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[
{
"content": "What type of creatures undergo Metamorphosis?\n\nContext: Metamorphosis is a biological process by which an animal physically develops including birth transformation or hatching, involving a conspicuous and relatively abrupt change in the animal's body structure through cell growth and differentia... |
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{
"content": "a ring x of diameter 16 inches is placed inside another ring y of diameter of 18 inches . how much fraction of the ring x surface is not covered by the ring y ?",
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"content": "To find the fraction of the surface of ring X that is not covered by ring Y, we need to com... |
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"content": "Let $N,O$ be functions such that $N(x) = 2\\sqrt{x}$, and $O(x) = x^2$. What is $N(O(N(O(N(O(3))))))$?",
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"content": "Notice that for any $x$, then $N(O(x)) = N(x^2) = 2\\sqrt{x^2} = 2x$. It follows that $$N(O(N(O(N(O(3)))))) = N(O(N(O(6)))) = N(O(12)) = \\boxed{24}... |
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{
"content": "Analyze polynomials $P(x) = 3x^3 - 5x + 4$ and $Q(x)$, which is formed by replacing each nonzero coefficient in $P(x)$ with their arithmetic mean. Determine if $P(x)$ and $Q(x)$ could have the same number of real roots and evaluate at $x = 1$.\nA) True for number of real roots, False for equal val... |
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{
"content": "Given the function $f(x)=x|x-2|$, find the solution set of the inequality $f(2-\\ln (x+1)) > f(3)$.",
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"content": "First, we analyze the function $f(x)$:\n$f(x) = x|x-2|$, which can be expressed as a piecewise function:\n\\[f(x)= \\begin{cases} x^2-2x & \\text{for } ... |
[
{
"content": "Name some popular American wine brands.",
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},
{
"content": "Josh, E&J Gallo, Justin, Chateau Ste. Michelle, Little Penguin, Franzia, Sutter Home",
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] |
[
{
"content": "Give me a list of the best golf movies I've never seen.",
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{
"content": "- Caddyshack\n- Tin Cup\n- The Greatest Game Ever Played\n- The Legend of Bagger Vance",
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] |
[
{
"content": "An express train overtakes a freight train. The speed of the express train is as many times greater than the speed of the freight train as the time spent passing by each other when traveling in the same direction is greater than the time they would spend traveling past each other in opposite direc... |
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{
"content": "Let $A,$ $R,$ $M,$ and $L$ be positive real numbers such that\n\\begin{align*}\n\\log_{10} (AL) + \\log_{10} (AM) &= 2, \\\\\n\\log_{10} (ML) + \\log_{10} (MR) &= 3, \\\\\n\\log_{10} (RA) + \\log_{10} (RL) &= 4.\n\\end{align*}Compute the value of the product $ARML.$",
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{
... |
[
{
"content": "An equilateral triangle has an area of $64\\sqrt{3}$ $\\text{cm}^2$. If each side of the triangle is decreased by 4 cm, by how many square centimeters is the area decreased?",
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{
"content": "We first consider an equilateral triangle with side length $s$. If we construc... |
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{
"content": "How many different triangles can be formed having a perimeter of 7 units if each side must have integral length?",
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"content": "Let $a,b,$ and $c$ represent the three side lengths of the triangle. The perimeter is $a+b+c=7,$ so $b+c=7-a$. We know by the Triangle Ineq... |
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{
"content": "In rectangle $ADEH$, points $B$ and $C$ trisect $\\overline{AD}$, and points $G$ and $F$ trisect $\\overline{HE}$. In addition, $AH=AC=2$. What is the area of quadrilateral $WXYZ$ shown in the figure?\n\n[asy]\nunitsize(1cm);\npair A,B,C,D,I,F,G,H,U,Z,Y,X;\nA=(0,0);\nB=(1,0);\nC=(2,0);\nD=(3,... |
