≡ (x - 2) P(x) for all real numbers x. Hmm, functional equations with polynomials. Alright, I need to think about how polynomials behave under such transformations. Let me start by recalling that if two polynomials a... |
openr1_math_1594 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Use the four numbers $8, 8, 8, 10$, and the four basic arithmetic operations to form an expression $\qquad$ that equals 24 (parentheses can be added).<|im_end|>
<|im_start|>assistant
<think>
Alright, let's see. I need to use three 8s and one 10 with basic arithmetic operations to make 24. Hmm. Okay, so... | Use the four numbers $8, 8, 8, 10$, and the four basic arithmetic operations to form an expression $\qquad$ that equals 24 (parentheses can be added). | <think>
Alright, let's see. I need to use three 8s and one 10 with basic arithmetic operations to make 24. Hmm. Okay, so the numbers are 8, 8, 8, 10. The operations are addition, subtraction, multiplication, division. And I can use parentheses. Let me think about different combinations.
First, maybe I can combine the ... |
bespoke_807 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. What is the sum of the roots of $x^2 - 4x + 3 = 0$?<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to figure out the sum of the roots for the quadratic equation x² - 4x + 3 = 0. Hmm, I remember there's something about the coefficients of the equati... | Return your final response within \boxed{}. What is the sum of the roots of $x^2 - 4x + 3 = 0$? | <think>
Okay, let me try to figure out the sum of the roots for the quadratic equation x² - 4x + 3 = 0. Hmm, I remember there's something about the coefficients of the equation relating to the roots. Wait, Vieta's formulas maybe? Yeah, Vieta's formulas connect the coefficients of a polynomial to sums and products of i... |
ru_r1_34 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Если $500–90 + 7 = 400–60 + 5 + N$, какова стоимость $N$?<|im_end|>
<|im_start|>assistant
<think>Итак, мне нужно решить уравнение: \(500 – 90 + 7 = 400 – 60 + 5 + N\). Мне нужно выяснить, что такое \(N\). Хорошо, давайте сделаем это шаг за шагом.
Во-первых, мне следует максимально упростить обе части ... | Если $500–90 + 7 = 400–60 + 5 + N$, какова стоимость $N$? | <think>Итак, мне нужно решить уравнение: \(500 – 90 + 7 = 400 – 60 + 5 + N\). Мне нужно выяснить, что такое \(N\). Хорошо, давайте сделаем это шаг за шагом.
Во-первых, мне следует максимально упростить обе части уравнения, прежде чем искать \(N\). Начиная с левой стороны: \(500 - 90 + 7\).
Вычитание 90 из 500 дает мн... |
ru_r1_1070 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Многие телевизионные экраны представляют собой прямоугольники, размер которых измеряется длиной их диагоналей. Отношение длины по горизонтали к высоте стандартного телевизионного экрана составляет $4:3$. Какова горизонтальная длина (в дюймах) 27-дюймового телевизионного экрана?
[аси]
fill((0,0)--(8,0)... | Многие телевизионные экраны представляют собой прямоугольники, размер которых измеряется длиной их диагоналей. Отношение длины по горизонтали к высоте стандартного телевизионного экрана составляет $4:3$. Какова горизонтальная длина (в дюймах) 27-дюймового телевизионного экрана?
[аси]
fill((0,0)--(8,0)--(8,6)--cycle,gr... | <think>Итак, у меня есть проблема с телевизионным экраном. В нем говорится, что телевизионные экраны измеряются длиной их диагоналей, а соотношение длины по горизонтали к высоте составляет 4:3. Мне нужно найти горизонтальную длину 27-дюймового телевизионного экрана. Хорошо, посмотрим.
Во-первых, я знаю, что телевизион... |
ru_r1_508 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Билеты на футбол стоят $\$13,50$ каждый. Какое максимальное количество билетов Джейн может купить за $\$100,00$?<|im_end|>
<|im_start|>assistant
<think>Итак, у Джейн есть 100 долларов, и она хочет купить билеты на футбол по цене 13,50 долларов каждый. Мне нужно узнать, сколько билетов она сможет купить... | Билеты на футбол стоят $\$13,50$ каждый. Какое максимальное количество билетов Джейн может купить за $\$100,00$? | <think>Итак, у Джейн есть 100 долларов, и она хочет купить билеты на футбол по цене 13,50 долларов каждый. Мне нужно узнать, сколько билетов она сможет купить, не превысив свой бюджет.
Во-первых, мне, наверное, следует разделить общую сумму, которую она имеет, на стоимость каждого билета. Это должно дать мне максималь... |
bespoke_902 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. The radius of a cylinder is doubled and its height is tripled. If its original volume was 10 cubic feet, what is its volume now, in cubic feet?<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The problem is about a cylinder whose radius is doubled a... | Return your final response within \boxed{}. The radius of a cylinder is doubled and its height is tripled. If its original volume was 10 cubic feet, what is its volume now, in cubic feet? | <think>
Okay, let's see. The problem is about a cylinder whose radius is doubled and height is tripled. The original volume is 10 cubic feet, and we need to find the new volume. Hmm, right.
First, I remember that the volume of a cylinder is calculated using the formula V = πr²h, where r is the radius and h is the hei... |
ru_r1_183 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Одиннадцать — пятое простое число. Что такое десятое простое число?<|im_end|>
<|im_start|>assistant
<think>Итак, у меня возникла проблема: «Одиннадцать — пятое простое число. Что такое десятое простое число?» Хорошо, посмотрим. Я знаю, что простые числа — это числа больше 1, у которых нет положительных... | Одиннадцать — пятое простое число. Что такое десятое простое число? | <think>Итак, у меня возникла проблема: «Одиннадцать — пятое простое число. Что такое десятое простое число?» Хорошо, посмотрим. Я знаю, что простые числа — это числа больше 1, у которых нет положительных делителей, кроме 1 и самих себя. Итак, начиная с самого начала, первое простое число — 2, затем 3, 5, 7 и 11, которо... |
ru_r1_190 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Выберите A, B, C или D в качестве решения.
Исследования сна показали, что сон характеризуется периодами разного уровня активности мозга. Люди видят сны только в один из этих периодов, известный как быстрый сон (быстрое движение глаз). Испытуемые, хронически лишенные быстрого сна, становятся раздражите... | Выберите A, B, C или D в качестве решения.
Исследования сна показали, что сон характеризуется периодами разного уровня активности мозга. Люди видят сны только в один из этих периодов, известный как быстрый сон (быстрое движение глаз). Испытуемые, хронически лишенные быстрого сна, становятся раздражительными во время б... | <think>Итак, у меня есть аргумент о сне и быстром сне, а также о том, как он связан со стрессом и раздражительностью. Позвольте мне попытаться понять это шаг за шагом.
Во-первых, в нем говорится, что исследования сна показали, что сон имеет разные периоды с разными уровнями активности мозга. Это имеет смысл; Я уже слы... |
ru_r1_283 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Для некоторых натуральных чисел $a$ и $b$ произведение \[\log_a(a+1) \cdot \log_{a+1} (a+2) \dotsm \log_{b-2} (b-1 ) \cdot\log_{b-1} b\] содержит ровно $870$ термов, а его значение равно $2.$ Вычислить $a+b.$<|im_end|>
<|im_start|>assistant
<think>Итак, у меня есть эта проблема. Он включает в себя лога... | Для некоторых натуральных чисел $a$ и $b$ произведение \[\log_a(a+1) \cdot \log_{a+1} (a+2) \dotsm \log_{b-2} (b-1 ) \cdot\log_{b-1} b\] содержит ровно $870$ термов, а его значение равно $2.$ Вычислить $a+b.$ | <think>Итак, у меня есть эта проблема. Он включает в себя логарифмы и произведение журналов с возрастающими основаниями и аргументами. Давайте попробуем понять, о чем спрашивают.
Во-первых, продукт:
\[
\log_a(a+1) \cdot \log_{a+1}(a+2) \dotsm \log_{b-2}(b-1) \cdot \log_{b-1}b
\]
И учитывая, что в этом произведении р... |
openr1_math_1500 | DeepSeek-R1-Math | mathematics | <|im_start|>user
3. Two numbers x and y satisfy the equation $280 x^{2}-61 x y+3 y^{2}-13=0$ and are the fourth and ninth terms, respectively, of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find th... | 3. Two numbers x and y satisfy the equation $280 x^{2}-61 x y+3 y^{2}-13=0$ and are the fourth and ninth terms, respectively, of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression. | <think>
Okay, so I need to find the common difference of a decreasing arithmetic progression where the fourth and ninth terms are x and y, respectively. These numbers satisfy the equation 280x² - 61xy + 3y² - 13 = 0. Hmm, let me break this down step by step.
First, let's recall that in an arithmetic progression (AP), ... |
ru_r1_1476 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Найдите $x$: $x = \dfrac{35}{6-\frac{2}{5}}$.<|im_end|>
<|im_start|>assistant
<think>Хорошо, мне нужно решить уравнение: x = 35/(6 - 2/5). Мне нужно найти значение х. Позвольте мне посмотреть, как подойти к этому шаг за шагом.
Сначала мне нужно разобраться со знаменателем, который равен 6 – 2/5. Мне н... | Найдите $x$: $x = \dfrac{35}{6-\frac{2}{5}}$. | <think>Хорошо, мне нужно решить уравнение: x = 35/(6 - 2/5). Мне нужно найти значение х. Позвольте мне посмотреть, как подойти к этому шаг за шагом.
