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[
{
"content": "If there are 4 peanuts in the first box, and Mary puts 6 more peanuts inside, resulting in 10 peanuts in the first box. Meanwhile, there are 7 peanuts in the second box, and John adds 5 more peanuts. How many peanuts are in both boxes combined?",
"role": "user"
},
{
"content": "After M... | If there are 4 peanuts in the first box, and Mary puts 6 more peanuts inside, resulting in 10 peanuts in the first box. Meanwhile, there are 7 peanuts in the second box, and John adds 5 more peanuts. How many peanuts are in both boxes combined? After Mary adds 6 more peanuts to the first box, there are 10 peanuts in th... |
[
{
"content": "What is the best way to get from one place to another?",
"role": "user"
},
{
"content": "The term \"best\" is relative to your individual preferences and circumstances. \n\nIn general, how to get from one place to another depends on several key factors, such as the distance between the... | What is the best way to get from one place to another? The term "best" is relative to your individual preferences and circumstances.
In general, how to get from one place to another depends on several key factors, such as the distance between these two places, available methods of transportations, and the amount of m... |
[
{
"content": "In the Cartesian coordinate system, the coordinates of the point $P(-2,1)$ with respect to the $x$-axis are ( ).\n\nA: $(2,1)$\n\nB: $(1,-2)$\n\nC: $(-2,-1)$\n\nD: $(2,-1)$",
"role": "user"
},
{
"content": "To find the coordinates of the point $P(-2,1)$ with respect to the $x$-axis, we... | In the Cartesian coordinate system, the coordinates of the point $P(-2,1)$ with respect to the $x$-axis are ( ).
A: $(2,1)$
B: $(1,-2)$
C: $(-2,-1)$
D: $(2,-1)$ To find the coordinates of the point $P(-2,1)$ with respect to the $x$-axis, we need to understand how reflection across the $x$-axis affects a point's coo... |
[
{
"content": "What is an operating model for AI?",
"role": "user"
},
{
"content": "It refers to the processes put in place to work with AI, from use case definition, to development to deployment and operation. Ultimately the objective of an operating model for AI is to streamline the value generatio... | What is an operating model for AI? It refers to the processes put in place to work with AI, from use case definition, to development to deployment and operation. Ultimately the objective of an operating model for AI is to streamline the value generation process from data to business results. Developing AI applications ... |
[
{
"content": "In the series A Song of Ice and Fire, who is the founder of House Blackfyre?",
"role": "user"
},
{
"content": "House Blackfyre, a branch that separated from House Targaryen, was founded by Daemon I Targaryen",
"role": "assistant"
}
] | In the series A Song of Ice and Fire, who is the founder of House Blackfyre? House Blackfyre, a branch that separated from House Targaryen, was founded by Daemon I Targaryen |
[
{
"content": "Classify if the following are hollywood or bollywood movies: Zanjeer; The Shawshank redemption; Sholay; Pride & prejudice",
"role": "user"
},
{
"content": "Zanjeer and Sholay are bollywood moves. The Shawshank redemption and Pride & Prejudice are hollywood movies.",
"role": "assist... | Classify if the following are hollywood or bollywood movies: Zanjeer; The Shawshank redemption; Sholay; Pride & prejudice Zanjeer and Sholay are bollywood moves. The Shawshank redemption and Pride & Prejudice are hollywood movies. |
[
{
"content": "A function $g$ is defined by $g(z) = (3 - 2i) z^2 + \\beta z + \\delta$ for all complex numbers $z$, where $\\beta$ and $\\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\\beta| + |\\delta|$?",
"role": "user"
... | A function $g$ is defined by $g(z) = (3 - 2i) z^2 + \beta z + \delta$ for all complex numbers $z$, where $\beta$ and $\delta$ are complex numbers and $i^2 = -1$. Suppose that $g(1)$ and $g(-i)$ are both real. What is the smallest possible value of $|\beta| + |\delta|$? Let $\beta = a + bi$ and $\delta = c + di$, where ... |
[
{
"content": "Points $P$, $Q$, and $R$ lie on the surface of a sphere with center $O$ and radius $25$. It is given that $PQ=15$, $QR=20$, $RP=25$, and that the distance from $O$ to $\\triangle PQR$ is $\\frac{x\\sqrt{y}}{z}$, where $x$, $y$, and $z$ are positive integers, $x$ and $z$ are relatively prime, and $... | Points $P$, $Q$, and $R$ lie on the surface of a sphere with center $O$ and radius $25$. It is given that $PQ=15$, $QR=20$, $RP=25$, and that the distance from $O$ to $\triangle PQR$ is $\frac{x\sqrt{y}}{z}$, where $x$, $y$, and $z$ are positive integers, $x$ and $z$ are relatively prime, and $y$ is not divisible by th... |
[
{
"content": "Why is Brazilian jiu jitsu an excellent martial art?",
"role": "user"
},
{
"content": "Brazilian jiu jitsu (BJJ) is an excellent martial art for a number of reasons. BJJ is, first and foremost, a generally safe martial art. BJJ does not emphasize or focus on striking techniques like ... | Why is Brazilian jiu jitsu an excellent martial art? Brazilian jiu jitsu (BJJ) is an excellent martial art for a number of reasons. BJJ is, first and foremost, a generally safe martial art. BJJ does not emphasize or focus on striking techniques like punching and kicking, rather, BJJ focuses on use of leverage and bod... |
[
{
"content": "A wheel has a circumference of 8 feet. If the time for a complete rotation of the wheel is shortened by 0.5 seconds, the speed increases by 6 miles per hour. Calculate the original speed $r$ of the wheel in miles per hour.\nA) 8\nB) 9\nC) 10\nD) 11\nE) 12",
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},
{
"content... | A wheel has a circumference of 8 feet. If the time for a complete rotation of the wheel is shortened by 0.5 seconds, the speed increases by 6 miles per hour. Calculate the original speed $r$ of the wheel in miles per hour.
A) 8
B) 9
C) 10
D) 11
E) 12 1. **Convert the circumference to miles**: The circumference of the w... |
[
{
"content": "Given this article, what is the largest size for a dead man zone?\n\nContext: The dead man zone is the area directly around a bushfire that is likely to burn within five minutes given the current wind conditions or an anticipated change in wind direction. The distance this zone extends from the fi... | Given this article, what is the largest size for a dead man zone?
Context: The dead man zone is the area directly around a bushfire that is likely to burn within five minutes given the current wind conditions or an anticipated change in wind direction. The distance this zone extends from the firefront is highly depend... |
[
{
"content": "Let the ellipse $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1(a＞b＞0)$$ have its left focus at F, its top vertex at A, and a line through point A perpendicular to AF intersects the ellipse and the positive x-axis at points P and Q, respectively, with AP:PQ = 8:5. \n(1) Find the eccentricity of ... | Let the ellipse $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a＞b＞0)$$ have its left focus at F, its top vertex at A, and a line through point A perpendicular to AF intersects the ellipse and the positive x-axis at points P and Q, respectively, with AP:PQ = 8:5.
(1) Find the eccentricity of the ellipse;
(2) Given ... |
[
{
"content": "Given the function $f(x)=\\cos (2x+\\varphi)$, where $(-\\pi\\leqslant \\varphi < \\pi)$, its graph is shifted to the right by $\\frac{\\pi}{2}$ units and overlaps with the graph of the function $y=\\sin x\\cos x + \\frac{\\sqrt{3}}{2}\\cos x$. What could be the value(s) of $|\\varphi|$?\n\nA: $\\... | Given the function $f(x)=\cos (2x+\varphi)$, where $(-\pi\leqslant \varphi < \pi)$, its graph is shifted to the right by $\frac{\pi}{2}$ units and overlaps with the graph of the function $y=\sin x\cos x + \frac{\sqrt{3}}{2}\cos x$. What could be the value(s) of $|\varphi|$?