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{
"content": "Given the function $f(x)=|2x+a|-|2x+3|$, $g(x)=|x-1|-3$. \n$(1)$ Solve the inequality: $|g(x)| < 2$; \n$(2)$ If for any $x_{1} \\in \\mathbb{R}$, there exists $x_{2} \\in \\mathbb{R}$ such that $f(x_{1})=g(x_{2})$ holds, find the range of the real number $a$.",
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"c... |
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"content": "Find the remainder when $3^{3^{3^3}}$ is divided by 1000.\n",
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"content": "Using the Carmichael function, we have $\\lambda(1000)=100$, so $3^{100}=1\\pmod{1000}$. Therefore, letting $N=3^{3^3}$, we seek to find an $n$ such that $N\\equiv n\\pmod{100}$ so that $3^N\\... |
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{
"content": "Tell me about the botanical garden?\n\nContext: A botanical garden or botanic garden[nb 1] is a garden with a documented collection of living plants for the purpose of scientific research, conservation, display, and education. Typically plants are labelled with their botanical names. It may contai... |
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{
"content": "Find the sum of all positive integral values of $n$ for which $\\frac{n+6}{n}$ is an integer.",
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},
{
"content": "The expression $\\frac{n+6}{n}$ can be simplified as $\\frac{n}{n}+\\frac{6}{n}$, or $1+\\frac{6}{n}$. So in order for this expression to have an integral val... |
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{
"content": "In $\\triangle ABC, \\overline{CA} = \\overline{CB}$. On $CB$ square $BCDE$ is constructed away from the triangle. If $x$ is the number of degrees in $\\angle DAB$, then\n$\\textbf{(A)}\\ x\\text{ depends upon }\\triangle ABC \\qquad \\textbf{(B)}\\ x\\text{ is independent of the triangle} \\\\ \\... |
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{
"content": "Calculate:\n\n(1) $4\\sqrt{2}+\\sqrt{8}-\\sqrt{18}$\n\n(2) $\\sqrt{1\\dfrac{1}{3}}\\div \\sqrt{2\\dfrac{1}{3}}\\times \\sqrt{1\\dfrac{2}{5}}$.",
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{
"content": "### Solution:\n\n#### For Problem (1):\n\nWe start by simplifying each term in the expression:\n\n- The term ... |
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{
"content": "Find the value of b from (15)^2 x 9^2 ÷ 356 = b. What is the value of b?",
"role": "user"
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{
"content": "To find the value of b, we need to calculate the expression on the left side of the equation:\n\n(15)^2 x 9^2 ÷ 356 = b\n\nFirst, calculate the squares:\n\n(15)^2 = 15 x 15 = 225\... |
[
{
"content": "Consider a hyperbola given by the equation $\\frac{y^2}{64} - \\frac{x^2}{16} = 1$. If the distance from a point $P$ on this hyperbola to one of its foci is $4$, find the distance from point $P$ to the other focus ( ).\n\nA: $20$\nB: $16$\nC: $12$\nD: $8$",
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"content... |
[
{
"content": "why do dogs like humans?",
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"content": "As humans we might not understand why dogs are such big fans of us. It could be that from a dogs perspective, humans seem to have an unlimited source of food at their disposal. Dogs may have developed their love for humans just... |
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{
"content": "In a population of 2,000 college students, 40 percent are freshmen, 30 percent are sophomores, 20 percent are juniors, and 10 percent are seniors. Among the freshmen, 60 percent are studying natural sciences, 30 percent are studying social sciences, and 10 percent are studying arts or humanities. ... |
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{
"content": "The sides of a triangle have lengths of $15$, $20$, and $25$. Find the length of the shortest altitude.",