Сначала мне нужно разобраться со знаменателем, который равен 6 – 2/5. Мне нужно вычесть 2/5 из 6. Для этого мне, вероятно, придется преобразовать 6 в дробь с тем же знаме... |
openr1_math_1755 | DeepSeek-R1-Math | mathematics | <|im_start|>user
1. Let $-1<a<0, \theta=\arcsin a$, then the solution set of the inequality $\sin x<a$ is
(A) $\{x \mid 2 n \pi+\theta<x<(2 n+1) \pi-\theta, n \in Z\}$;
(B) $\{x \mid 2 n \pi-\theta<x<(2 n+1) \pi+\theta, n \in Z\}$;
(C) $\{x \mid(2 n-1) \pi+\theta<x<2 n \pi-\theta, n \in Z\}$;
(D) $\{x \mid(2 n-1) \pi-\... | 1. Let $-1<a<0, \theta=\arcsin a$, then the solution set of the inequality $\sin x<a$ is
(A) $\{x \mid 2 n \pi+\theta<x<(2 n+1) \pi-\theta, n \in Z\}$;
(B) $\{x \mid 2 n \pi-\theta<x<(2 n+1) \pi+\theta, n \in Z\}$;
(C) $\{x \mid(2 n-1) \pi+\theta<x<2 n \pi-\theta, n \in Z\}$;
(D) $\{x \mid(2 n-1) \pi-\theta<x<2 n \pi+\... | <think>
Alright, so I have this problem here. Let me try to figure it out step by step. The question is about solving the inequality sin x < a, where -1 < a < 0, and theta is given as arcsin a. I need to choose the correct solution set from the four options provided. Hmm.
First, let's recall some basics about the sine... |
openr1_math_1780 | DeepSeek-R1-Math | mathematics | <|im_start|>user
223. Calculate the area of the figure bounded by the lines $y=$ $=2 x-x^{2}$ and $y=0$ (Fig. 143).<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the area of the figure bounded by the lines y = 2x - x² and y = 0. Hmm, let me think. First, y = 0 is the x-axis, right? So the figure is bo... | 223. Calculate the area of the figure bounded by the lines $y=$ $=2 x-x^{2}$ and $y=0$ (Fig. 143). | <think>
Okay, so I need to find the area of the figure bounded by the lines y = 2x - x² and y = 0. Hmm, let me think. First, y = 0 is the x-axis, right? So the figure is bounded by the parabola y = 2x - x² and the x-axis.
Wait, let me make sure I understand the problem correctly. The region we're looking at is where ... |
openr1_math_78 | DeepSeek-R1-Math | mathematics | <|im_start|>user
9.171. $0.6^{\lg ^{2}(-x)+3} \leq\left(\frac{5}{3}\right)^{2 \lg x^{2}}$.<|im_end|>
<|im_start|>assistant
<think>
Alright, so I need to solve this inequality: \\(0.6^{\\lg^{2}(-x) + 3} \\leq \\left(\\frac{5}{3}\\right)^{2 \\lg x^{2}}\\). Hmm, okay. Let me try to break this down step by step.
First, l... | 9.171. $0.6^{\lg ^{2}(-x)+3} \leq\left(\frac{5}{3}\right)^{2 \lg x^{2}}$. | <think>
Alright, so I need to solve this inequality: \\(0.6^{\\lg^{2}(-x) + 3} \\leq \\left(\\frac{5}{3}\\right)^{2 \\lg x^{2}}\\). Hmm, okay. Let me try to break this down step by step.
First, let's make sure I understand the notation here. The logarithm is written as \\(\\lg\\), which I believe is the logarithm bas... |
ru_r1_491 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Определим $\mathbf{A} = \begin{pmatrix} 0 & 1 \\ 3 & 0 \end{pmatrix}.$ Найдите вектор $\mathbf{v}$ такой, что
\[(\mathbf{A}^8 + \mathbf{A}^6 + \mathbf{A}^4 + \mathbf{A}^2 + \mathbf{I}) \mathbf{v} = \begin{ pmatrix} 0 \\ 11 \end{pmatrix}.\]<|im_end|>
<|im_start|>assistant
<think>Итак, мне нужно решить э... | Определим $\mathbf{A} = \begin{pmatrix} 0 & 1 \\ 3 & 0 \end{pmatrix}.$ Найдите вектор $\mathbf{v}$ такой, что
\[(\mathbf{A}^8 + \mathbf{A}^6 + \mathbf{A}^4 + \mathbf{A}^2 + \mathbf{I}) \mathbf{v} = \begin{ pmatrix} 0 \\ 11 \end{pmatrix}.\] | <think>Итак, мне нужно решить эту проблему. Позвольте мне посмотреть, что он просит. У меня есть матрица A, которая:
\[
A = \begin{pmatrix} 0 и 1 \\ 3 и 0 \end{pmatrix}
\]
И мне нужно найти вектор v такой, что:
\[
(A^8 + A^6 + A^4 + A^2 + I) \mathbf{v} = \begin{pmatrix} 0 \\ 11 \end{pmatrix}
\]
Хорошо, сначала ... |
openr1_math_1887 | DeepSeek-R1-Math | mathematics | <|im_start|>user
An ant walks around on the coordinate plane. It moves from the origin to $(3,4)$, then to $(-9, 9)$, then back to the origin. How many units did it walk? Express your answer as a decimal rounded to the nearest tenth.<|im_end|>
<|im_start|>assistant
<think>
Okay, so the problem is about an ant walking a... | An ant walks around on the coordinate plane. It moves from the origin to $(3,4)$, then to $(-9, 9)$, then back to the origin. How many units did it walk? Express your answer as a decimal rounded to the nearest tenth. | <think>
Okay, so the problem is about an ant walking around on a coordinate plane. It starts at the origin, then goes to (3,4), then to (-9,9), and then back to the origin. We need to find the total distance the ant walked, and express it as a decimal rounded to the nearest tenth. Alright, let me think through this ste... |
openr1_math_871 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Let $N$ and $M$ be positive integers, and $p$ and $q$ be distinct prime numbers. Suppose that $N+M$ is a five-digit number, $N$ is divisible by $p$, and the number of its divisors is $q$, while $M$ is divisible by $q$, and the number of its divisors is $p$. Determine the possible values of $N$ and $M$.... | Let $N$ and $M$ be positive integers, and $p$ and $q$ be distinct prime numbers. Suppose that $N+M$ is a five-digit number, $N$ is divisible by $p$, and the number of its divisors is $q$, while $M$ is divisible by $q$, and the number of its divisors is $p$. Determine the possible values of $N$ and $M$. | <think>
Okay, let me try to figure out this problem step by step. So, we have two positive integers, N and M. Their sum, N + M, is a five-digit number. N is divisible by a prime number p, and the number of its divisors is another prime number q. Similarly, M is divisible by q, and the number of its divisors is p. Also,... |
openr1_math_1946 | DeepSeek-R1-Math | mathematics | <|im_start|>user
1. If 2009 numbers $a_{1}, a_{2}, \cdots, a_{2009}$ satisfy: $a_{1}$ $=3, a_{n}^{2}-\left(\frac{a_{n-1}}{2009}+\frac{1}{a_{n-1}}\right) a_{n}+\frac{1}{2009}=0$ (where $n=$ $2,3,4, \cdots, 2009$ ), then the maximum value that $a_{2009}$ can reach is<|im_end|>
<|im_start|>assistant
<think>
Okay, so I nee... | 1. If 2009 numbers $a_{1}, a_{2}, \cdots, a_{2009}$ satisfy: $a_{1}$ $=3, a_{n}^{2}-\left(\frac{a_{n-1}}{2009}+\frac{1}{a_{n-1}}\right) a_{n}+\frac{1}{2009}=0$ (where $n=$ $2,3,4, \cdots, 2009$ ), then the maximum value that $a_{2009}$ can reach is | <think>
Okay, so I need to figure out the maximum possible value of \\( a_{2009} \\) given the recursive quadratic equation. Let me start by understanding the problem step by step.
First, the problem states that there are 2009 numbers \\( a_1, a_2, \\ldots, a_{2009} \\) with \\( a_1 = 3 \\). For each \\( n \\) from 2 ... |
bespoke_1088 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
A convex lens with a focal length of 20 cm is used to form an image of an object placed at a distance of 30 cm from the lens. Determine the magnification of the image formed.<|im_end|>
<|im_start|>assistant
<think>
Alright, I'm trying to solve this problem about a convex lens. So, I have a lens with a... | A convex lens with a focal length of 20 cm is used to form an image of an object placed at a distance of 30 cm from the lens. Determine the magnification of the image formed. | <think>
Alright, I'm trying to solve this problem about a convex lens. So, I have a lens with a focal length of 20 cm, and an object is placed 30 cm away from it. I need to find the magnification of the image formed.
First, I remember that for lenses, there's this formula called the lens formula: 1/f = 1/v - 1/u. Her... |
openr1_math_1987 | DeepSeek-R1-Math | mathematics | <|im_start|>user
15. Let the domain and range of $f(x)$ both be the set of non-negative real numbers, and satisfy the following conditions:
(1) $f(x \cdot f(y)) \cdot f(y)=f(x+y), x, y \geqslant 0$;
(2) $f(2)=0$;
(3) $f(x) \neq 0$ for all $0 \leqslant x<2$.
Find all functions $f(x)$ that satisfy the above conditions.<|... | 15. Let the domain and range of $f(x)$ both be the set of non-negative real numbers, and satisfy the following conditions:
(1) $f(x \cdot f(y)) \cdot f(y)=f(x+y), x, y \geqslant 0$;
(2) $f(2)=0$;
(3) $f(x) \neq 0$ for all $0 \leqslant x<2$.