A: $\frac{\pi}{6}$
B: $\frac{\pi}{3}$
C: $\f... |
[
{
"content": "Best place to visit for a day from Bangalore",
"role": "user"
},
{
"content": "1. Myshore\n2. Coorg\n3. Ooty",
"role": "assistant"
}
] | Best place to visit for a day from Bangalore 1. Myshore
2. Coorg
3. Ooty |
[
{
"content": "A certain stock yields 8%. The market value of the stock is $150. What is the percentage of the stock's original value?",
"role": "user"
},
{
"content": "To determine the percentage of the stock's original value, we need to know the original value of the stock. Without this information... | A certain stock yields 8%. The market value of the stock is $150. What is the percentage of the stock's original value? To determine the percentage of the stock's original value, we need to know the original value of the stock. Without this information, we cannot calculate the percentage of the original value based on ... |
[
{
"content": "What is Wimbledon?\n\nContext: The Wimbledon Championships, commonly known simply as Wimbledon, is the oldest tennis tournament in the world and is widely regarded as the most prestigious. It has been held at the All England Club in Wimbledon, London, since 1877 and is played on outdoor grass cour... | What is Wimbledon?
Context: The Wimbledon Championships, commonly known simply as Wimbledon, is the oldest tennis tournament in the world and is widely regarded as the most prestigious. It has been held at the All England Club in Wimbledon, London, since 1877 and is played on outdoor grass courts, with retractable roo... |
[
{
"content": "In a class of 94 students, 41 are taking French, 22 are taking German. Of the students taking French or German, some are taking both courses. There are 40 students not enrolled in either course. How many students are taking both French and German?",
"role": "user"
},
{
"content": "Let'... | In a class of 94 students, 41 are taking French, 22 are taking German. Of the students taking French or German, some are taking both courses. There are 40 students not enrolled in either course. How many students are taking both French and German? Let's denote the number of students taking both French and German as \( ... |
[
{
"content": "What sport did Troy Stradford play in college?",
"role": "user"
},
{
"content": "Troy Stradford played football at Boston College in the 1980s and went on to play professionally for the Miami Dolphins in the NFL.",
"role": "assistant"
}
] | What sport did Troy Stradford play in college? Troy Stradford played football at Boston College in the 1980s and went on to play professionally for the Miami Dolphins in the NFL. |
[
{
"content": "There are 11 square formations, each consisting of the same number of soldiers. If 1 general is added, a larger square formation can be formed. What is the minimum number of soldiers in one of the original square formations?",
"role": "user"
},
{
"content": "\n1. **Define the variables... | There are 11 square formations, each consisting of the same number of soldiers. If 1 general is added, a larger square formation can be formed. What is the minimum number of soldiers in one of the original square formations?
1. **Define the variables**:
Let \( a \) represent the number of soldiers in one original ... |
[
{
"content": "Is it safe for dogs to eat chocolate?",
"role": "user"
},
{
"content": "No, chocolate is considered dangerous for dogs to consume.",
"role": "assistant"
}
] | Is it safe for dogs to eat chocolate? No, chocolate is considered dangerous for dogs to consume. |
[
{
"content": "Let $a,$ $b,$ $c$ be distinct complex numbers such that\n\\[\\frac{a}{1 - b} = \\frac{b}{1 - c} = \\frac{c}{1 - a} = k.\\]Find the sum of all possible values of $k.$",
"role": "user"
},
{
"content": "From the given equation,\n\\begin{align*}\na &= k(1 - b), \\\\\nb &= k(1 - c), \\\\\nc... | Let $a,$ $b,$ $c$ be distinct complex numbers such that
\[\frac{a}{1 - b} = \frac{b}{1 - c} = \frac{c}{1 - a} = k.\]Find the sum of all possible values of $k.$ From the given equation,
\begin{align*}
a &= k(1 - b), \\
b &= k(1 - c), \\
c &= k(1 - a).
\end{align*}Then
\begin{align*}
a &= k(1 - b) \\
&= k(1 - k(1 - c)) \... |
[
{
"content": "Which episode does Jon Snow lead a battle where he is outnumbered by Ramsey Bolton?",
"role": "user"
},
{
"content": "\"Battle of the Bastards\" is the ninth episode of the sixth season of Game of Thrones which had the massive battle between Jon Snow and Ramsey Bolton. Ramsey Bolton ul... | Which episode does Jon Snow lead a battle where he is outnumbered by Ramsey Bolton? "Battle of the Bastards" is the ninth episode of the sixth season of Game of Thrones which had the massive battle between Jon Snow and Ramsey Bolton. Ramsey Bolton ultimately lost, thanks to Lord Baelish bringing fighters from the Vale ... |
[
{
"content": "When the greatest common divisor and least common multiple of two integers are multiplied, the product is 180. How many different values could be the greatest common divisor of the two integers?",
"role": "user"
},
{
"content": "We know that $\\gcd(a,b) \\cdot \\mathop{\\text{lcm}}[a,b... | When the greatest common divisor and least common multiple of two integers are multiplied, the product is 180. How many different values could be the greatest common divisor of the two integers? We know that $\gcd(a,b) \cdot \mathop{\text{lcm}}[a,b] = ab$ for all positive integers $a$ and $b$. Hence, in this case, $ab... |
[
{
"content": "In 1995, a village had a population that was a perfect square. Twelve years later, after an increase of 180 people, its population was 12 more than a perfect square. Twenty-four years after 1995, with an increase of another 180 people, the population once again equaled a perfect square. Determine ... | In 1995, a village had a population that was a perfect square. Twelve years later, after an increase of 180 people, its population was 12 more than a perfect square. Twenty-four years after 1995, with an increase of another 180 people, the population once again equaled a perfect square. Determine the closest percent gr... |
[
{
"content": "What are the FAANG companies?",
"role": "user"
},
{
"content": "Five of the best performing tech stocks of the last decade are Facebook, Amazon, Apple, Netflix, and Google. Those five stocks have been pulled together into an the acronym FAANG.",
"role": "assistant"
}
] | What are the FAANG companies? Five of the best performing tech stocks of the last decade are Facebook, Amazon, Apple, Netflix, and Google. Those five stocks have been pulled together into an the acronym FAANG. |
[
{
"content": "What is a whiteboard?\n\nContext: A whiteboard (also known by the terms marker board, dry-erase board, dry-wipe board, and pen-board) is a glossy, usually white surface for making non-permanent markings. Whiteboards are analogous to blackboards, but with a smoother surface allowing for rapid marki... | What is a whiteboard?