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"content": "First notice that this is a right triangle, so two of the altitudes are the legs, whose lengths are $15$ and $20$. The third altitude, whose length... |
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{
"content": "Jia, Yi, Bing, and Ding, four friends, go shopping together at malls A and B. It is known that Jia and Yi can each buy up to two pieces of clothing, while Bing and Ding can each buy up to one piece. If they altogether buy two pieces of clothing, one from mall A and the other from mall B, then the ... |
[
{
"content": "Tell me whether these Pokemon are fairy or poison type: Clefable, Jigglypuff, Rapidash, Arbok, Nidorino, Zubat",
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{
"content": "Clefable: Fairy\nJigglypuff: Fairy\nRapidash: Fairy\nArbok: Poison\nNidorino: Poison\nZubat: Poison",
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] |
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{
"content": "Find four numbers that form a geometric progression, where the sum of the first and last terms is -49, and the sum of the two middle terms is 14.",
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{
"content": "To find four numbers in a geometric progression where the sum of the extreme terms is \\(-49\\) and the su... |
[
{
"content": "What is the largest base-4 number that has four digits? Express your answer in base 10.",
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},
{
"content": "The largest base-4 number that has four digits is $3333_4$, which is equal to $3 \\cdot 4^3 + 3 \\cdot 4^2 + 3 \\cdot 4 + 3 = \\boxed{255}$.",
"role": "assista... |
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{
"content": "What is the greatest 3-digit base 8 positive integer that is divisible by 5? (Express your answer in base 8.)",
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{
"content": "The greatest 3-digit base 8 positive integer is $777_8$, which is equal to $7 \\cdot 8^2 + 7 \\cdot 8 + 7 = 511$. This number leaves a remai... |
[
{
"content": "Where is Jasienica?",
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{
"content": "Jasienica [jaɕeˈnit͡sa] is a village in the administrative district of Gmina Łoniów, within Sandomierz County, Świętokrzyskie Voivodeship, in south-central Poland. It lies approximately 3 kilometres (2 mi) south-east of Łoniów, 21 k... |
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{
"content": "In Chemistry class, Samantha finds that she can make a certain solution by mixing $.04$ liters of chemical A with $.02$ liters of water (this gives her $.06$ liters of the solution). She wants to make a total of $.48$ liters of this new solution. To do so, how many liters of water will she use?",... |
[
{
"content": "A uniform ladder of mass $m$ and length $\\mathcal{L}$ is resting on a wall. A man of mass $m$ climbs up the ladder and is in perfect equilibrium with the ladder when he is $\\frac{2}{3}\\mathcal{L}$ the way up the ladder. The ladder makes an angle of $ \\theta = 30^\\circ $ with the hor... |
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{
"content": "Let \\( a \\) be a nonzero real number. In the Cartesian coordinate system \\( xOy \\), the quadratic curve \\( x^2 + ay^2 + a^2 = 0 \\) has a focal distance of 4. Determine the value of \\( a \\).",
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{
"content": "\nGiven a non-zero real number \\( a \\), consider the... |
[
{
"content": "A circular table has a radius of 6 units. Eight rectangular place mats are placed on it. Each mat has a width of 1 unit and a length of $y$ units, positioned so that two corners of the mat touch the table's edge, with these corners being endpoints of the side of length $y$. Additionally, the mats ... |
[
{
"content": "When did Massachusetts become a state?",
"role": "user"
},
{
"content": "Massachusetts became a state on February 6, 1788.",
"role": "assistant"
}
] |
[
{
"content": "Rationalize the denominator of $\\frac{\\sqrt{8}+\\sqrt{3}}{\\sqrt{2}+\\sqrt{3}}$. Express your answer in simplest form.",
"role": "user"
},
{
"content": "Multiplying top and bottom by the conjugate, we have $\\frac{\\sqrt{8}+\\sqrt{3}}{\\sqrt{2}+\\sqrt{3}} = \\frac{(\\sqrt{8}+\\sqrt{3... |