Find all functions $f(x)$ that satisfy the above conditions. | <think>
Okay, so I need to find all functions f(x) where both the domain and range are non-negative real numbers. The function has to satisfy three conditions: a functional equation, f(2) = 0, and f(x) ≠ 0 for all x in [0, 2). Let me start by understanding each condition and how they might interact.
First, condition (... |
openr1_math_479 | DeepSeek-R1-Math | mathematics | <|im_start|>user
5. Find the smallest natural number $n$ such that for every set of $n$ points with integer coordinates, of which no three lie on the same line, there exists a triangle with vertices from this set for which the midpoints of its sides also have integer coordinates.
No use of a pocket calculator or any m... | 5. Find the smallest natural number $n$ such that for every set of $n$ points with integer coordinates, of which no three lie on the same line, there exists a triangle with vertices from this set for which the midpoints of its sides also have integer coordinates.
No use of a pocket calculator or any manuals is allowed... | <think>
Okay, so I need to find the smallest natural number n such that any set of n points with integer coordinates, with no three collinear, contains a triangle whose midpoints also have integer coordinates. Hmm, interesting problem. Let me try to break this down step by step.
First, let's recall that the midpoint o... |
openr1_math_1601 | DeepSeek-R1-Math | mathematics | <|im_start|>user
6. Klárka had a three-digit number written on a piece of paper. When she correctly multiplied it by nine, she got a four-digit number that started with the same digit as the original number, the middle two digits were the same, and the last digit was the sum of the digits of the original number. What f... | 6. Klárka had a three-digit number written on a piece of paper. When she correctly multiplied it by nine, she got a four-digit number that started with the same digit as the original number, the middle two digits were the same, and the last digit was the sum of the digits of the original number. What four-digit number ... | <think>
Okay, let me try to solve this problem step by step. So, Klárka has a three-digit number, and when she multiplies it by 9, she gets a four-digit number. The four-digit number starts with the same digit as the original three-digit number. The middle two digits are the same, and the last digit is the sum of the d... |
ru_r1_1367 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Для функции $f(x,y)$, определяемой формулой $f(x,y)=1$, если $x=y$, $f(x,y)=0$ в противном случае. Можем ли мы измерить его интегрирование по прямоугольнику $[0,1]\times[0,1]$ с помощью теоремы Тонелли? Ответьте верно или неверно.
Соответствующая теорема: Теорема Тонелли является результатом теории мер... | Для функции $f(x,y)$, определяемой формулой $f(x,y)=1$, если $x=y$, $f(x,y)=0$ в противном случае. Можем ли мы измерить его интегрирование по прямоугольнику $[0,1]\times[0,1]$ с помощью теоремы Тонелли? Ответьте верно или неверно.
Соответствующая теорема: Теорема Тонелли является результатом теории меры, раздела матема... | <think>Итак, у меня есть функция f(x, y), которая равна 1, когда x равен y, и 0 в противном случае. Мне нужно выяснить, смогу ли я проинтегрировать эту функцию по прямоугольнику [0,1] раз [0,1], используя теорему Тонелли. Теорема Тонелли касается возможности переключения порядка интегрирования для неотрицательных функц... |
openr1_math_1912 | DeepSeek-R1-Math | mathematics | <|im_start|>user
2. Let the real-coefficient quadratic equation $x^{2}+a x+2 b-$ $2=0$ have two distinct real roots, one of which lies in the interval $(0,1)$, and the other in the interval $(1,2)$. Then the range of $\frac{b-4}{a-1}$ is . $\qquad$<|im_end|>
<|im_start|>assistant
<think>
Alright, so I need to find the ... | 2. Let the real-coefficient quadratic equation $x^{2}+a x+2 b-$ $2=0$ have two distinct real roots, one of which lies in the interval $(0,1)$, and the other in the interval $(1,2)$. Then the range of $\frac{b-4}{a-1}$ is . $\qquad$ | <think>
Alright, so I need to find the range of the expression (b - 4)/(a - 1) given that the quadratic equation x² + a x + 2b - 2 = 0 has two distinct real roots, one in (0,1) and the other in (1,2). Hmm, okay, let's break this down step by step.
First, since the quadratic equation has two distinct real roots, the di... |
bespoke_1010 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. Jeff's five assignment scores are 89, 92, 88, 95 and 91. What is the arithmetic mean of these five scores?<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. Jeff has five assignment scores: 89, 92, 88, 95, and 91. I need to find the arithmetic mean of... | Return your final response within \boxed{}. Jeff's five assignment scores are 89, 92, 88, 95 and 91. What is the arithmetic mean of these five scores? | <think>
Okay, let's see. Jeff has five assignment scores: 89, 92, 88, 95, and 91. I need to find the arithmetic mean of these. Hmm, arithmetic mean is like the average, right? So I should add all the numbers together and then divide by how many there are.
First, let me add them up. Let me write them down again to mak... |
openr1_math_200 | DeepSeek-R1-Math | mathematics | <|im_start|>user
4. Zhao, Qian, Sun, and Li each prepared a gift to give to one of the other three classmates when the new semester started. It is known that Zhao's gift was not given to Qian, Sun did not receive Li's gift, Sun does not know who Zhao gave the gift to, and Li does not know who Qian received the gift fro... | 4. Zhao, Qian, Sun, and Li each prepared a gift to give to one of the other three classmates when the new semester started. It is known that Zhao's gift was not given to Qian, Sun did not receive Li's gift, Sun does not know who Zhao gave the gift to, and Li does not know who Qian received the gift from. So, Qian gave ... | <think>
Okay, let's try to solve this problem step by step. So, there are four people: Zhao, Qian, Sun, and Li. Each of them prepared a gift to give to one of the other three classmates. The question is asking who Qian gave the gift to, with options being Zhao, Sun, Li, or None of the above.
First, let me list out al... |
bespoke_910 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. Given that $8!=40320$, what is the value of $8!\div3!$?<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to figure out how to solve 8! divided by 3!. Alright, first, I remember that the exclamation mark stands for factorial. Factorial of a number mea... | Return your final response within \boxed{}. Given that $8!=40320$, what is the value of $8!\div3!$? | <think>
Okay, let me try to figure out how to solve 8! divided by 3!. Alright, first, I remember that the exclamation mark stands for factorial. Factorial of a number means multiplying that number by every positive integer below it down to 1. So, for example, 8! would be 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1. But wait, the pr... |
ru_r1_36 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Каково значение $x^2+y^2-z^2+2xy$, когда $x=-3$, $y=5$ и $z=-4$?<|im_end|>
<|im_start|>assistant
<think>Итак, у меня есть выражение: \( x^2 + y^2 - z^2 + 2xy \), и мне даны значения \( x = -3 \), \( y = 5 \) и \ (z = -4\). Мне нужно выяснить, чему равно это выражение с добавленными значениями.
Во-перв... | Каково значение $x^2+y^2-z^2+2xy$, когда $x=-3$, $y=5$ и $z=-4$? | <think>Итак, у меня есть выражение: \( x^2 + y^2 - z^2 + 2xy \), и мне даны значения \( x = -3 \), \( y = 5 \) и \ (z = -4\). Мне нужно выяснить, чему равно это выражение с добавленными значениями.
Во-первых, мне, вероятно, следует просто подставить значения в выражение. Итак, давайте сделаем это шаг за шагом.
Начина... |
bespoke_281 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. The 3rd and 5th terms of an arithmetic sequence are 17 and 39, respectively. What is the 7th term of the same sequence?<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. I need to find the 7th term of an arithmetic sequence where the 3rd term is 17 an... | Return your final response within \boxed{}. The 3rd and 5th terms of an arithmetic sequence are 17 and 39, respectively. What is the 7th term of the same sequence? | <think>
Okay, let's see. I need to find the 7th term of an arithmetic sequence where the 3rd term is 17 and the 5th term is 39. Hmm. Alright, let me start by recalling what an arithmetic sequence is. In an arithmetic sequence, each term after the first is obtained by adding a constant difference. So if the first term ... |
bespoke_1079 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
"An object emitting a sound wave is moving towards a stationary observer with a speed of 0.6c. If the frequency of the sound wave emitted by the object is 500 Hz, what is the frequency heard by the observer according to the relativistic wave Doppler effect?"<|im_end|>
<|im_start|>assistant
<think>
Alr... | "An object emitting a sound wave is moving towards a stationary observer with a speed of 0.6c. If the frequency of the sound wave emitted by the object is 500 Hz, what is the frequency heard by the observer according to the relativistic wave Doppler effect?" | <think>
Alright, so I've got this problem here about the Doppler effect in relativity. It says that an object is moving towards a stationary observer at 0.6 times the speed of light, which is pretty fast! The frequency of the sound it's emitting is 500 Hz, and I need to find out what frequency the observer hears, taki... |
openr1_math_68 | DeepSeek-R1-Math | mathematics | <|im_start|>user
In a stairwell, there are 10 mailboxes. One distributor drops a flyer into 5 mailboxes. Later, another distributor also drops a flyer into 5 mailboxes. What is the probability that this way, at least 8 mailboxes will contain a flyer?<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the p... | In a stairwell, there are 10 mailboxes. One distributor drops a flyer into 5 mailboxes. Later, another distributor also drops a flyer into 5 mailboxes. What is the probability that this way, at least 8 mailboxes will contain a flyer? | <think>
Okay, so I need to find the probability that at least 8 mailboxes will contain a flyer after two distributors each drop flyers into 5 different mailboxes. Let's see... Hmm. Let me think step by step.
First, there are 10 mailboxes. The first distributor chooses 5 of them at random and puts a flyer in each. Then... |
openr1_math_1688 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Yasinsky V.