Context: A whiteboard (also known by the terms marker board, dry-erase board, dry-wipe board, and pen-board) is a glossy, usually white surface for making non-permanent markings. Whiteboards are analogous to blackboards, but with a smoother surface allowing for rapid marking and erasing of markin... |
[
{
"content": "For all positive integers \\( n \\) less than or equal to \\( 3000 \\), define \\( b_n \\) as follows:\n\\[\nb_n =\\left\\{\n\\begin{array}{lr} \n15, & \\text{if } n \\text{ is divisible by } 15 \\text{ and } 17;\\\\ \n17, & \\text{if } n \\text{ is divisible by } 17 \\text{ and } 13;\\\\ \n13, & ... | For all positive integers \( n \) less than or equal to \( 3000 \), define \( b_n \) as follows:
\[
b_n =\left\{
\begin{array}{lr}
15, & \text{if } n \text{ is divisible by } 15 \text{ and } 17;\\
17, & \text{if } n \text{ is divisible by } 17 \text{ and } 13;\\
13, & \text{if } n \text{ is divisible by } 13 \text{ ... |
[
{
"content": "Identify which animal species is alive or extinct: Repenomamus, Jaguar",
"role": "user"
},
{
"content": "Jaguar is alive, Repenomamus is extinct.",
"role": "assistant"
}
] | Identify which animal species is alive or extinct: Repenomamus, Jaguar Jaguar is alive, Repenomamus is extinct. |
[
{
"content": "At a restaurant that serves only vegan food, tell me whether each of the following items could be found in the kitchen or not: Butter, Eggs, Tofu, Chicken, Kale, Spinach, Peppers, Steak, Bread, Chickpeas, Pork",
"role": "user"
},
{
"content": "Butter: Could not be found in the kitchen\... | At a restaurant that serves only vegan food, tell me whether each of the following items could be found in the kitchen or not: Butter, Eggs, Tofu, Chicken, Kale, Spinach, Peppers, Steak, Bread, Chickpeas, Pork Butter: Could not be found in the kitchen
Eggs: Could not be found in the kitchen
Tofu: Could be found in the ... |
[
{
"content": "If \\(7x = 28\\) and \\(x + w = 9\\), what is the value of \\(xw\\)?\n(A) 9\n(B) 20\n(C) 18\n(D) 52\n(E) -252",
"role": "user"
},
{
"content": "\n1. To determine the value of \\( x \\), we start by solving the equation \\( 7x = 28 \\).\n\n\\[\nx = \\frac{28}{7}\n\\]\n\n\\[\nx = 4\n\\]\... | If \(7x = 28\) and \(x + w = 9\), what is the value of \(xw\)?
(A) 9
(B) 20
(C) 18
(D) 52
(E) -252
1. To determine the value of \( x \), we start by solving the equation \( 7x = 28 \).
\[
x = \frac{28}{7}
\]
\[
x = 4
\]
2. Next, we use the value of \( x \) to find \( w \) from the equation \( x + w = 9 \).
\[
4 + ... |
[
{
"content": "What is a, \"one-timer\" in ice hockey?",
"role": "user"
},
{
"content": "Ice hockey is a sport typically played with one goalie and five skaters on each of two opposing teams. A one-timer is an expression used to describe a shot taken by a player directly from a pass of another playe... | What is a, "one-timer" in ice hockey? Ice hockey is a sport typically played with one goalie and five skaters on each of two opposing teams. A one-timer is an expression used to describe a shot taken by a player directly from a pass of another player in a single motion. This does not necessarily have to be a shot on ... |
[
{
"content": "In an election with only two candidates, a candidate secured 70% of the valid votes and won by a certain majority. The total number of valid votes was 435. What was the majority of votes by which the candidate won?",
"role": "user"
},
{
"content": "Let's denote the winning candidate's ... | In an election with only two candidates, a candidate secured 70% of the valid votes and won by a certain majority. The total number of valid votes was 435. What was the majority of votes by which the candidate won? Let's denote the winning candidate's votes as \( W \) and the losing candidate's votes as \( L \).
Since... |
[
{
"content": "In the plane, the triangles $ABD$ and $BCE$ are outside the triangle $ABC$ and are also isosceles triangles right at $D$ and $C$ respectively. The point $H$ is the intersection of $CD$ and $AE$ . Show that $BH$ and $AE$ are perpendicular.",
"role": "user"
},
{
"conte... | In the plane, the triangles $ABD$ and $BCE$ are outside the triangle $ABC$ and are also isosceles triangles right at $D$ and $C$ respectively. The point $H$ is the intersection of $CD$ and $AE$ . Show that $BH$ and $AE$ are perpendicular. 1. **Identify the given information and setup the problem:**
... |
[
{
"content": "What are a couple date night ideas?",
"role": "user"
},
{
"content": "Date nights are a great way to connect with your significant other. You could go to dinner and a movie. This option may sound cliché but try to be creative by choosing a restaurant you haven't been to and see a mov... | What are a couple date night ideas? Date nights are a great way to connect with your significant other. You could go to dinner and a movie. This option may sound cliché but try to be creative by choosing a restaurant you haven't been to and see a movie at a small neighborhood movie theater. Another great option is a... |
[
{
"content": "What are the four primary states of matter? Answer in bullet point list, and capitalize each words",
"role": "user"
},
{
"content": "- Solid\n- Liquid\n- Gas\n- Plasma",
"role": "assistant"
}
] | What are the four primary states of matter? Answer in bullet point list, and capitalize each words - Solid
- Liquid
- Gas
- Plasma |
[
{
"content": "Name five NFL teams in the NFC.",
"role": "user"
},
{
"content": "Five NFL teams in the NFC are the Carolina Panthers, New Orleans Saints, Philadelphia Eagles, New York Giants, and Detroit Lions.",
"role": "assistant"
}
] | Name five NFL teams in the NFC. Five NFL teams in the NFC are the Carolina Panthers, New Orleans Saints, Philadelphia Eagles, New York Giants, and Detroit Lions. |
[
{
"content": "In the Cartesian coordinate system, if the line $ax-y+1=0$ passes through the focus of the parabola $y^2=4x$, then the real number $a=$ ____.",
"role": "user"
},
{
"content": "Firstly, the focus of the parabola $y^2=4x$ has the coordinates $(1,0)$, following the standard form of the pa... | In the Cartesian coordinate system, if the line $ax-y+1=0$ passes through the focus of the parabola $y^2=4x$, then the real number $a=$ ____. Firstly, the focus of the parabola $y^2=4x$ has the coordinates $(1,0)$, following the standard form of the parabolic equation where the focus is at $(\frac{p}{2},0)$ when the eq... |
[
{
"content": "A group of science students went on a field trip along with their teachers and another group of language students. They took transportation using the following vehicles: 3 vans, 5 buses, and 2 minibuses. The vans had a capacity of 7 students each, the buses could hold 25 students and 2 teachers ea... | A group of science students went on a field trip along with their teachers and another group of language students. They took transportation using the following vehicles: 3 vans, 5 buses, and 2 minibuses. The vans had a capacity of 7 students each, the buses could hold 25 students and 2 teachers each, and the minibuses ... |
[
{
"content": "The opposite of $2023$ is\n\nA: $-\\frac{1}{{2023}}$\n\nB: $\\frac{1}{{2023}}$\n\nC: $-2023$\n\nD: $2023$",
"role": "user"
},
{
"content": "To find the opposite of a number, we simply change its sign. If the number is positive, its opposite is negative, and vice versa. For the number $... | The opposite of $2023$ is
A: $-\frac{1}{{2023}}$
B: $\frac{1}{{2023}}$
C: $-2023$
D: $2023$ To find the opposite of a number, we simply change its sign. If the number is positive, its opposite is negative, and vice versa. For the number $2023$, which is positive, its opposite is obtained by changing its sign to neg... |
[
{
"content": "In town A, there are \\( n \\) girls and \\( n \\) boys, and each girl knows each boy. In town B, there are \\( n \\) girls \\( g_{1}, g_{2}, \\ldots, g_{n} \\) and \\( 2n-1 \\) boys \\( b_{1}, b_{2}, \\ldots, b_{2n-1} \\). The girl \\( g_{i}, i=1,2, \\ldots, n \\), knows the boys \\( b_{1}, b_{2}... | In town A, there are \( n \) girls and \( n \) boys, and each girl knows each boy. In town B, there are \( n \) girls \( g_{1}, g_{2}, \ldots, g_{n} \) and \( 2n-1 \) boys \( b_{1}, b_{2}, \ldots, b_{2n-1} \). The girl \( g_{i}, i=1,2, \ldots, n \), knows the boys \( b_{1}, b_{2}, \ldots, b_{2i-1} \), and no others. Fo... |
[
{
"content": "There are many CDs in the store. The rock and roll CDs are $5 each, the pop CDs are $10 each, the dance CDs are a certain price each, and the country CDs are $7 each. Julia wants to buy 4 of each, but she only has 75 dollars and is short by $25. How much does each dance CD cost?",
"role": "use... | There are many CDs in the store. The rock and roll CDs are $5 each, the pop CDs are $10 each, the dance CDs are a certain price each, and the country CDs are $7 each. Julia wants to buy 4 of each, but she only has 75 dollars and is short by $25. How much does each dance CD cost? Let's calculate the total cost of the CD... |
[
{
"content": "Which TV show features a team of teenagers that power up with powers given to them by Zordon to save the world from monsters?",
"role": "user"
},
{
"content": "The Mighty Morphin Power Rangers.",
"role": "assistant"
}
] | Which TV show features a team of teenagers that power up with powers given to them by Zordon to save the world from monsters? The Mighty Morphin Power Rangers. |
[
{
"content": "Which company designed SoundBridge?\n\nContext: SoundBridge is a hardware device from Roku, Inc. designed to play internet radio or digital audio streamed across a home network, over either Wi-Fi or ethernet. SoundBridge devices directly browsed the Radio Roku guide. As of 2008 all Roku SoundBridg... | Which company designed SoundBridge?