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{
"content": "Consider a sequence of numbers \\((a_1, a_2, \\ldots, a_{2n})\\). Define the operation\n\\[ S\\left(\\left(a_1, a_2, \\ldots, a_{2n}\\right)\\right) = \\left(a_1 a_2, a_2 a_3, \\ldots, a_{2n-1} a_{2n}, a_{2n} a_1\\right). \\]\n\nProve that whatever the sequence \\((a_1, a_2, \\ldots, a_{2n})\\) is... |
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{
"content": "A pirate is counting the loot he has stolen from S.S. AOPS, on which everything is counted in base 5. A treasure chest contains $3124_{5}$ dollars worth of jewelry, $3122_{5}$ dollars worth of gold coins, and $124_{5}$ dollars worth of rubbing alcohol. What is the total dollar amount that this p... |
[
{
"content": "Calculate the following expression: $5 \\cdot 7 - (3 \\cdot 2 + 5 \\cdot 4) \\div 2$.",
"role": "user"
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{
"content": "First, handle the operations inside the parentheses, following the order of operations (PEMDAS/BODMAS):\n1. Calculate the multiplications inside the parentheses:\n ... |
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{
"content": "Let $g(x)$ be a function piecewise defined as \\[g(x) = \\left\\{\n\\begin{array}{cl}\n-x & x\\le 0, \\\\\n2x-41 & x>0.\n\\end{array}\n\\right.\\] If $a$ is negative, find $a$ so that $g(g(g(10.5)))=g(g(g(a)))$.",
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{
"content": "First we must find $g(g(g(10.5)))$. We h... |
[
{
"content": "Andrey, Boris, and Denis were eating candies, each at his own constant rate. While Andrey ate 4 candies, Boris managed to eat only 3. Denis ate candies the fastest: he ate 7 candies while Andrey ate 6. In total, they ate 70 candies. How many candies did each of them eat?",
"role": "user"
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... |
[
{
"content": "The perimeter of rhombus $ABCD$ is $8\\sqrt{5}$, if the diagonal $AC=4$, then the area of this rhombus is ( )\n\nA: $8$\n\nB: $16$\n\nC: $16\\sqrt{5}$\n\nD: $32$",
"role": "user"
},
{
"content": "Given that quadrilateral $ABCD$ is a rhombus, we have the following properties:\n\n1. All ... |
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{
"content": "Who was Singhari Shyamasundar Kar\n\nContext: Desaraja Adiguru Singhari Shyamasundar Kar (Odia: ସିଂହାରୀ ଶ୍ୟାମସୁନ୍ଦର କର, romanized: Singhāri Syāmasundara Kara; 4 March 1908 – 16 March 1975) was a renowned Odissi musician, Guru, singer, scholar and composer. Born to a sebayata (servitor) family of t... |
[
{
"content": "In the xy-coordinate system, there are two points (m, n) and (m + 1, n + 0.5) on the line with the equation x = 2y + 5. What is the x-coordinate of the second point?",
"role": "user"
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{
"content": "To find the x-coordinate of the second point, we need to substitute the y-coordinate o... |
[
{
"content": "Suppose that $a$ is inversely proportional to $b$. Let $a_1,a_2$ be two nonzero values of $a$ such that $\\frac{a_1}{a_2}=\\frac{2}{3}$. Let the corresponding $b$ values be $b_1,b_2$. If $b_1,b_2$ are nonzero, find the value of $\\frac{b_1}{b_2}$.",
"role": "user"
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{
"content": "I... |
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{
"content": "what is the perimeter , in meters , of a rectangular garden 4 meters wide that has the same area as a rectangular playground 16 meters long and 12 meters wide ?",
"role": "user"
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{
"content": "First, let's find the area of the rectangular playground. The area of a rectangle is found ... |
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{
"content": "Goldie makes $5 an hour for pet-sitting. Last week, she worked for 20 hours while this week, she worked for 30 hours. How much did Goldie earn in two weeks for pet-sitting?",
"role": "user"
},
{
"content": "To calculate Goldie's earnings for two weeks, we need to add the amount she ear... |
[
{
"content": "Name some largest US states",
"role": "user"
},
{
"content": "- Alaska \n- Texas \n- California\n- Montana \n- New Mexico",