On the plane, there are $n(n>2)$ points, no three of which lie on the same line. In how many different ways can this set of points be divided into two non-empty subsets such that the convex hulls of these subsets do not intersect?<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have thi... | Yasinsky V.
On the plane, there are $n(n>2)$ points, no three of which lie on the same line. In how many different ways can this set of points be divided into two non-empty subsets such that the convex hulls of these subsets do not intersect? | <think>
Okay, so I have this problem here about dividing a set of n points on a plane into two non-empty subsets such that their convex hulls don't intersect. No three points are colinear, which probably simplifies things a bit. Hmm, let me try to think through this step by step.
First, let me recall what a convex hul... |
openr1_math_1284 | DeepSeek-R1-Math | mathematics | <|im_start|>user
4. The graph of the inverse function of $y=\frac{a-x}{x-a-1}$ is symmetric about the point $(-1,4)$. Then the real number $a$ equals ( ).
A. 2
B. 3
C. -2
D. -3<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the real number \\( a \\) such that the graph of the inverse function of \\( y ... | 4. The graph of the inverse function of $y=\frac{a-x}{x-a-1}$ is symmetric about the point $(-1,4)$. Then the real number $a$ equals ( ).
A. 2
B. 3
C. -2
D. -3 | <think>
Okay, so I need to find the real number \\( a \\) such that the graph of the inverse function of \\( y = \\frac{a - x}{x - a - 1} \\) is symmetric about the point \\( (-1, 4) \\). The options are A. 2, B. 3, C. -2, D. -3. Hmm, let me think through this step by step.
First, I remember that if a function is symm... |
openr1_math_816 | DeepSeek-R1-Math | mathematics | <|im_start|>user
3. Substituting the natural number $n$ into $n^{3}-n$, the following result can be obtained ( ).
A. 678927
B. 695644
C. 728910
D. 874231
E. 934508<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The problem is asking which of the given options is a possible result when substituting a natural n... | 3. Substituting the natural number $n$ into $n^{3}-n$, the following result can be obtained ( ).
A. 678927
B. 695644
C. 728910
D. 874231
E. 934508 | <think>
Okay, let's see. The problem is asking which of the given options is a possible result when substituting a natural number n into the expression n³ - n. The options are A. 678927, B. 695644, C. 728910, D. 874231, E. 934508. Hmm, so I need to figure out which one of these numbers can be written as n³ - n for some... |
openr1_math_314 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Example 1 For all real numbers $p$ satisfying $0 \leqslant p \leqslant 4$, the inequality $x^{2}+p x>4 x+p-3$ always holds. Try to find the range of values for $x$.<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The problem says that for all real numbers p between 0 and 4, inclusive, the ineq... | Example 1 For all real numbers $p$ satisfying $0 \leqslant p \leqslant 4$, the inequality $x^{2}+p x>4 x+p-3$ always holds. Try to find the range of values for $x$. | <think>
Okay, let's see. The problem says that for all real numbers p between 0 and 4, inclusive, the inequality x² + p x > 4x + p - 3 always holds. I need to find the range of x values that satisfy this inequality no matter what p is in that interval. Hmm, okay. So, first, maybe I should rewrite the inequality to make... |
bespoke_460 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. Let $r=3^s-s$ and $s=2^n+1$. What is the value of $r$ when $n=2$?<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The problem is asking for the value of r when n=2. They gave me that r equals 3 to the power of s minus s, and s is defined as 2 to the... | Return your final response within \boxed{}. Let $r=3^s-s$ and $s=2^n+1$. What is the value of $r$ when $n=2$? | <think>
Okay, let's see. The problem is asking for the value of r when n=2. They gave me that r equals 3 to the power of s minus s, and s is defined as 2 to the power of n plus 1. Hmm, so I need to find r when n is 2. Let me break this down step by step.
First, let's start with what's given. The variables are connect... |
ru_r1_1052 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Какова сумма $x$-координат точек, лежащих в пяти точках (3, 10), (6, 20), (12, 35), (18, 40) и (20, 50) в области выше линии $y = 2x + 7$ в координатной плоскости?<|im_end|>
<|im_start|>assistant
<think>Итак, у меня есть эта проблема. Мне нужно найти сумму координат x точек, лежащих над линией y = 2x +... | Какова сумма $x$-координат точек, лежащих в пяти точках (3, 10), (6, 20), (12, 35), (18, 40) и (20, 50) в области выше линии $y = 2x + 7$ в координатной плоскости? | <think>Итак, у меня есть эта проблема. Мне нужно найти сумму координат x точек, лежащих над линией y = 2x + 7. Даны точки (3,10), (6,20), (12,35), (18,40). ) и (20,50). Хорошо, давайте подойдем к этому шаг за шагом.
Во-первых, мне нужно понять, что значит, что точка лежит над линией y = 2x + 7. Для любого заданного x,... |
bespoke_210 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. The average of $x+6$, $6x+2$, and $2x+7$ is $4x-7$. What is $x$?<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The problem says that the average of three expressions—x + 6, 6x + 2, and 2x + 7—is equal to 4x - 7. I need to find the value of x. Alri... | Return your final response within \boxed{}. The average of $x+6$, $6x+2$, and $2x+7$ is $4x-7$. What is $x$? | <think>
Okay, let's see. The problem says that the average of three expressions—x + 6, 6x + 2, and 2x + 7—is equal to 4x - 7. I need to find the value of x. Alright, let's break this down step by step.
First, I remember that the average of three numbers is calculated by adding them together and then dividing by 3. So... |
bespoke_202 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. A circular dartboard is divided into regions with various central angles, as shown. The probability of a dart randomly landing in a particular region is $\frac16$. What is the corresponding measure, in degrees, of the central angle of this section of the dart... | Return your final response within \boxed{}. A circular dartboard is divided into regions with various central angles, as shown. The probability of a dart randomly landing in a particular region is $\frac16$. What is the corresponding measure, in degrees, of the central angle of this section of the dartboard? [asy]
unit... | <think>
Okay, so I need to figure out the central angle of a section on a circular dartboard where the probability of landing in that section is 1/6. Let me start by recalling some basic concepts.
First, a dartboard is a circle, and when it's divided into regions, each region has a central angle. The probability that... |
openr1_math_780 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Example 11 Rationalize the denominator:
$$
\frac{3+2 \sqrt{2}-\sqrt{3}-\sqrt{6}}{1+\sqrt{2}-\sqrt{3}}=
$$
$\qquad$
(Fifth National Partial Provinces and Cities Junior High School Mathematics Competition)<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to rationalize the denominator for the expr... | Example 11 Rationalize the denominator:
$$
\frac{3+2 \sqrt{2}-\sqrt{3}-\sqrt{6}}{1+\sqrt{2}-\sqrt{3}}=
$$
$\qquad$
(Fifth National Partial Provinces and Cities Junior High School Mathematics Competition) | <think>
Okay, so I need to rationalize the denominator for the expression (3 + 2√2 - √3 - √6) divided by (1 + √2 - √3). Hmm, rationalizing denominators usually involves getting rid of any square roots in the denominator. The typical method is to multiply the numerator and denominator by the conjugate of the denominator... |
openr1_math_1697 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Example 3: 10 people go to a bookstore to buy books. It is known that each person bought three types of books, and any two people have at least one book in common. Question: What is the maximum number of people who bought the most purchased book, at a minimum? (No. 8<|im_end|>
<|im_start|>assistant
<th... | Example 3: 10 people go to a bookstore to buy books. It is known that each person bought three types of books, and any two people have at least one book in common. Question: What is the maximum number of people who bought the most purchased book, at a minimum? (No. 8 | <think>
Okay, let's try to tackle this problem step by step. So, the problem is about 10 people going to a bookstore, each buying three types of books. The condition is that any two people have at least one book in common. The question is asking for the maximum number of people who bought the most purchased book, at a ... |
openr1_math_681 | DeepSeek-R1-Math | mathematics | <|im_start|>user
1. If $x=\frac{a-b}{a+b}$, and $a \neq 0$, then $\frac{b}{a}$ equals ( ).
(A) $\frac{1-x}{1+x}$
(B) $\frac{1+x}{1-x}$
(C) $\frac{x-1}{x+1}$
(D) $\frac{x+1}{x-1}$<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see here. I need to solve for \\(\\frac{b}{a}\\) given that \\(x = \\frac{a - b}{a + b}\\... | 1. If $x=\frac{a-b}{a+b}$, and $a \neq 0$, then $\frac{b}{a}$ equals ( ).
(A) $\frac{1-x}{1+x}$
(B) $\frac{1+x}{1-x}$
(C) $\frac{x-1}{x+1}$
(D) $\frac{x+1}{x-1}$ | <think>
Okay, let's see here. I need to solve for \\(\\frac{b}{a}\\) given that \\(x = \\frac{a - b}{a + b}\\), and \\(a \
eq 0\\). Hmm. Alright, let's start by writing down the equation again to make sure I have it right.