Context: SoundBridge is a hardware device from Roku, Inc. designed to play internet radio or digital audio streamed across a home network, over either Wi-Fi or ethernet. SoundBridge devices directly browsed the Radio Roku guide. As of 2008 all Roku SoundBridge products were disconti... |
[
{
"content": "Points $P_{1}(-1$,$y_{1})$, $P_{2}(\\frac{5}{2}$,$y_{2})$, $P_{3}(6$,$y_{3})$ are all on the graph of the quadratic function $y=mx^{2}-2mx+1\\left(m \\gt 0\\right)$. The relationship between $y_{1}$, $y_{2}$, and $y_{3}$ is ( ).\n\nA: $y_{1} \\gt y_{2} \\gt y_{3}$\n\nB: $y_{3} \\gt y_{2} \\gt y_{1... | Points $P_{1}(-1$,$y_{1})$, $P_{2}(\frac{5}{2}$,$y_{2})$, $P_{3}(6$,$y_{3})$ are all on the graph of the quadratic function $y=mx^{2}-2mx+1\left(m \gt 0\right)$. The relationship between $y_{1}$, $y_{2}$, and $y_{3}$ is ( ).
A: $y_{1} \gt y_{2} \gt y_{3}$
B: $y_{3} \gt y_{2} \gt y_{1}$
C: $y_{2} \gt y_{3} \gt y_{1}$... |
[
{
"content": "In a $6 \\times 6$ chessboard, some $1 \\times 2$ dominoes are placed, each covering exactly two squares. If there are still 14 squares uncovered, determine if at least one more domino can be placed on the board.",
"role": "user"
},
{
"content": "\n1. **Strategy for approaching the pro... | In a $6 \times 6$ chessboard, some $1 \times 2$ dominoes are placed, each covering exactly two squares. If there are still 14 squares uncovered, determine if at least one more domino can be placed on the board.
1. **Strategy for approaching the problem** \\
The problem involves placing $1 \times 2$ dominos on a $6... |
[
{
"content": "A picky princess is choosing a groom. One hundred grooms are courting her, each better than the last, and no two are equal. However, they propose in a random order. We will call a groom outstanding if the princess likes him more than all who have proposed before him. The first groom is also consid... | A picky princess is choosing a groom. One hundred grooms are courting her, each better than the last, and no two are equal. However, they propose in a random order. We will call a groom outstanding if the princess likes him more than all who have proposed before him. The first groom is also considered outstanding since... |
[
{
"content": "in the rectangular coordinate system , points ( 4 , 0 ) and ( – 4 , 0 ) both lie on circle c . what is the maximum possible value of the radius of c",
"role": "user"
},
{
"content": "The points (4, 0) and (-4, 0) are both on the x-axis and are 8 units apart. Since they both lie on circ... | in the rectangular coordinate system , points ( 4 , 0 ) and ( – 4 , 0 ) both lie on circle c . what is the maximum possible value of the radius of c The points (4, 0) and (-4, 0) are both on the x-axis and are 8 units apart. Since they both lie on circle C, they must be endpoints of a diameter of the circle. The diamet... |
[
{
"content": "Bill is stocking the kitchenware section of the Walmart. He needs to stack 60 pots. On each shelf, he can stack five pots vertically and a certain number of sets of vertically stacked pots side-by-side. He needs 4 shelves to stock all the pots. How many sets of vertically stacked pots can he place... | Bill is stocking the kitchenware section of the Walmart. He needs to stack 60 pots. On each shelf, he can stack five pots vertically and a certain number of sets of vertically stacked pots side-by-side. He needs 4 shelves to stock all the pots. How many sets of vertically stacked pots can he place side-by-side on each ... |
[
{
"content": "The area of the smallest square that will contain a circle of radius 4 is what?",
"role": "user"
},
{
"content": "The smallest square has side 8 and area $8^2=\\boxed{64}$. [asy]\ndraw(Circle((12,12), 12));\ndraw((0,0)--(0,24)--(24,24)--(24,0)--cycle);\ndraw((0,12)--(24,12));\ndot((12,... | The area of the smallest square that will contain a circle of radius 4 is what? The smallest square has side 8 and area $8^2=\boxed{64}$. [asy]
draw(Circle((12,12), 12));
draw((0,0)--(0,24)--(24,24)--(24,0)--cycle);
draw((0,12)--(24,12));
dot((12,12));
label("4",(18,12),N);
label("8",(12, 0),S);
[/asy] |
[
{
"content": "The number $N$ is between $7$ and $15$. The average of $4$, $8$, and $N$ could be\n$\\text{(A)}\\ 6 \\qquad \\text{(B)}\\ 7 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 11 \\qquad \\text{(E)}\\ 13$\n$\\text{(A)}\\ 6 \\qquad \\text{(B)}\\ 7 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 11 \\qquad ... | The number $N$ is between $7$ and $15$. The average of $4$, $8$, and $N$ could be
$\text{(A)}\ 6 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 11 \qquad \text{(E)}\ 13$
$\text{(A)}\ 6 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 11 \qquad \text{(E)}\ 13$ 1. **Identify the range of $N$**: ... |
[
{
"content": "How do you plan a trip?",
"role": "user"
},
{
"content": "There are many different ways to plan a trip. In addition, there are many different factors to include in order to plan a trip. First of all, it is important to know the type of travel. Is this a business trip? Is this family tr... | How do you plan a trip? There are many different ways to plan a trip. In addition, there are many different factors to include in order to plan a trip. First of all, it is important to know the type of travel. Is this a business trip? Is this family trip?
Next you need to decide on how many days you want to stay in yo... |
[
{
"content": "a can finish a piece of work in 4 days . b can do it in 14 days . they work together for two days and then a goes away . in how many days will b finish the work ?",
"role": "user"
},
{
"content": "Let's calculate the amount of work A and B can do together in one day.\n\nA can finish th... | a can finish a piece of work in 4 days . b can do it in 14 days . they work together for two days and then a goes away . in how many days will b finish the work ? Let's calculate the amount of work A and B can do together in one day.
A can finish the work in 4 days, so A's work rate is 1/4 of the work per day.