"role": "assistant"
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] |
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{
"content": "Given vectors $\\overrightarrow {a}=(x, 2)$ and $\\overrightarrow {b}=(2, -5)$, and the angle between $\\overrightarrow {a}$ and $\\overrightarrow {b}$ is obtuse, find the range of the real number $x$.",
"role": "user"
},
{
"content": "Since $\\overrightarrow {a}$ and $\\overrightarrow... |
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{
"content": "What is the sum of the greatest common factor of 3 and 6 and the least common multiple of 3 and 6?",
"role": "user"
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{
"content": "The greatest common factor of 3 and $6=2\\cdot3$ is 3. The least common multiple of 3 and $6=2\\cdot3$ is $2\\cdot3=6$. Their sum is $3+6=\\boxed{9}$.",
... |
[
{
"content": "What is a baseball hat?",
"role": "user"
},
{
"content": "A baseball hat is a fabric hat with a bill on the front that shades the eyes from the sun. It is referred to as a baseball hat because all baseball players wear this style of hat while in the field playing defense, and they have... |
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{
"content": "Let $\\mathbf{M}$ be a matrix, and let $\\mathbf{v}$ and $\\mathbf{w}$ be vectors, such that\n\\[\\mathbf{M} \\mathbf{v} = \\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix} \\quad \\text{and} \\quad \\mathbf{M} \\mathbf{w} = \\begin{pmatrix} -2 \\\\ -5 \\end{pmatrix}.\\]Compute $\\mathbf{M} (\\mathbf{v} + ... |
[
{
"content": "Who invented Boltzmann Machines?",
"role": "user"
},
{
"content": "Boltzmann machines were invented in 1985 by Geoffrey Hinton and Terrence Sejnowski.",
"role": "assistant"
}
] |
[
{
"content": "In an orchard, there are 200 apples, 150 oranges, and 100 pears. Among these fruits, 40 percent of apples, 25 percent of oranges, and 35 percent of pears are rotten. Further, 70 percent of the rotten apples, 50 percent of the rotten oranges, and 80 percent of the rotten pears have a strong smell. ... |
[
{
"content": "Five congruent triangles, each of which is half a square, are placed together edge to edge in three different ways as shown to form shapes P, Q, and R.\n\nWhich of the following lists gives the shapes in ascending order of the lengths of their perimeters?\nA. P, Q, R\nB. Q, P, R\nC. R, Q, P\nD. R,... |
[
{
"content": "Summarize the important contributions of Thomas Edison from this paragraph in a bulleted list.\n\nContext: Thomas Alva Edison (February 11, 1847 – October 18, 1931) was an American inventor and businessman. He developed many devices in fields such as electric power generation, mass communication, ... |
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{
"content": "The equation of the tangent line to the curve $y=\\frac{e^x}{x+1}$ at the point $(1, \\frac{e}{2})$ is\n\nA: $y=\\frac{e}{4}x$\n\nB: $y=\\frac{e}{2}x$\n\nC: $y=\\frac{e}{4}x+\\frac{e}{4}$\n\nD: $y=\\frac{e}{2}x+\\frac{3e}{4}$",
"role": "user"
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{
"content": "Given the function $y=\\fr... |
[
{
"content": "if the area of a triangle with base 6 cm is equal to the area of a square with side 6 cm , then the altitude of the triangle is",
"role": "user"
},
{
"content": "To find the altitude of the triangle, we first need to find the area of the square, since the area of the triangle is given ... |
[
{
"content": "The sum of 4 consecutive even numbers is a certain value. The largest number is 38. What is the sum of these numbers?",
"role": "user"
},
{
"content": "Let's denote the four consecutive even numbers as E1, E2, E3, and E4, with E4 being the largest number, which is 38.\n\nSince these ar... |
[
{
"content": "Who painted the ‘Girl with the Pearl Earring’ and the ‘Milkmaid’?",
"role": "user"
},
{
"content": "The ‘Girl with the Pearl Earring’ and the ‘Milkmaid’ were painted by Johannes Vermeer.",
"role": "assistant"
}
] |
[
{
"content": "Mary sees three breeding balls with a certain number of snakes each and 6 additional pairs of snakes. She saw a total of 36 snakes. How many snakes were in each breeding ball?",
"role": "user"
},
{