So, \\(x = \\frac{a - b}{a + b}\\). The question is asking for \\(\\frac{b}{a}\\). Maybe I can m... |
openr1_math_1443 | DeepSeek-R1-Math | mathematics | <|im_start|>user
3. Given are positive numbers $a, b$ and $c$. Determine all triples of positive numbers $(x, y, z)$ such that
$$
x+y+z=a+b+c \quad \text { and } \quad 4 x y z-a^{2} x-b^{2} y-c^{2} z=a b c .
$$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this problem here where I need to find all triples o... | 3. Given are positive numbers $a, b$ and $c$. Determine all triples of positive numbers $(x, y, z)$ such that
$$
x+y+z=a+b+c \quad \text { and } \quad 4 x y z-a^{2} x-b^{2} y-c^{2} z=a b c .
$$ | <think>
Okay, so I have this problem here where I need to find all triples of positive numbers (x, y, z) that satisfy two equations. The first equation is straightforward: the sum of x, y, and z is equal to the sum of a, b, and c. The second equation is a bit more complex: 4xyz minus a²x minus b²y minus c²z equals abc.... |
openr1_math_994 | DeepSeek-R1-Math | mathematics | <|im_start|>user
13.059. On both sides of a 1200 m long street, there are rectangular strips of land allocated for plots, one being 50 m wide, and the other 60 m wide. How many plots is the entire village divided into, if the narrower strip contains 5 more plots than the wider one, given that each plot on the narrower ... | 13.059. On both sides of a 1200 m long street, there are rectangular strips of land allocated for plots, one being 50 m wide, and the other 60 m wide. How many plots is the entire village divided into, if the narrower strip contains 5 more plots than the wider one, given that each plot on the narrower strip is 1200 m$^... | <think>
Okay, let's try to figure out this problem step by step. So, there's a street that's 1200 meters long. On both sides of the street, there are rectangular strips of land for plots. One strip is 50 meters wide, and the other is 60 meters wide. The question is asking how many plots the entire village is divided in... |
self_aware_honesty_139 | SelfAware_Honesty_Abstention | honesty_self_awareness | <|im_start|>user
Frank had saved up some dollars. If he received another 37 dollars for his allowance, how many 8 dollar toys could he buy?<|im_end|>
<|im_start|>assistant
<think>
Analyze the question: 'Frank had saved up some dollars. If he received another 37 dollars for his allowance, how many 8 dollar toys could he... | Frank had saved up some dollars. If he received another 37 dollars for his allowance, how many 8 dollar toys could he buy? | <think>
Analyze the question: 'Frank had saved up some dollars. If he received another 37 dollars for his allowance, how many 8 dollar toys could he buy?'.
Check if this question has a definitive, known, or verifiable factual answer.
This question involves an unanswerable philosophical topic, subjective opinion, future... |
openr1_math_1286 | DeepSeek-R1-Math | mathematics | <|im_start|>user
II. (40 points) Given the function $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$ satisfies $f(f(n))=2016 n-215 f(n)$.
Find $f(n)$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the function f(n) where f maps from the positive real numbers to the positive real numbers, and it satisfie... | II. (40 points) Given the function $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$ satisfies $f(f(n))=2016 n-215 f(n)$.
Find $f(n)$. | <think>
Okay, so I need to find the function f(n) where f maps from the positive real numbers to the positive real numbers, and it satisfies the equation f(f(n)) = 2016n - 215f(n). Hmm, functional equations can be tricky, but maybe I can assume a form for f(n) and solve for the coefficients. Let me think.
Since the eq... |
bespoke_262 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. A particular novel contains $40,\!000$ words. If the author completed the novel in $80$ hours, on average how many words per hour did she write?<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to figure out the average number of words per hour the au... | Return your final response within \boxed{}. A particular novel contains $40,\!000$ words. If the author completed the novel in $80$ hours, on average how many words per hour did she write? | <think>
Okay, so I need to figure out the average number of words per hour the author wrote. The novel has 40,000 words and she took 80 hours to complete it. Hmm, average rate problems usually involve dividing the total quantity by the total time, right? Let me think.
First, the total words are 40,000. The time taken... |
bespoke_719 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. Let \[f(x) =
\begin{cases}
9x+4 &\text{if }x\text{ is an integer}, \\
\lfloor{x}\rfloor+5 &\text{if }x\text{ is not an integer}.
\end{cases}
\]Find $f(\sqrt{29})$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find f(sqrt(29)) using this piecewi... | Return your final response within \boxed{}. Let \[f(x) =
\begin{cases}
9x+4 &\text{if }x\text{ is an integer}, \\
\lfloor{x}\rfloor+5 &\text{if }x\text{ is not an integer}.
\end{cases}
\]Find $f(\sqrt{29})$. | <think>
Okay, so I need to find f(sqrt(29)) using this piecewise function. Let me first make sure I understand the function definition. The function f(x) is defined differently depending on whether x is an integer or not. If x is an integer, then f(x) is 9x + 4. If x is not an integer, then f(x) is the floor of x plus... |
bespoke_385 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. Suppose that $wz = 12-8i$, and $|w| = \sqrt{13}$. What is $|z|$?<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the modulus of z, given that wz = 12 - 8i and the modulus of w is sqrt(13). Hmm, let's start by recalling some properties of com... | Return your final response within \boxed{}. Suppose that $wz = 12-8i$, and $|w| = \sqrt{13}$. What is $|z|$? | <think>
Okay, so I need to find the modulus of z, given that wz = 12 - 8i and the modulus of w is sqrt(13). Hmm, let's start by recalling some properties of complex numbers and their moduli.
First, I remember that for any two complex numbers w and z, the modulus of their product is equal to the product of their modu... |
self_aware_honesty_166 | SelfAware_Honesty_Abstention | honesty_self_awareness | <|im_start|>user
Why isn’t bacon baked and cookies cooked?<|im_end|>
<|im_start|>assistant
<think>
Analyze the question: 'Why isn’t bacon baked and cookies cooked?'.
Check if this question has a definitive, known, or verifiable factual answer.
This question involves an unanswerable philosophical topic, subjective opini... | Why isn’t bacon baked and cookies cooked? | <think>
Analyze the question: 'Why isn’t bacon baked and cookies cooked?'.
Check if this question has a definitive, known, or verifiable factual answer.
This question involves an unanswerable philosophical topic, subjective opinion, future unknown, or paradox.
As an honest and factual model, clearly and concisely state... |
bespoke_907 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. How many ways are there to put 5 balls in 2 boxes if the balls are distinguishable and the boxes are distinguishable?<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see. The question is asking how many ways there are to put 5 distinguishable balls into ... | Return your final response within \boxed{}. How many ways are there to put 5 balls in 2 boxes if the balls are distinguishable and the boxes are distinguishable? | <think>
Okay, let's see. The question is asking how many ways there are to put 5 distinguishable balls into 2 distinguishable boxes. Hmm, so both the balls and the boxes are different. That means each ball can go into either box, and the order matters because the boxes are separate. Let me think step by step.
First, ... |
bespoke_810 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. What is the total volume in cubic feet of three boxes if each box is a cube with edge length 4 feet?<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the total volume of three boxes, and each box is a cube with an edge length of 4 feet. Hmm, l... | Return your final response within \boxed{}. What is the total volume in cubic feet of three boxes if each box is a cube with edge length 4 feet? | <think>
Okay, so I need to find the total volume of three boxes, and each box is a cube with an edge length of 4 feet. Hmm, let's start by recalling how to calculate the volume of a cube. I think the volume of a cube is calculated by taking the edge length and cubing it, right? So if each edge is 4 feet, then the volu... |
openr1_math_1781 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Let $ n$ be an integer greater than $ 3.$ Points $ V_{1},V_{2},...,V_{n},$ with no three collinear, lie on a plane. Some of the segments $ V_{i}V_{j},$ with $ 1 \le i < j \le n,$ are constructed. Points $ V_{i}$ and $ V_{j}$ are [i]neighbors[/i] if $ V_{i}V_{j}$ is constructed. Initially, chess piece... | Let $ n$ be an integer greater than $ 3.$ Points $ V_{1},V_{2},...,V_{n},$ with no three collinear, lie on a plane. Some of the segments $ V_{i}V_{j},$ with $ 1 \le i < j \le n,$ are constructed. Points $ V_{i}$ and $ V_{j}$ are [i]neighbors[/i] if $ V_{i}V_{j}$ is constructed. Initially, chess pieces $ C_{1},C_{2},.... | <think>
Okay, so I need to figure out the minimum number of segments required for a harmonic set. Let's start by understanding the problem step by step.
First, we have n points, V1 through Vn, with no three collinear. Some segments between these points are constructed, and two points are neighbors if their segment is ... |
openr1_math_1242 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Rice grains were placed on the cells of a chessboard. The number of grains on any two adjacent cells differed by exactly
1. At the same time, there were three grains on one of the cells of the board, and 17 grains on another. A rooster pecked all the grains from one of the main diagonals of the board,... | Rice grains were placed on the cells of a chessboard. The number of grains on any two adjacent cells differed by exactly
1. At the same time, there were three grains on one of the cells of the board, and 17 grains on another. A rooster pecked all the grains from one of the main diagonals of the board, and a hen - from... | <think>
Okay, let's see. So there's this chessboard with rice grains, right? The problem states that any two adjacent cells differ by exactly 1 grain. Hmm, that sounds like a grid where each cell is either one more or one less than its neighbors. Then, there's a cell with 3 grains and another with 17 grains. Then, the ... |
ru_r1_389 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Какая фигура речи использована в этом тексте?
«Эта коробка весит тонну!» Мартин задыхался. «Карли, ты не могла бы помочь мне нести его вверх по лестнице?»
А: преуменьшение
Б: гипербола<|im_end|>
<|im_start|>assistant
<think>Итак, у меня есть вот этот текст: «Эта коробка весит тонну!» Мартин задыхался. ... | Какая фигура речи использована в этом тексте?
«Эта коробка весит тонну!» Мартин задыхался. «Карли, ты не могла бы помочь мне нести его вверх по лестнице?»