B can f... |
[
{
"content": "The base six number $125_{6}$ is equal to the base $b$ number $221_{b}$. What is the positive value of $b$?",
"role": "user"
},
{
"content": "First, convert the base six number $125_6$ to base ten:\n$$ 125_{6} = 1 \\cdot 6^2 + 2 \\cdot 6^1 + 5 \\cdot 6^0 = 1 \\cdot 36 + 2 \\cdot 6 + 5 ... | The base six number $125_{6}$ is equal to the base $b$ number $221_{b}$. What is the positive value of $b$? First, convert the base six number $125_6$ to base ten:
$$ 125_{6} = 1 \cdot 6^2 + 2 \cdot 6^1 + 5 \cdot 6^0 = 1 \cdot 36 + 2 \cdot 6 + 5 \cdot 1 = 36 + 12 + 5 = 53_{10}. $$
Next, convert the base $b$ number $22... |
[
{
"content": "Petya has $n^{3}$ white $1 \\times 1 \\times 1$ cubes. He wants to assemble them into an $n \\times n \\times n$ cube that is completely white on the outside. What is the minimum number of cube faces Vasya needs to paint to prevent Petya from achieving this? Solve the problem for a) $n=2$; b) $n=3... | Petya has $n^{3}$ white $1 \times 1 \times 1$ cubes. He wants to assemble them into an $n \times n \times n$ cube that is completely white on the outside. What is the minimum number of cube faces Vasya needs to paint to prevent Petya from achieving this? Solve the problem for a) $n=2$; b) $n=3$. To determine the minimu... |
[
{
"content": "Joan picked some oranges, and Sara picked 10 oranges. Alyssa picked 30 pears. In total, 47 oranges were picked. How many oranges did Joan pick?",
"role": "user"
},
{
"content": "If Sara picked 10 oranges and the total number of oranges picked was 47, then Joan must have picked the diff... | Joan picked some oranges, and Sara picked 10 oranges. Alyssa picked 30 pears. In total, 47 oranges were picked. How many oranges did Joan pick? If Sara picked 10 oranges and the total number of oranges picked was 47, then Joan must have picked the difference between the total and what Sara picked.
So, Joan picked 47 (... |
[
{
"content": "Asia found a beautiful homecoming dress in a foreign online store. The dress was originally priced at €300. The store had a sale going on, offering a 20% discount on dresses. Then, Asia also had a special member coupon that granted her an additional 10% discount on the already reduced price. She b... | Asia found a beautiful homecoming dress in a foreign online store. The dress was originally priced at €300. The store had a sale going on, offering a 20% discount on dresses. Then, Asia also had a special member coupon that granted her an additional 10% discount on the already reduced price. She bought the dress, and t... |
[
{
"content": "Why is a pomegranate known as a hand grenade?",
"role": "user"
},
{
"content": "When pomegranates become overripe, they will literally burst on the tree, like a grenade. In Spanish, they are known as granadas for this reason.",
"role": "assistant"
}
] | Why is a pomegranate known as a hand grenade? When pomegranates become overripe, they will literally burst on the tree, like a grenade. In Spanish, they are known as granadas for this reason. |
[
{
"content": "What are some physical activities I could do on a sunny day?",
"role": "user"
},
{
"content": "On a sunny day, you can take advantage of the nice weather by doing outdoor activities. For solo activities, you can go on a run, a hike or a bike ride. For group activities, consider playing... | What are some physical activities I could do on a sunny day? On a sunny day, you can take advantage of the nice weather by doing outdoor activities. For solo activities, you can go on a run, a hike or a bike ride. For group activities, consider playing frisbee or playing tennis. |
[
{
"content": "an equal number of desks and bookcases are to be placed along a library wall that is 15 meters long . each desk is 2 meters long , and each bookshelf is 1.5 meters long . if the maximum possible number of desks and bookcases are to be placed along the wall , then the space along the wall that is l... | an equal number of desks and bookcases are to be placed along a library wall that is 15 meters long . each desk is 2 meters long , and each bookshelf is 1.5 meters long . if the maximum possible number of desks and bookcases are to be placed along the wall , then the space along the wall that is left over will be how m... |
[
{
"content": "How many island does Hawaii have?",
"role": "user"
},
{
"content": "Hawaii has 8 islands in total,",
"role": "assistant"
}
] | How many island does Hawaii have? Hawaii has 8 islands in total, |
[
{
"content": "If w is 40 percent less than u, u is 40 percent less than y, and z is 46 percent less than y, then z is greater than w by what percent of w?",
"role": "user"
},
{
"content": "Let's start by expressing the relationships between w, u, and y in terms of equations.\n\nIf w is 40 percent le... | If w is 40 percent less than u, u is 40 percent less than y, and z is 46 percent less than y, then z is greater than w by what percent of w? Let's start by expressing the relationships between w, u, and y in terms of equations.
If w is 40 percent less than u, we can write:
w = u - 0.40u
w = 0.60u (since 1 - 0.40 = 0.... |
[
{
"content": "Square $ABCD$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 12$ and the $x$-axis, as shown below. Find the area of square $ABCD.$\n\n[asy]\nunitsize(0.8 cm);\n\nreal parab (real x) {\n return(x^2 - 8*x + 12);\n}\n\npair A, B, C, D;\nreal x = -1 + sqrt(5);\n\nA = (4 - x,0);\nB ... | Square $ABCD$ is inscribed in the region bound by the parabola $y = x^2 - 8x + 12$ and the $x$-axis, as shown below. Find the area of square $ABCD.$
[asy]
unitsize(0.8 cm);
real parab (real x) {
return(x^2 - 8*x + 12);
}
pair A, B, C, D;
real x = -1 + sqrt(5);
A = (4 - x,0);
B = (4 + x,0);
C = (4 + x,-2*x);
D = ... |
[
{
"content": "A function $f(x)$ satisfies the following conditions: ① $f(x)$ is a monotonically increasing function on the interval $\\left[a,b\\right]$; ② The range of $f(x)$ on $\\left[a,b\\right]$ is $\\left[ka,kb\\right]$. The interval $\\left[a,b\\right]$ is called a $k$-level \"harmonic interval\" of $y=f... | A function $f(x)$ satisfies the following conditions: ① $f(x)$ is a monotonically increasing function on the interval $\left[a,b\right]$; ② The range of $f(x)$ on $\left[a,b\right]$ is $\left[ka,kb\right]$. The interval $\left[a,b\right]$ is called a $k$-level "harmonic interval" of $y=f(x)$. If the function $f(x)=x^{2... |
[
{
"content": "Which team's have the most NCAA Division I men's basketball championships?",
"role": "user"
},
{
"content": "1. UCLA has 11 championship wins\n2. Kentucky has 8 championship wins\n3. North Carolina has 6 championship wins\n4. Due has 5 championship wins\n5. Indian has 5 championship wi... | Which team's have the most NCAA Division I men's basketball championships? 1. UCLA has 11 championship wins
2. Kentucky has 8 championship wins
3. North Carolina has 6 championship wins
4. Due has 5 championship wins
5. Indian has 5 championship wins |
[
{
"content": "A liter of cream contains a certain percentage of butterfat. To obtain one liter of mixture with 6.5 percent butterfat, 3 liters of skim milk containing 5.5 percent butterfat must be added to the mixture. What is the percentage of butterfat in the cream?",
"role": "user"
},
{
"content"... | A liter of cream contains a certain percentage of butterfat. To obtain one liter of mixture with 6.5 percent butterfat, 3 liters of skim milk containing 5.5 percent butterfat must be added to the mixture. What is the percentage of butterfat in the cream? Let's denote the percentage of butterfat in the cream as x%. We w... |
[
{
"content": "What is the smallest positive four-digit integer equivalent to 6 mod 7?",
"role": "user"
},
{
"content": "An integer that is equivalent to 6 mod 7 can be written in the form $7k+6$.\n\n$1000$ is the smallest four-digit integer, so we want to solve the inequality $7k+6 \\ge 1000$. This ... | What is the smallest positive four-digit integer equivalent to 6 mod 7? An integer that is equivalent to 6 mod 7 can be written in the form $7k+6$.