"content": "Let's denote the number of snakes in each breeding ball as x. Since there ar... |
[
{
"content": "Given this paragraph about Love Island UK Season 9, Kai Fagan and Sanam Harrinanan won the final vote by what percent?\n\nContext: On 13 March 2023, the series was won by Kai Fagan and Sanam Harrinanan with 43.75% of the final vote, becoming the first couple formed during the Casa Amor twist to wi... |
[
{
"content": "If $128^3 = 16^x$, what is the value of $2^{-x}$? Express your answer as a common fraction.",
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},
{
"content": "First, express 128 and 16 as powers of 2. We know that $128 = 2^7$ and $16 = 2^4$. Rewrite the given equation using these bases:\n$$ (2^7)^3 = (2^4)^x. $$\n\nS... |
[
{
"content": "Convert the point \\( (\\rho, \\theta, \\phi) = (3, \\frac{3\\pi}{2}, \\frac{\\pi}{3}) \\) in spherical coordinates to rectangular coordinates.",
"role": "user"
},
{
"content": "Given \\( \\rho = 3, \\theta = \\frac{3\\pi}{2}, \\phi = \\frac{\\pi}{3} \\), we apply the conversion formul... |
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{
"content": "There were 349 pieces of candy in a bowl. Talitha took some pieces and Solomon took 153 pieces. Now, there are 88 pieces of candy remaining in the bowl. How many pieces did Talitha take?",
"role": "user"
},
{
"content": "To find out how many pieces of candy Talitha took, we need to sub... |
[
{
"content": "Categorize the below names into fishes and insects.\nsalmon, ant, bee, rohu, pike, wasp, beetle, carp, butterfly, cod, moth",
"role": "user"
},
{
"content": "Insects - ant, bee, wasp, beetle, butterfly, moth\nFish - salmon, rohu, carp, pike, cod",
"role": "assistant"
}
] |
[
{
"content": "Cut a triangle into three parts that can be assembled into a rectangle.",
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{
"content": "1. Consider a triangle \\( \\triangle ABC \\).\n2. To cut the triangle into three parts such that they can be reassembled to form a rectangle, follow these steps:\n\n a. Draw a ... |
[
{
"content": "Andy and Dawn spent the weekend cleaning the house. When putting away the laundry Andy took six minutes more than two times the number of minutes it took Dawn to wash the dishes. It took Andy 46 minutes to put away the laundry. How many minutes did it take Dawn to wash the dishes?",
"role": "u... |
[
{
"content": "How many elements of order \\( k \\) are there in \\(\\mathbb{Z} / p \\mathbb{Z}^{*}\\)?",
"role": "user"
},
{
"content": ": \n\n1. **Let \\( x \\) be a generator of \\( \\mathbb{Z} / p \\mathbb{Z}^{*} \\):**\n\n A generator \\( x \\) of the multiplicative group \\( \\mathbb{Z} / p \... |
[
{
"content": "Give me a bulleted list of the the first five books in the bible old testament?",
"role": "user"
},
{
"content": "The first five books of the bible old testament are called the Pentateuch or the books of law\n1) The Book of Genesis - is an account of the creation of the world, the earl... |
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{
"content": "Which of the following events is a certain event?\n\nA: A vehicle randomly passing through an intersection encounters a red light.\n\nB: Two out of 400 people have their birthdays on the same day.\n\nC: Three line segments can form a triangle.\n\nD: When buying a movie ticket, the seat number is a... |
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{
"content": "Let $z$ be a complex number such that\n\\[z^2 + |z|^2 = 3 - 5i.\\]Find $|z|^2.$",
"role": "user"
},
{
"content": "Let $z = a + bi,$ where $a$ and $b$ are real numbers. Then $z^2 = (a + bi)^2 = a^2 + 2abi - b^2$ and $|z|^2 = a^2 + b^2,$ so\n\\[a^2 + 2abi - b^2 + a^2 + b^2 = 3 - 5i.\\]E... |
[
{
"content": "Consider the function $f(x) = 2x^2 - 4x + 9$. Evaluate $2f(3) + 3f(-3)$.",
"role": "user"
},
{
"content": "We have $f(3) = 2(3^2) - 4\\cdot 3 + 9 = 18 - 12 + 9 = 15$ and $f(-3) = 2(-3)^2 - 4(-3) + 9 = 18 +12+9 = 39$. So, we have $2f(3)+3f(-3) = 2(15) + 3(39) = 30 + 117 = \\boxed{147}$... |
[
{
"content": "Given the sequence $\\{a_n\\}$ where $a_1=2$, $a_2=4$, and $a_{n+1}+2a_{n-1}=3a_{n}$ for $n\\geqslant 2$.\n\n$(1)$ Prove that the sequence $\\{a_{n+1}-a_{n}\\}$ is a geometric sequence;\n\n$(2)$ Find the general formula for the sequence $\\{a_n\\}$;\n\n$(3)$ Let $b_n=a_n-1$, and $S_n=\\frac{a_1}{b... |