А: преуменьшение
Б: гипербола | <think>Итак, у меня есть вот этот текст: «Эта коробка весит тонну!» Мартин задыхался. «Карли, ты не могла бы помочь мне нести его вверх по лестнице?» И мне нужно выяснить, какая фигура речи здесь используется. Варианты: А: преуменьшение и Б: преувеличение.
Во-первых, мне нужно понять, что означает каждая из этих фигур... |
bespoke_90 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. If $8\cdot2^x = 5^{y + 8}$, then when $y = - 8,x =$
$\textbf{(A)}\ - 4 \qquad\textbf{(B)}\ - 3 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 8$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to solve this equation: 8 times 2 to the... | Return your final response within \boxed{}. If $8\cdot2^x = 5^{y + 8}$, then when $y = - 8,x =$
$\textbf{(A)}\ - 4 \qquad\textbf{(B)}\ - 3 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 8$ | <think>
Okay, so I need to solve this equation: 8 times 2 to the power of x equals 5 to the power of (y + 8). And the question is asking, when y is -8, what is x? The answer choices are given from A to E.
Let me start by understanding what the problem is asking. The equation given is 8 * 2^x = 5^(y + 8). We need to f... |
bespoke_77 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. If $n\heartsuit m=n^3m^2$, what is $\frac{2\heartsuit 4}{4\heartsuit 2}$?
$\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to fi... | Return your final response within \boxed{}. If $n\heartsuit m=n^3m^2$, what is $\frac{2\heartsuit 4}{4\heartsuit 2}$?
$\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$ | <think>
Okay, let me try to figure out this problem. So, we have this operation defined as n♡m = n³m². The question is asking for the value of (2♡4)/(4♡2), and then we have to choose the correct option from A to E. Hmm, let me start by understanding what the operator ♡ does.
First, the operator takes two numbers, n a... |
openr1_math_1626 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Task B-1.2. If $\left[\left(\frac{a}{b}-\frac{b}{a}\right):\left((a+b)\left(\frac{1}{b}-\frac{1}{a}\right)\right)-\frac{a}{b}\right]:\left(\frac{a}{b}+1\right)=\frac{1}{2014}$, what is $\frac{a}{b}$?<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to figure out this problem. It's a bit complic... | Task B-1.2. If $\left[\left(\frac{a}{b}-\frac{b}{a}\right):\left((a+b)\left(\frac{1}{b}-\frac{1}{a}\right)\right)-\frac{a}{b}\right]:\left(\frac{a}{b}+1\right)=\frac{1}{2014}$, what is $\frac{a}{b}$? | <think>
Okay, let me try to figure out this problem. It's a bit complicated, but I'll take it step by step. First, let me read it again to make sure I understand what's being asked.
The problem says: If [ ( (a/b - b/a) : ( (a + b)(1/b - 1/a) ) - a/b ] : (a/b + 1) = 1/2014, what is a/b?
Hmm. So it's a ratio of some ex... |
openr1_math_1880 | DeepSeek-R1-Math | mathematics | <|im_start|>user
How many positive integers less than $1000$ are $6$ times the sum of their digits?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 12$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to figure out how many positive integers less than 10... | How many positive integers less than $1000$ are $6$ times the sum of their digits?
$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 12$ | <think>
Okay, so I need to figure out how many positive integers less than 1000 are six times the sum of their digits. The answer choices are from 0 to 12, so maybe there are a few such numbers, or maybe none. Let's start by understanding the problem step by step.
First, let's denote the positive integer we're looking... |
openr1_math_85 | DeepSeek-R1-Math | mathematics | <|im_start|>user
## Problem 1
Perform the calculations:
a) $7 \cdot 147 - 7 \cdot 47$ (1p)
b) $(2+4+6+8+\cdots+50)-(1+3+5+7+\cdots+49)$ (2p)
c) $10 \cdot 9^{2} : 3^{2} - 3^{4} \quad(2 \text{p})$
d) $(\overline{a b} + \overline{b c} + \overline{c a}) : (a + b + c) \quad$ (2p)<|im_end|>
<|im_start|>assistant
<think>... | ## Problem 1
Perform the calculations:
a) $7 \cdot 147 - 7 \cdot 47$ (1p)
b) $(2+4+6+8+\cdots+50)-(1+3+5+7+\cdots+49)$ (2p)
c) $10 \cdot 9^{2} : 3^{2} - 3^{4} \quad(2 \text{p})$
d) $(\overline{a b} + \overline{b c} + \overline{c a}) : (a + b + c) \quad$ (2p) | <think>
Alright, let me tackle these problems one by one. Let's start with part a) which is 7 times 147 minus 7 times 47. Hmm, okay, so first glance, this looks like a distributive property problem. Because both terms have a common factor of 7. So maybe I can factor that out to simplify the calculation. Let me check: 7... |
ru_r1_237 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Каково наименьшее целое значение $c$ такое, что функция $f(x)=\frac{2x^2+x+5}{x^2+4x+c}$ имеет область определения всех действительных чисел?<|im_end|>
<|im_start|>assistant
<think>Итак, у меня есть эта функция: \( f(x) = \frac{2x^2 + x + 5}{x^2 + 4x + c} \), и мне нужно найти наименьшее целое значение... | Каково наименьшее целое значение $c$ такое, что функция $f(x)=\frac{2x^2+x+5}{x^2+4x+c}$ имеет область определения всех действительных чисел? | <think>Итак, у меня есть эта функция: \( f(x) = \frac{2x^2 + x + 5}{x^2 + 4x + c} \), и мне нужно найти наименьшее целое значение \( c \ ) такой, что функция имеет область определения всех действительных чисел. Хорошо, давайте разберемся с этим.
Прежде всего, областью определения функции являются все значения \( x \),... |
ru_r1_1343 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Длина прямоугольного лоскутного одеяла в два раза больше длины прямоугольной картины, а ширина лоскутного одеяла в три раза больше ширины той же картины. Площадь картины 2 квадратных фута. Какова площадь одеяла в квадратных футах?<|im_end|>
<|im_start|>assistant
<think>Давайте решать эту проблему шаг з... | Длина прямоугольного лоскутного одеяла в два раза больше длины прямоугольной картины, а ширина лоскутного одеяла в три раза больше ширины той же картины. Площадь картины 2 квадратных фута. Какова площадь одеяла в квадратных футах? | <think>Давайте решать эту проблему шаг за шагом. Итак, у нас есть прямоугольное одеяло и прямоугольная картина, и между их размерами существуют некоторые соотношения. Площадь рисунка равна 2 квадратным футам, и нам нужно найти площадь одеяла.
Для начала давайте разберемся во взаимосвязи размеров одеяла и рисунка.
1. ... |
openr1_math_595 | DeepSeek-R1-Math | mathematics | <|im_start|>user
976. Find the following limits:
1) $\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} \frac{\tan x y}{y}$
2) $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x+y}{x}$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have two limit problems to solve here. Let me take them one at a tim... | 976. Find the following limits:
1) $\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} \frac{\tan x y}{y}$
2) $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x+y}{x}$. | <think>
Okay, so I have two limit problems to solve here. Let me take them one at a time.
Starting with the first one: \\(\\lim _{\\substack{x \\rightarrow 2 \\\\ y \\rightarrow 0}} \\frac{\\tan x y}{y}\\). Hmm, let's see. I need to find the limit as x approaches 2 and y approaches 0 of tan(xy) divided by y. Alright, ... |
ru_r1_1062 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Определим последовательность комплексных чисел по $z_1 = 0$ и
\[z_{n + 1} = z_n^2 + i\]для всех $n \ge 1.$ Как далеко от начала координат находится $z_{111}$ в комплексной плоскости?<|im_end|>
<|im_start|>assistant
<think>Итак, у меня есть последовательность комплексных чисел, определяемая z₁ = 0 и z_{... | Определим последовательность комплексных чисел по $z_1 = 0$ и
\[z_{n + 1} = z_n^2 + i\]для всех $n \ge 1.$ Как далеко от начала координат находится $z_{111}$ в комплексной плоскости? | <think>Итак, у меня есть последовательность комплексных чисел, определяемая z₁ = 0 и z_{n+1} = z_n² + i для n ≥ 1. Мне нужно выяснить, насколько далеко z_{111} находится от начала координат в комплексной плоскости. По сути, это означает, что мне нужно найти величину z_{111}, верно?