$1000$ is the smallest four-digit integer, so we want to solve the inequality $7k+6 \ge 1000$. This inequality has solution $k \ge 142$, so since $k$ must be an integer, t... |
[
{
"content": "Each day, Jenny ate $20\\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of second day, 32 remained. How many jellybeans were in the jar originally?",
"role": "user"
},
{
"content": "Since Jenny ate $20\\%$ of the jellybeans remaining each day, $80\\%... | Each day, Jenny ate $20\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of second day, 32 remained. How many jellybeans were in the jar originally? Since Jenny ate $20\%$ of the jellybeans remaining each day, $80\%$ of the jellybeans are left at the end of each day. If $x$ is the numb... |
[
{
"content": "Two circles touch each other externally. Find the length of their common external tangent (between the points of tangency), given that the radii are 16 and 25.",
"role": "user"
},
{
"content": ": \n1. Let's name the centers of the given circles $O$ and $O_2$, having radii 16 and 25, re... | Two circles touch each other externally. Find the length of their common external tangent (between the points of tangency), given that the radii are 16 and 25. :
1. Let's name the centers of the given circles $O$ and $O_2$, having radii 16 and 25, respectively.
2. Let $M$ and $M_2$ be the points of tangency of the cir... |
[
{
"content": "Given the function $f(x)=x^{2}+\\\\ln x$, if the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$ is $y=ax+b$, then $a+b=$ _____ .",
"role": "user"
},
{
"content": "Since $f(x)=x^{2}+\\\\ln x$,\n\nThen $f(1)=1^{2}+\\\\ln 1=1$, so the point of tangency is $(1,1... | Given the function $f(x)=x^{2}+\\ln x$, if the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$ is $y=ax+b$, then $a+b=$ _____ . Since $f(x)=x^{2}+\\ln x$,
Then $f(1)=1^{2}+\\ln 1=1$, so the point of tangency is $(1,1)$.
The derivative of the function is $f′(x)=2x+ \\dfrac {1}{x}$, so $f′(1)... |
[
{
"content": "When did Felix Luna die?\n\nContext: Félix César Luna (30 September 1925 – 5 November 2009) was an Argentine writer, lyricist and historian.",
"role": "user"
},
{
"content": "Felix Luna died on November 5th, 2009",
"role": "assistant"
}
] | When did Felix Luna die?
Context: Félix César Luna (30 September 1925 – 5 November 2009) was an Argentine writer, lyricist and historian. Felix Luna died on November 5th, 2009 |
[
{
"content": "A circle centered at $A$ with a radius of $1$ and a circle centered at $B$ with a radius of $4$ are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. What is the radius of the third circle? [asy]\ndraw((-3,0)--(7.5,0));\ndraw(Circ... | A circle centered at $A$ with a radius of $1$ and a circle centered at $B$ with a radius of $4$ are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. What is the radius of the third circle? [asy]
draw((-3,0)--(7.5,0));
draw(Circle((-1,1),1),linewidth(0... |
[
{
"content": "Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 5y^2 = 6,$ as shown below. Find $r.$\n[asy]\nsize(7cm);\ndraw(scale(sqrt(6), sqrt(6)/sqrt(5))* unitcircle);\ndraw((0,-1.5)--(0,1.7),EndArrow);\ndraw((-3,0)--(3,0),EndArrow);\ndraw(Circle( (s... | Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 5y^2 = 6,$ as shown below. Find $r.$
[asy]
size(7cm);
draw(scale(sqrt(6), sqrt(6)/sqrt(5))* unitcircle);
draw((0,-1.5)--(0,1.7),EndArrow);
draw((-3,0)--(3,0),EndArrow);
draw(Circle( (sqrt(0.96),0), sqrt(0.96) ));... |
[
{
"content": "What are good native plants for a garden in Ohio?",
"role": "user"
},
{
"content": "Native plants that will grow well in Ohio include Purple Coneflower, Thin Leaf Mountain Mint, Sawtooth Sunflower, and Little Bluestem Grass. There's quite a few plants that grow well in Ohio and are nat... | What are good native plants for a garden in Ohio? Native plants that will grow well in Ohio include Purple Coneflower, Thin Leaf Mountain Mint, Sawtooth Sunflower, and Little Bluestem Grass. There's quite a few plants that grow well in Ohio and are native to the area. The Missouri Botanical Garden has great resources o... |
[
{
"content": "For integers $a$ and $T,$ $T \\neq 0,$ a parabola whose general equation is $y = ax^2 + bx + c$ passes through the points $A = (0,0),$ $B = (2T,0),$ and $C = (2T + 1,28).$ Let $N$ be the sum of the coordinates of the vertex point. Determine the largest value of $N.$",
"role": "user"
},
{... | For integers $a$ and $T,$ $T \neq 0,$ a parabola whose general equation is $y = ax^2 + bx + c$ passes through the points $A = (0,0),$ $B = (2T,0),$ and $C = (2T + 1,28).$ Let $N$ be the sum of the coordinates of the vertex point. Determine the largest value of $N.$ Since the parabola passes through the points $(0,0)$... |
[
{
"content": "What is the Fallout game?\n\nContext: Fallout: A Post Nuclear Role Playing Game is a 1997 role-playing video game developed and published by Interplay Productions. In a mid-22nd century post-apocalyptic and retro-futuristic world, decades after a global nuclear war between the United States and Ch... | What is the Fallout game?
Context: Fallout: A Post Nuclear Role Playing Game is a 1997 role-playing video game developed and published by Interplay Productions. In a mid-22nd century post-apocalyptic and retro-futuristic world, decades after a global nuclear war between the United States and China, Fallout's protagoni... |
[
{
"content": "Find the remainder of the Euclidean division of \\( 2018^{2019^{2020}} \\) by 11.",
"role": "user"
},
{
"content": "1. Consider the given expression \\(2018^{2019^{2020}}\\) modulo 11. To simplify the problem, we will make use of Fermat's Little Theorem, which states that if \\(p\\) is... | Find the remainder of the Euclidean division of \( 2018^{2019^{2020}} \) by 11. 1. Consider the given expression \(2018^{2019^{2020}}\) modulo 11. To simplify the problem, we will make use of Fermat's Little Theorem, which states that if \(p\) is a prime number and \(a\) is any integer not divisible by \(p\), then \(a^... |
[
{
"content": "Let $F(0) = 0,$ $F(1) = \\frac{3}{2},$ and\n\\[F(n) = \\frac{5}{2} F(n - 1) - F(n - 2)\\]for $n \\ge 2.$ Find\n\\[\\sum_{n = 0}^\\infty \\frac{1}{F(2^n)}.\\]",
"role": "user"
},
{
"content": "We claim that $F(n) = 2^n - \\frac{1}{2^n}$ for all nonnegative integers $n.$ We prove this ... | Let $F(0) = 0,$ $F(1) = \frac{3}{2},$ and
\[F(n) = \frac{5}{2} F(n - 1) - F(n - 2)\]for $n \ge 2.$ Find
\[\sum_{n = 0}^\infty \frac{1}{F(2^n)}.\] We claim that $F(n) = 2^n - \frac{1}{2^n}$ for all nonnegative integers $n.$ We prove this by strong induction.