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{
"content": "Simplify $\\dfrac{125}{5000} \\cdot 40.$",
"role": "user"
},
{
"content": "First, notice that both 125 and 5000 can be divided by 125, and 40 can be simplified by itself. We can start by simplifying the fraction and multiplication:\n\\[\n\\dfrac{125}{5000} = \\dfrac{1}{40}\n\\]\nNow, m... |
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{
"content": "In trapezoid $ABCD$ the lengths of the bases $AB$ and $CD$ are 8 and 17 respectively. The legs of the trapezoid are extended beyond $A$ and $B$ to meet at point $E$. What is the ratio of the area of triangle $EAB$ to the area of trapezoid $ABCD$? Express your answer as a common fraction.",
"ro... |
[
{
"content": "In the Cartesian coordinate system $xOy$, for any point $P (x,y)$ not on the coordinate axes, we call the point $P'$ $\\left( \\frac{1}{x},\\frac{1}{y} \\right)$ the \"inverse point\" of point $P$. On the line $y=-x+1$, there are two points $A$ and $B$, whose \"inverse points\" $A'$ and $B'$ are b... |
[
{
"content": "Let $\\textrm{B}$ be a digit. If the 7-digit number $658274\\textrm{B}$ is divisible by 2, 4, 5, 7, and 8, what is $\\textrm{B}$?",
"role": "user"
},
{
"content": "For divisibility by 5, the units digit $\\textrm{B}$ must be 0 or 5. To satisfy the condition for divisibility by 2, the u... |
[
{
"content": "Jesse received $50 as a gift to buy what she wants. She goes to the mall and falls in love with a novel that costs her $7. Then she went to lunch because she was very hungry and spent twice as much as the novel cost her. How much money did Jesse have left after going to the mall?",
"role": "us... |
[
{
"content": "Amandine and Brennon play a turn-based game, with Amadine starting.\nOn their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers.\nFor example, if $3, 5$ have been selected so far, only $1, 2, 4, 7$ are avai... |
[
{
"content": "Given that the length of the conjugate axis of the hyperbola $x^{2}-my^{2}=1$ is three times that of the transverse axis, find the value of the real number $m$.",
"role": "user"
},
{
"content": "[Analysis]\n\nBy utilizing the standard equation of a hyperbola, we can derive the relation... |
[
{
"content": "Compute $0.18\\div0.003.$",
"role": "user"
},
{
"content": "We have \\[0.18 \\div 0.003 = \\frac{0.18}{0.003} = \\frac{0.18}{0.003}\\cdot \\frac{1000}{1000}\n=\\frac{180}{3} = \\boxed{60}.\\]",
"role": "assistant"
}
] |
[
{
"content": "What is the sum of the positive factors of 60?",
"role": "user"
},
{
"content": "First, perform the prime factorization of 60. We find that $60 = 2^2 \\cdot 3 \\cdot 5$. \n\nTo find the sum of all positive divisors of 60, we can use the formula derived from its prime factorization. Eac... |
[
{
"content": "Let $\\theta$ be the angle between the planes $2x + y - 2z + 3 = 0$ and $6x + 3y + 2z - 5 = 0.$ Find $\\cos \\theta.$",
"role": "user"
},
{
"content": "The two planes intersect at a line, as shown below.\n\n[asy]\nunitsize(0.4 cm);\n\npair[] A, B, C, P;\npair M;\n\nA[1] = (3,3);\nA[2]... |
[
{
"content": "Find all real numbers $a$ such that the equation\n\\[x^3 - ax^2 - 2ax + a^2 - 1 = 0\\]has exactly one real solution in $x.$",
"role": "user"
},
{
"content": "Writing the equation as a quadratic in $a,$ we get\n\\[a^2 - (x^2 + 2x) a + (x^3 - 1) = a^2 - (x^2 + 2x) a + (x - 1)(x^2 + x + 1... |
[
{
"content": "Evaluate $2000^3-1999\\cdot 2000^2-1999^2\\cdot 2000+1999^3$",
"role": "user"
},
{
"content": "Let $a = 1999$ and $b = 2000.$ Then\n\\begin{align*}\n2000^3 - 1999 \\cdot 2000^2 - 1999^2 \\cdot 2000 + 1999^3 &= b^3 - ab^2 - a^2 b + a^3 \\\\\n&= b^2 (b - a) - a^2 (b - a) \\\\\n&= (b^2 -... |
[
{
"content": "Find the number of positive solutions to\n\\[\\sin (\\arccos( \\tan (\\arcsin x) ) ) ) = x.\\]",
"role": "user"
},
{
"content": "Let $\\theta = \\arcsin x,$ so $0 < \\theta < \\frac{\\pi}{2}$ and $\\sin \\theta = x.$ Then\n\\[\\cos \\theta = \\sqrt{1 - x^2},\\]so\n\\[\\tan \\theta = \... |
[
{