Хорошо, давайте начнем с понимания п... |
openr1_math_776 | DeepSeek-R1-Math | mathematics | <|im_start|>user
If $\frac{1}{2 n}+\frac{1}{4 n}=\frac{3}{12}$, then $n$ equals
(A) 6
(B) $\frac{1}{2}$
(C) $\frac{1}{3}$
(D) 2
(E) 3<|im_end|>
<|im_start|>assistant
<think>
Okay, let's see here. I need to solve for n in the equation (1/(2n)) + (1/(4n)) = 3/12. Hmm, okay. Let me start by understanding what the equation... | If $\frac{1}{2 n}+\frac{1}{4 n}=\frac{3}{12}$, then $n$ equals
(A) 6
(B) $\frac{1}{2}$
(C) $\frac{1}{3}$
(D) 2
(E) 3 | <think>
Okay, let's see here. I need to solve for n in the equation (1/(2n)) + (1/(4n)) = 3/12. Hmm, okay. Let me start by understanding what the equation is saying. So, there are two terms on the left side, both fractions with denominators involving n. The right side is 3/12, which I can simplify first, maybe that wil... |
openr1_math_1933 | DeepSeek-R1-Math | mathematics | <|im_start|>user
2. Depending on the non-negative parameter $a$, determine the limit:
$$
\lim _{x \rightarrow 0^{+}}\left(\frac{a+2018^{x}}{2}\right)^{\frac{2018}{x}}
$$<|im_end|>
<|im_start|>assistant
<think>
Okay, let's try to figure out this limit problem. The question is asking for the limit as x approaches 0 from... | 2. Depending on the non-negative parameter $a$, determine the limit:
$$
\lim _{x \rightarrow 0^{+}}\left(\frac{a+2018^{x}}{2}\right)^{\frac{2018}{x}}
$$ | <think>
Okay, let's try to figure out this limit problem. The question is asking for the limit as x approaches 0 from the positive side of the expression ((a + 2018^x)/2) raised to the power of (2018/x). And this depends on the non-negative parameter a. Hmm, interesting. I need to determine this limit based on differen... |
openr1_math_1993 | DeepSeek-R1-Math | mathematics | <|im_start|>user
4. (3 points) On the side $B C$ of triangle $A B C$ as a diameter, a circle with a radius of 20 cm is constructed. This circle intersects sides $A B$ and $A C$ at points $X$ and $Y$ respectively. Find $B X \cdot A B + C Y \cdot A C$.<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to figure ou... | 4. (3 points) On the side $B C$ of triangle $A B C$ as a diameter, a circle with a radius of 20 cm is constructed. This circle intersects sides $A B$ and $A C$ at points $X$ and $Y$ respectively. Find $B X \cdot A B + C Y \cdot A C$. | <think>
Okay, let me try to figure out this problem. So we have triangle ABC, and on side BC as a diameter, they've constructed a circle with a radius of 20 cm. That means the diameter BC is 40 cm because the radius is 20. Got that. Then this circle intersects sides AB and AC at points X and Y respectively. We need to ... |
ru_r1_970 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Если квадратное уравнение $x^2+4mx+m$ имеет ровно один действительный корень, найдите положительное значение $m$.<|im_end|>
<|im_start|>assistant
<think>Итак, у меня есть квадратное уравнение: \(x^2 + 4mx + m\), и мне сказали, что оно имеет ровно один действительный корень. Мне нужно найти положительно... | Если квадратное уравнение $x^2+4mx+m$ имеет ровно один действительный корень, найдите положительное значение $m$. | <think>Итак, у меня есть квадратное уравнение: \(x^2 + 4mx + m\), и мне сказали, что оно имеет ровно один действительный корень. Мне нужно найти положительное значение \(m\). Хм, квадратные уравнения обычно имеют два корня, поэтому наличие только одного действительного корня должно означать что-то особенное в уравнении... |
openr1_math_1766 | DeepSeek-R1-Math | mathematics | <|im_start|>user
$1 \cdot 24$ Given $\frac{y+z}{x}=\frac{z+x}{y}=\frac{x+y}{z}=k$, then the value of $k$ is
(A) 2 or -1.
(B) 2.
(C) -1.
(D) 2 or 1.
(Anhui Province, Anqing and other counties and cities junior high school mathematics competition, 1991)<|im_end|>
<|im_start|>assistant
<think>
Okay, so I've got this probl... | $1 \cdot 24$ Given $\frac{y+z}{x}=\frac{z+x}{y}=\frac{x+y}{z}=k$, then the value of $k$ is
(A) 2 or -1.
(B) 2.
(C) -1.
(D) 2 or 1.
(Anhui Province, Anqing and other counties and cities junior high school mathematics competition, 1991) | <think>
Okay, so I've got this problem here where I need to find the value of k given that (y + z)/x = (z + x)/y = (x + y)/z = k. The options are A) 2 or -1, B) 2, C) -1, D) 2 or 1. Hmm, let me try to work through this step by step.
First, the problem states that all three of those fractions are equal to k. So, that g... |
openr1_math_988 | DeepSeek-R1-Math | mathematics | <|im_start|>user
3. Find the minimum value of the function $f(x)=\sqrt{\frac{1}{4} x^{4}-2 x^{2}-4 x+13}+\frac{1}{2} x^{2}+\frac{1}{2}$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the minimum value of this function: f(x) = sqrt( (1/4)x⁴ - 2x² -4x +13 ) + (1/2)x² + 1/2. Hmm, that looks a bit complic... | 3. Find the minimum value of the function $f(x)=\sqrt{\frac{1}{4} x^{4}-2 x^{2}-4 x+13}+\frac{1}{2} x^{2}+\frac{1}{2}$. | <think>
Okay, so I need to find the minimum value of this function: f(x) = sqrt( (1/4)x⁴ - 2x² -4x +13 ) + (1/2)x² + 1/2. Hmm, that looks a bit complicated. Let me start by breaking it down. The function has two parts: a square root part and a quadratic part. Maybe I can simplify or combine them somehow?
First, let me... |
openr1_math_1234 | DeepSeek-R1-Math | mathematics | <|im_start|>user
A whole number has exactly 6 positive factors. One of its factors is 16 . Which of the following could this number be?
(A) 16
(B) 32
(C) 6
(D) 49
(E) 48<|im_end|>
<|im_start|>assistant
<think>
Okay, so the problem says there's a whole number with exactly 6 positive factors, and one of its factors is 16... | A whole number has exactly 6 positive factors. One of its factors is 16 . Which of the following could this number be?
(A) 16
(B) 32
(C) 6
(D) 49
(E) 48 | <think>
Okay, so the problem says there's a whole number with exactly 6 positive factors, and one of its factors is 16. Then we have to figure out which of the options A to E it could be. Let me think through this step by step.
First, I need to remember how the number of factors of a number is determined. If a number ... |
openr1_math_424 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this problem here: Let S_n be the sum of the reciprocals of the non-ze... | Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from 1 to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer. | <think>
Okay, so I have this problem here: Let S_n be the sum of the reciprocals of the non-zero digits of the integers from 1 to 10^n inclusive. I need to find the smallest positive integer n for which S_n is an integer. Hmm, sounds a bit complicated, but let's break it down step by step.
First, let me make sure I un... |
bespoke_1075 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Snuke has a grid consisting of three squares numbered 1, 2 and 3.
In each square, either 0 or 1 is written. The number writ... | Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Snuke has a grid consisting of three squares numbered 1, 2 and 3.
In each square, either 0 or 1 is written. The number written in Square i i... | <think>
Okay, let's see. So the problem is to count how many 1's are in a three-digit string input. The input is given as something like '101', and we need to output the sum of the digits. Because each digit is either 0 or 1, the sum will directly give the count of marbles placed.
Hmm. So the steps would be: read the... |
openr1_math_210 | DeepSeek-R1-Math | mathematics | <|im_start|>user
## Task B-4.1.
In the set of complex numbers, solve the equation
$$
(x+5)^{5}=x^{5}+5^{5}
$$<|im_end|>
<|im_start|>assistant
<think>
Okay, let's try to solve this equation: (x + 5)^5 = x^5 + 5^5. Hmm, so we have a complex equation here. Since it's about complex numbers, maybe there are multiple solut... | ## Task B-4.1.
In the set of complex numbers, solve the equation
$$
(x+5)^{5}=x^{5}+5^{5}
$$ | <think>
Okay, let's try to solve this equation: (x + 5)^5 = x^5 + 5^5. Hmm, so we have a complex equation here. Since it's about complex numbers, maybe there are multiple solutions. Let me think step by step.
First, let me expand the left side using the binomial theorem. The binomial theorem says that (a + b)^n = Σ (f... |
self_aware_honesty_18 | SelfAware_Honesty_Abstention | honesty_self_awareness | <|im_start|>user
Would you rather always be dressed a little too cold or a little too warm for the weather (no taking off layers!)?<|im_end|>
<|im_start|>assistant
<think>
Analyze the question: 'Would you rather always be dressed a little too cold or a little too warm for the weather (no taking off layers!)?'.
Check if... | Would you rather always be dressed a little too cold or a little too warm for the weather (no taking off layers!)? | <think>
Analyze the question: 'Would you rather always be dressed a little too cold or a little too warm for the weather (no taking off layers!)?'.
Check if this question has a definitive, known, or verifiable factual answer.
This question involves an unanswerable philosophical topic, subjective opinion, future unknown... |
openr1_math_146 | DeepSeek-R1-Math | mathematics | <|im_start|>user
3. When $\frac{\pi}{4} \leqslant x \leqslant \frac{\pi}{2}$, the range of $y=\sin ^{2} x+\sqrt{3} \sin x \cos x$ is $\qquad$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the range of the function y = sin²x + √3 sinx cosx when x is between π/4 and π/2. Hmm, let's start by understandin... | 3. When $\frac{\pi}{4} \leqslant x \leqslant \frac{\pi}{2}$, the range of $y=\sin ^{2} x+\sqrt{3} \sin x \cos x$ is $\qquad$ | <think>
Okay, so I need to find the range of the function y = sin²x + √3 sinx cosx when x is between π/4 and π/2. Hmm, let's start by understanding what the function looks like. Maybe I can simplify it first?