The result for $n = 0$ and $n = 1.$ Assume that the result... |
[
{
"content": "Give me some strategies to prevent procrastination",
"role": "user"
},
{
"content": "Procrastination is a huge problem for individuals in today's society. Though there are many strategies to beat procrastination, the effectiveness of these strategies is largely dependent on the individ... | Give me some strategies to prevent procrastination Procrastination is a huge problem for individuals in today's society. Though there are many strategies to beat procrastination, the effectiveness of these strategies is largely dependent on the individual and his/her/their circumstances. Here are some popular strategie... |
[
{
"content": "Each subsequent number is chosen independently of the previous ones. If it matches one of the already chosen numbers, you should move clockwise to the first unchosen number.\n\nIn the end, $k$ different numbers are obtained.\n\na) Is it true that the appearance of each specific number in such a se... | Each subsequent number is chosen independently of the previous ones. If it matches one of the already chosen numbers, you should move clockwise to the first unchosen number.
In the end, $k$ different numbers are obtained.
a) Is it true that the appearance of each specific number in such a selection is equally likely?... |
[
{
"content": "Let $f$ be a function taking the nonnegative integers to the nonnegative integers, such that\n\\[2f(a^2 + b^2) = [f(a)]^2 + [f(b)]^2\\]for all nonnegative integers $a$ and $b.$\n\nLet $n$ be the number of possible values of $f(25),$ and let $s$ be the sum of the possible values of $f(25).$ Find $... | Let $f$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[2f(a^2 + b^2) = [f(a)]^2 + [f(b)]^2\]for all nonnegative integers $a$ and $b.$
Let $n$ be the number of possible values of $f(25),$ and let $s$ be the sum of the possible values of $f(25).$ Find $n \times s.$ Setting $a = 0... |
[
{
"content": "After spending some amount on rent, Rs. 1500 on milk, Rs. 4500 on groceries, Rs. 2500 on children's education, Rs. 2000 on petrol, and Rs. 5200 on miscellaneous expenses, Mr. Kishore saved 10% of his monthly salary. He saved Rs. 2300. How much did he spend on rent?",
"role": "user"
},
{
... | After spending some amount on rent, Rs. 1500 on milk, Rs. 4500 on groceries, Rs. 2500 on children's education, Rs. 2000 on petrol, and Rs. 5200 on miscellaneous expenses, Mr. Kishore saved 10% of his monthly salary. He saved Rs. 2300. How much did he spend on rent? Let's denote Mr. Kishore's monthly salary as S.
Accor... |
[
{
"content": "What is the greatest product obtainable from two integers whose sum is 2024?",
"role": "user"
},
{
"content": "Let the two integers be $x$ and $2024-x$. The product we want to maximize is $2024x - x^2$. To find the maximum, we can complete the square:\n\n1. Rewrite the expression: $202... | What is the greatest product obtainable from two integers whose sum is 2024? Let the two integers be $x$ and $2024-x$. The product we want to maximize is $2024x - x^2$. To find the maximum, we can complete the square:
1. Rewrite the expression: $2024x - x^2 = -x^2 + 2024x$.
2. Complete the square:
\[
-x^2 + 2024... |
[
{
"content": "Let $A = (-4, 0),$ $B = (-3, 2),$ $C = (3, 2),$ and $D = (4, 0).$ Suppose that point $P$ satisfies \\[PA + PD = 10,\\] and \\[PB + PC = 8.\\] Find the $y-$coordinate of $P$, which can be expressed in the form $\\frac{-a + b \\sqrt{c}}{d},$ where $a, b, c, d$ are positive integers. Find $a + b + c ... | Let $A = (-4, 0),$ $B = (-3, 2),$ $C = (3, 2),$ and $D = (4, 0).$ Suppose that point $P$ satisfies \[PA + PD = 10,\] and \[PB + PC = 8.\] Find the $y-$coordinate of $P$, which can be expressed in the form $\frac{-a + b \sqrt{c}}{d},$ where $a, b, c, d$ are positive integers. Find $a + b + c + d.$ Since $PA + PD = 10$, ... |
[
{
"content": "Extract the names from the cast of the film Replikator, and separate them with a comma.\n\nContext: Replikator is a 1994 science fiction film directed by G. Philip Jackson and starring Michael St. Gerard, Brigitte Bako, Ned Beatty and Ilona Staller.\n\nCast\n\n Michael St. Gerard as Ludo Ludovi... | Extract the names from the cast of the film Replikator, and separate them with a comma.
Context: Replikator is a 1994 science fiction film directed by G. Philip Jackson and starring Michael St. Gerard, Brigitte Bako, Ned Beatty and Ilona Staller.
Cast
Michael St. Gerard as Ludo Ludovic
Brigitte Bako as Kathy... |
[
{
"content": "Mari made 4 more than a certain multiple of decorated buttons as Kendra. Sue made half as many as Kendra. Sue made 6 buttons. Mari made 64 buttons. What is the multiple of Kendra's buttons that Mari made?",
"role": "user"
},
{
"content": "Let's denote the certain multiple of decorated ... | Mari made 4 more than a certain multiple of decorated buttons as Kendra. Sue made half as many as Kendra. Sue made 6 buttons. Mari made 64 buttons. What is the multiple of Kendra's buttons that Mari made? Let's denote the certain multiple of decorated buttons that Kendra made as "x".
According to the information give... |
[
{
"content": "Solve the system of differential equations\n\n$$\n\\left\\{\\begin{array}{l}\nx^{\\prime}=2 x-4 y \\\\\ny^{\\prime}=x-3 y\n\\end{array}\\right.\n$$",
"role": "user"
},
{
"content": "\nTo solve the system of differential equations\n\n\\[ \n\\left\\{\n\\begin{array}{l}\nx' = 2x - 4y \\\\... | Solve the system of differential equations
$$
\left\{\begin{array}{l}
x^{\prime}=2 x-4 y \\
y^{\prime}=x-3 y
\end{array}\right.
$$
To solve the system of differential equations
\[
\left\{
\begin{array}{l}
x' = 2x - 4y \\
y' = x - 3y
\end{array}
\right.