"content": "If $(ax+b)(bx+a)=24x^2+\\Box\\cdot x+24$, where $a$, $b$, and $\\Box$ are distinct integers, what is the minimum possible value of $\\Box$, the coefficient of $x$?",
"role": "user"
},
{
"content": "We expand the left side of the equation: \n\\[\n(ax+b)(bx+a) = abx^2 + (a^2 + b^2)x + ab... |
[
{
"content": "How many whole numbers are between $\\sqrt[3]{10}$ and $\\sqrt[3]{200}$?",
"role": "user"
},
{
"content": "Since $2^3=8$ and $3^3=27$, we know that $2<\\sqrt[3]{10}<3$. Then we find that $5^3=125$ and $6^3=216$, so $5<\\sqrt[3]{200}<6$. We have $\\sqrt[3]{10}<3$ and $5<\\sqrt[3]{200}$.... |
[
{
"content": "Let $X,$ $Y,$ and $Z$ be points such that $\\frac{XZ}{XY} = \\frac{ZY}{XY} = \\frac{1}{2}.$ If $Y = (1, 7)$, $Z = (-1, -7)$, then what is the sum of the coordinates of $X$?",
"role": "user"
},
{
"content": "From the problem, we can see that $XZ = ZY$ and $XZ + ZY = XY$, which means tha... |
[
{
"content": "Calculate the sum of the coefficients in the polynomial $2(4x^{8} - 5x^5 + 9x^3 - 6) + 8(x^6 - 4x^3 + 6)$ when it is fully simplified.",
"role": "user"
},
{
"content": "To find the sum of the coefficients in the polynomial \n$$2(4x^{8} - 5x^5 + 9x^3 - 6) + 8(x^6 - 4x^3 + 6),$$ \nwe sub... |
[
{
"content": "The Malibu Country Club needs to drain its pool for refinishing. The hose they use to drain it can remove 60 cubic feet of water per minute. The pool is 80 feet wide by a certain length by 10 feet deep and is currently at 100% capacity. It takes 2000 minutes to drain the pool. How long is the pool... |
[
{
"content": "The plane is colored in white and black. Show that there exist two points of the same color that are exactly $1 \\mathrm{~cm}$ apart.",
"role": "user"
},
{
"content": "\nThe goal is to show that in any two-colored plane (with each point colored either black or white), there exist at le... |
[
{
"content": "What are the options for you to use as an egg substitute when you are vegan?",
"role": "user"
},
{
"content": "When you are vegan you don’t eat any animal products or by-products. Depending on what you are making and what you are looking for from a taste perspective there are several o... |
[
{
"content": "How many minutes in a football match?",
"role": "user"
},
{
"content": "There are 90 minutes in a football match. This is played in two 45 minute halves where teams swap sides of the pitch at half-time. This is for football, also called soccer, which is one of the most popular sports i... |
[
{
"content": "Given the sequence $\\{a\\_n\\}$ with first term $a\\_1 = \\frac{1}{4}$ and common ratio $q= \\frac{1}{4}$ of a geometric sequence, and $b\\_n + 2 = 3 \\log_{\\frac{1}{4}} a\\_n$ for all $n \\in \\mathbb{N}^*$.\n\n(I) Prove that the sequence $\\{b\\_n\\}$ is an arithmetic sequence.\n(II) Let $c\\_... |
[
{
"content": "What are good activities for two year old boys?",
"role": "user"
},
{
"content": "Two year old boys tend to be active, curious and social. Enjoyable activities for this age include going to the park, dancing to music, reading a book, and taking a walk around the neighborhood. Local com... |
[
{
"content": "Give me a list of things I can do to sleep better at night",
"role": "user"
},
{
"content": "Here are some things you can do to sleep better at night:\n- Don't look at screens within an hour of going to bed\n- Keep the room your sleeping in as dark as possible\n- Don't drink many fluid... |
[
{
"content": "Based on the following passage when do schools close and resume after summer holidays in the United States?\n\nContext: Schools and universities typically have a summer break to take advantage of the warmer weather and longer days. In almost all countries, children are out of school during this ti... |
[
{
"content": "Identify which car manufacturer is Chinese or American: Ford, Higer",
"role": "user"
},
{
"content": "Ford is American, Higer is Chinese",
"role": "assistant"
}
] |
[
{
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $\\frac{\\cos A}{a} + \\frac{\\cos B}{b} = \\frac{2\\sqrt{3}\\sin C}{3a}$.\n$(1)$ Find the magnitude of angle $B$;\n$(2)$ If the area of triangle $ABC$ is $\\frac{\\sqrt{3}}{2... |
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