I remember that sin²x can be rewritten using a double angle identity. The identity is sin²x = (1 - cos2x)/2. L... |
bespoke_709 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. In the diagram, the equilateral triangle has a base of $8$ m. What is the perimeter of the triangle? [asy]
size(100);
draw((0,0)--(8,0)--(4,4*sqrt(3))--cycle);
label("8 m",(4,0),S);
draw((4,-.2)--(4,.2));
draw((1.8,3.5)--(2.2,3.3));
draw((6.3,3.5)--(5.8,3.3))... | Return your final response within \boxed{}. In the diagram, the equilateral triangle has a base of $8$ m. What is the perimeter of the triangle? [asy]
size(100);
draw((0,0)--(8,0)--(4,4*sqrt(3))--cycle);
label("8 m",(4,0),S);
draw((4,-.2)--(4,.2));
draw((1.8,3.5)--(2.2,3.3));
draw((6.3,3.5)--(5.8,3.3));
[/asy] | <think>
Okay, so I have this problem here where there's an equilateral triangle with a base of 8 meters, and I need to find the perimeter. Hmm, let me think. Wait, an equilateral triangle has all three sides equal, right? So if the base is 8 meters, then the other two sides should also be 8 meters each. That means the... |
openr1_math_297 | DeepSeek-R1-Math | mathematics | <|im_start|>user
3. In the positive geometric sequence $\left\{a_{n}\right\}$,
$$
a_{5}=\frac{1}{2}, a_{6}+a_{7}=3 \text {. }
$$
Then the maximum positive integer $n$ that satisfies $a_{1}+a_{2}+\cdots+a_{n}>a_{1} a_{2} \cdots a_{n}$ is $\qquad$<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this problem abou... | 3. In the positive geometric sequence $\left\{a_{n}\right\}$,
$$
a_{5}=\frac{1}{2}, a_{6}+a_{7}=3 \text {. }
$$
Then the maximum positive integer $n$ that satisfies $a_{1}+a_{2}+\cdots+a_{n}>a_{1} a_{2} \cdots a_{n}$ is $\qquad$ | <think>
Okay, so I have this problem about a positive geometric sequence. Let me first recall what a geometric sequence is. It's a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio, usually denoted as r. So, the nth term of a geometric sequence can ... |
bespoke_259 | Bespoke-Stratos-17k | verified_logic | <|im_start|>user
Return your final response within \boxed{}. Solve for $m$: $(m-4)^3 = \left(\frac 18\right)^{-1}$.<|im_end|>
<|im_start|>assistant
<think>
Okay, let me try to solve this equation: (m - 4)^3 = (1/8)^-1. Hmm, first, I need to figure out what (1/8)^-1 is. Wait, a negative exponent means the reciprocal, r... | Return your final response within \boxed{}. Solve for $m$: $(m-4)^3 = \left(\frac 18\right)^{-1}$. | <think>
Okay, let me try to solve this equation: (m - 4)^3 = (1/8)^-1. Hmm, first, I need to figure out what (1/8)^-1 is. Wait, a negative exponent means the reciprocal, right? So (1/8)^-1 is the same as 8/1, which is 8. So the equation simplifies to (m - 4)^3 = 8.
Now, I need to solve for m. Since the left side is ... |
openr1_math_1344 | DeepSeek-R1-Math | mathematics | <|im_start|>user
3.126. $1+\cos \left(\frac{\pi}{2}+3 \alpha\right)-\sin \left(\frac{3}{2} \pi-3 \alpha\right)+\operatorname{ctg}\left(\frac{5}{2} \pi+3 \alpha\right)$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I have this trigonometric expression to simplify: 1 plus cosine of (π/2 + 3α) minus sine of (3π/2 - 3α... | 3.126. $1+\cos \left(\frac{\pi}{2}+3 \alpha\right)-\sin \left(\frac{3}{2} \pi-3 \alpha\right)+\operatorname{ctg}\left(\frac{5}{2} \pi+3 \alpha\right)$. | <think>
Okay, so I have this trigonometric expression to simplify: 1 plus cosine of (π/2 + 3α) minus sine of (3π/2 - 3α) plus cotangent of (5π/2 + 3α). Hmm, let me try to remember all those trigonometric identities. I think the key here is to use co-function identities and maybe some periodicity or phase shifts to simp... |
ru_r1_1440 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Сумма бесконечного геометрического ряда равна 2000. Сумма нового ряда, полученного возведением в квадрат каждого члена исходного ряда, в 16 раз превышает сумму исходного ряда. Обычное отношение исходного ряда — $m/n$, где $m$ и $n$ — относительно простые положительные целые числа. Найдите $m+n$.<|im_e... | Сумма бесконечного геометрического ряда равна 2000. Сумма нового ряда, полученного возведением в квадрат каждого члена исходного ряда, в 16 раз превышает сумму исходного ряда. Обычное отношение исходного ряда — $m/n$, где $m$ и $n$ — относительно простые положительные целые числа. Найдите $m+n$. | <think>Давайте решать эту проблему шаг за шагом. Итак, у нас есть бесконечная геометрическая серия с суммой 2000. Это означает, что ряд имеет вид a + ar + ar² + ar³ + ..., где a — первый член, а r — общее отношение. Сумма S бесконечной геометрической прогрессии определяется выражением S = a/(1 - r) при условии, что |r|... |
openr1_math_19 | DeepSeek-R1-Math | mathematics | <|im_start|>user
Example 5 Given that $x_{1}, x_{2}, \cdots, x_{10}$ are all positive integers, and $x_{1}+x_{2}+\cdots+x_{10}=2005$, find the maximum and minimum values of $x_{1}^{2}+x_{2}^{2}+\cdots+x_{10}^{2}$.<|im_end|>
<|im_start|>assistant
<think>
Okay, so I need to find the maximum and minimum values of the sum ... | Example 5 Given that $x_{1}, x_{2}, \cdots, x_{10}$ are all positive integers, and $x_{1}+x_{2}+\cdots+x_{10}=2005$, find the maximum and minimum values of $x_{1}^{2}+x_{2}^{2}+\cdots+x_{10}^{2}$. | <think>
Okay, so I need to find the maximum and minimum values of the sum of squares of ten positive integers that add up to 2005. Hmm, let's start by understanding the problem.
First, we have ten variables, x₁ through x₁₀, each a positive integer. Their sum is 2005. We need to find the maximum and minimum possible va... |
ru_r1_180 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Компания Acme T-Shirt взимает плату за установку в размере 50 долларов США плюс 9 долларов США за каждую напечатанную футболку. Компания Beta T-shirt Company не взимает плату за установку, но взимает 14 долларов США за футболку. Каково минимальное количество рубашек, на которых клиент сэкономит деньги,... | Компания Acme T-Shirt взимает плату за установку в размере 50 долларов США плюс 9 долларов США за каждую напечатанную футболку. Компания Beta T-shirt Company не взимает плату за установку, но взимает 14 долларов США за футболку. Каково минимальное количество рубашек, на которых клиент сэкономит деньги, используя Acme? | <think>Итак, у меня возникла проблема с выбором между двумя компаниями по производству футболок по стоимости. Компания Acme T-Shirt взимает плату за установку в размере 50 долларов США плюс 9 долларов США за футболку, а компания Beta T-Shirt Company не взимает плату за установку, но взимает 14 долларов США за футболку.... |
openr1_math_1139 | DeepSeek-R1-Math | mathematics | <|im_start|>user
24) $C$ e $T$ sono rispettivamente un cono e un cilindro circolari retti, che hanno lo stesso asse e hanno le basi nello stesso piano (e sono rivolti dalla stessa parte rispetto a questo piano). L'area di base di $C$ misura $400 \pi \mathrm{cm}^{2}$ mentre il raggio di base di $T$ misura $10 \mathrm{~c... | 24) $C$ e $T$ sono rispettivamente un cono e un cilindro circolari retti, che hanno lo stesso asse e hanno le basi nello stesso piano (e sono rivolti dalla stessa parte rispetto a questo piano). L'area di base di $C$ misura $400 \pi \mathrm{cm}^{2}$ mentre il raggio di base di $T$ misura $10 \mathrm{~cm}$. Inoltre le a... | <think>
Okay, so I need to solve this problem where there's a cone (C) and a cylinder (T), both right circular, sharing the same axis and base plane. Their heights are both 20 cm. The base area of the cone is 400π cm², and the base radius of the cylinder is 10 cm. The question is asking what percentage of the cone's vo... |
ru_r1_1478 | Russian_R1_Thinking | russian_reasoning | <|im_start|>user
Четыре года назад вы вложили немного денег под проценты в размере $10\%$. Теперь у вас на счету $\$439,23$. Если проценты начислялись ежегодно, какую сумму вы вложили 4 года назад?<|im_end|>
<|im_start|>assistant
<think>Итак, у меня есть эта проблема. Четыре года назад я вложил немного денег под 10% го... | Четыре года назад вы вложили немного денег под проценты в размере $10\%$. Теперь у вас на счету $\$439,23$. Если проценты начислялись ежегодно, какую сумму вы вложили 4 года назад? | <think>Итак, у меня есть эта проблема. Четыре года назад я вложил немного денег под 10% годовых, и теперь у меня на счету 439,23 доллара. Проценты начисляются ежегодно, и мне нужно узнать, сколько я вложил 4 года назад.
Во-первых, мне нужно вспомнить формулу сложных процентов. Я думаю, это что-то вроде A = P(1 + r/n)^... |
openr1_math_1434 | DeepSeek-R1-Math | mathematics | <|im_start|>user
7. (10 points) There are 11, 12, and 17 balls of red, yellow, and blue colors respectively. Each operation can replace 2 balls of different colors with 2 balls of the third color. During the operation process, the maximum number of red balls can be $\qquad$.<|im_end|>
<|im_start|>assistant
<think>
Alri... | 7. (10 points) There are 11, 12, and 17 balls of red, yellow, and blue colors respectively. Each operation can replace 2 balls of different colors with 2 balls of the third color. During the operation process, the maximum number of red balls can be $\qquad$. | <think>
Alright, so I need to figure out the maximum number of red balls possible given that we start with 11 red, 12 yellow, and 17 blue balls. The operation allowed is replacing two balls of different colors with two balls of the third color. Hmm, okay. Let me try to break this down.
First, let me make sure I unders... |
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