\]
we first represent this system using matrix notation. Let \... |
[
{
"content": "James has a rainwater collection barrel with a maximum capacity of 80 gallons. For each inch of rain, he collects 15 gallons. On Monday it rained 4 inches, on Tuesday it rained 3 inches, and on Wednesday it rained 2.5 inches. James can sell water for $1.2 per gallon on Monday, $1.5 per gallon on T... | James has a rainwater collection barrel with a maximum capacity of 80 gallons. For each inch of rain, he collects 15 gallons. On Monday it rained 4 inches, on Tuesday it rained 3 inches, and on Wednesday it rained 2.5 inches. James can sell water for $1.2 per gallon on Monday, $1.5 per gallon on Tuesday, and $0.8 per g... |
[
{
"content": "Mongolia TST 2011 Test 1 #2\n\nLet $p$ be a prime number. Prove that: $\\sum_{k=0}^p (-1)^k \\dbinom{p}{k} \\dbinom{p+k}{k} \\equiv -1 (\\mod p^3)$ (proposed by B. Batbayasgalan, inspired by Putnam olympiad problem)\n\nNote: I believe they meant to say $p>2$ as well.",
"role": "user"
},
... | Mongolia TST 2011 Test 1 #2
Let $p$ be a prime number. Prove that: $\sum_{k=0}^p (-1)^k \dbinom{p}{k} \dbinom{p+k}{k} \equiv -1 (\mod p^3)$ (proposed by B. Batbayasgalan, inspired by Putnam olympiad problem)
Note: I believe they meant to say $p>2$ as well. To prove the given statement, we will use combinatorial i... |
[
{
"content": "In a modified walking game, I stand still on move 1, and for each subsequent move $n$ where $2 \\le n \\le 30$, I take two steps forward if $n$ is prime and three steps backward if $n$ is composite. After completing move 30, I walk directly back to my original starting point. How many steps long i... | In a modified walking game, I stand still on move 1, and for each subsequent move $n$ where $2 \le n \le 30$, I take two steps forward if $n$ is prime and three steps backward if $n$ is composite. After completing move 30, I walk directly back to my original starting point. How many steps long is my walk back? Let us c... |
[
{
"content": "Find $10110_2\\times10100_2\\div10_2$. Express your answer in base 2.",
"role": "user"
},
{
"content": "It is much easier to find $10110_2\\div10_2$ and then multiply by $10100_2$ than it is to do the calculations in the original order. For $10110_2\\div10_2$, since the last digit of $... | Find $10110_2\times10100_2\div10_2$. Express your answer in base 2. It is much easier to find $10110_2\div10_2$ and then multiply by $10100_2$ than it is to do the calculations in the original order. For $10110_2\div10_2$, since the last digit of $10110_2$ is a 0, we can simply take it off to get $1011_2$. This is simi... |
[
{
"content": "Each square in the grid shown is 1 cm by 1 cm. What is the area of the shaded figure, in cm²?",
"role": "user"
},
{
"content": "\n1. **Determine the total area of the grid:**\n\n The grid is composed of squares measuring \\(1 \\, \\text{cm} \\times 1 \\, \\text{cm}\\). There are \\(3... | Each square in the grid shown is 1 cm by 1 cm. What is the area of the shaded figure, in cm²?
1. **Determine the total area of the grid:**
The grid is composed of squares measuring \(1 \, \text{cm} \times 1 \, \text{cm}\). There are \(3\) rows and \(9\) columns.
The total number of squares is:
\[
3 \t... |
[
{
"content": "A circle of radius 1 is tangent to a circle of radius 2. The sides of $\\triangle ABC$ are tangent to the circles as shown, and the sides $\\overline{AB}$ and $\\overline{AC}$ are congruent. What is the area of $\\triangle ABC$?\n\n[asy]\nunitsize(0.7cm);\npair A,B,C;\nA=(0,8);\nB=(-2.8,0);\nC=(2.... | A circle of radius 1 is tangent to a circle of radius 2. The sides of $\triangle ABC$ are tangent to the circles as shown, and the sides $\overline{AB}$ and $\overline{AC}$ are congruent. What is the area of $\triangle ABC$?
[asy]
unitsize(0.7cm);
pair A,B,C;
A=(0,8);
B=(-2.8,0);
C=(2.8,0);
draw(A--B--C--cycle,linewid... |
[
{
"content": "Construct a parallelogram given its perimeter, one diagonal, and the angle between the given diagonal and one side of the parallelogram.",
"role": "user"
},
{
"content": "\n1. **Given Data and Definitions:**\n - Let \\( P \\) be the perimeter of the parallelogram.\n - Let \\( d \\)... | Construct a parallelogram given its perimeter, one diagonal, and the angle between the given diagonal and one side of the parallelogram.
1. **Given Data and Definitions:**
- Let \( P \) be the perimeter of the parallelogram.
- Let \( d \) be the given diagonal of the parallelogram.
- Let \( \alpha \) be the a... |
[
{
"content": "Add $4_6 + 14_6.$ Express your answer in base $6.$",
"role": "user"
},
{
"content": "When adding the numbers, we notice that $4+4$ leaves a residue of $2$ when divided by $6.$ Thus, the sum will have a rightmost digit of $2,$ and we must carry-over. This yields that $$\\begin{array}{c@... | Add $4_6 + 14_6.$ Express your answer in base $6.$ When adding the numbers, we notice that $4+4$ leaves a residue of $2$ when divided by $6.$ Thus, the sum will have a rightmost digit of $2,$ and we must carry-over. This yields that $$\begin{array}{c@{}c@{\;}c@{}c@{}c} & & & \stackrel{1}{} & \stackrel{}{4}_6 \\ &+ & & ... |
[
{
"content": "It is known that $\\sin y = 2 \\cos x + \\frac{5}{2} \\sin x$ and $\\cos y = 2 \\sin x + \\frac{5}{2} \\cos x$. Find $\\sin 2x$.",
"role": "user"
},
{
"content": "\n1. Given the equations:\n\n \\[\n \\sin y = 2 \\cos x + \\frac{5}{2} \\sin x\n \\]\n \\[\n \\cos y = 2 \\sin x ... | It is known that $\sin y = 2 \cos x + \frac{5}{2} \sin x$ and $\cos y = 2 \sin x + \frac{5}{2} \cos x$. Find $\sin 2x$.
1. Given the equations:
\[
\sin y = 2 \cos x + \frac{5}{2} \sin x
\]
\[
\cos y = 2 \sin x + \frac{5}{2} \cos x
\]
2. Square both equations and add them together:
\[
(\sin y... |
[
{
"content": "Find the area of the triangle with vertices $(-1,4),$ $(7,0),$ and $(11,5).$",
"role": "user"
},
{
"content": "Let $A = (-1,4),$ $B = (7,0),$ and $C = (11,5).$ Let $\\mathbf{v} = \\overrightarrow{CA} = \\begin{pmatrix} -1 - 11 \\\\ 4 - 5 \\end{pmatrix} = \\begin{pmatrix} -12 \\\\ -1 \... | Find the area of the triangle with vertices $(-1,4),$ $(7,0),$ and $(11,5).$ Let $A = (-1,4),$ $B = (7,0),$ and $C = (11,5).$ Let $\mathbf{v} = \overrightarrow{CA} = \begin{pmatrix} -1 - 11 \\ 4 - 5 \end{pmatrix} = \begin{pmatrix} -12 \\ -1 \end{pmatrix}$ and $\mathbf{w} = \overrightarrow{CB} = \begin{pmatrix} 7 - 11 ... |
[
{
"content": "Give me a list of the Adventure motorbikes that I can buy on my LAMS (Learner Approved Motorcycle Scheme) restriction, living in Australia",
"role": "user"
},
{
"content": "This is not a comprehensive list but you can use this as a starting point:\nKawasaki Versys-X 300\nKawazaki KLR65... | Give me a list of the Adventure motorbikes that I can buy on my LAMS (Learner Approved Motorcycle Scheme) restriction, living in Australia This is not a comprehensive list but you can use this as a starting point:
Kawasaki Versys-X 300
Kawazaki KLR650
Honda CRF300 Rally
BMW G 310 GS
KTM 390 Adventure
Honda CB500XA
Yama... |
[
{
"content": "A truncated cone has horizontal bases with radii 24 and 5. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?",
"role": "user"
},
{
"content": "Let's denote the radii of the bottom and top bases of the truncated cone as ... | A truncated cone has horizontal bases with radii 24 and 5. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere? Let's denote the radii of the bottom and top bases of the truncated cone as \( R = 24 \) and \( r = 5 \), respectively. The sphere is tangent to... |
[
{
"content": "Evaluate $\\log_464$.",
"role": "user"
},
{
"content": "We have $4^3=64$, so $\\log_4 64 = \\boxed{3}$.",
"role": "assistant"
}
] | Evaluate $\log_464$. We have $4^3=64$, so $\log_4 64 = \boxed{3}$. |
[
{
"content": "The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. The maximum possible area of such a triangle can be written in the form $p\\sqrt{q}-r$, where $p$, $q$, and $r$ are positive integers, and $q$ is not div... | The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r$, where $p$, $q$, and $r$ are positive integers, and $q$ is not divisible by the square of... |
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