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[ { "content": "A right circular cone has a volume of $12\\pi$ cubic centimeters. The height of the cone is 4 cm. How many centimeters is the circumference of the base of the cone, in terms of $\\pi$?", "role": "user" }, { "content": " To find the circumference of the base of the cone, I'll first use ...	[ { "content": "A right circular cone has a volume of $12\\pi$ cubic centimeters. The height of the cone is 4 cm. How many centimeters is the circumference of the base of the cone, in terms of $\\pi$?", "role": "user" }, { "content": " Hmm, okay, so I have this problem about a right circular cone. Let...
[ { "content": "Determine the smallest positive real number $r$ such that there exist differentiable functions $f\\colon \\mathbb{R} \\to \\mathbb{R}$ and $g\\colon \\mathbb{R} \\to \\mathbb{R}$ satisfying \\begin{enumerate} \\item[(a)] $f(0) > 0$, \\item[(b)] $g(0) = 0$, \\item[(c)] $\|f'(x)\| \\leq \|g(x)\|$ for al...	[ { "content": "Determine the smallest positive real number $r$ such that there exist differentiable functions $f\\colon \\mathbb{R} \\to \\mathbb{R}$ and $g\\colon \\mathbb{R} \\to \\mathbb{R}$ satisfying \\begin{enumerate} \\item[(a)] $f(0) > 0$, \\item[(b)] $g(0) = 0$, \\item[(c)] $\|f'(x)\| \\leq \|g(x)\|$ for al...
[ { "content": "In the quadrilateral $ABCD$ , $AB = BC = CD$ and $\\angle BMC = 90^\\circ$ , where $M$ is the midpoint of $AD$ . Determine the acute angle between the lines $AC$ and $BD$ .", "role": "user" }, { "content": " To solve this problem, we need to analyze the given quadrilateral \\...	[ { "content": "In the quadrilateral $ABCD$ , $AB = BC = CD$ and $\\angle BMC = 90^\\circ$ , where $M$ is the midpoint of $AD$ . Determine the acute angle between the lines $AC$ and $BD$ .", "role": "user" }, { "content": " \nAlright, so I have this geometry problem here. Let me try to under...
[ { "content": "Let $x_1=97$, and for $n>1$, let $x_n=\\frac{n}{x_{n-1}}$. Calculate the product $x_1x_2x_3x_4x_5x_6x_7x_8$.", "role": "user" }, { "content": " To solve the problem, I'll start by understanding the recursive sequence given. \n\nWe have \\( x_1 = 97 \\) and for each \\( n > 1 \\), \\( x...	[ { "content": "Let $x_1=97$, and for $n>1$, let $x_n=\\frac{n}{x_{n-1}}$. Calculate the product $x_1x_2x_3x_4x_5x_6x_7x_8$.", "role": "user" }, { "content": " \nAlright, let me try to figure out this problem. So, we have a sequence where x₁ is 97, and for each n greater than 1, xₙ is equal to n divid...
[ { "content": "Using the digits $0$, $1$, $2$, $3$, $4$ to form a four-digit number without repeating any digit, determine the total number of four-digit numbers less than $2340$.", "role": "user" }, { "content": " To determine the total number of four-digit numbers less than 2340 using the digits 0,...	[ { "content": "Using the digits $0$, $1$, $2$, $3$, $4$ to form a four-digit number without repeating any digit, determine the total number of four-digit numbers less than $2340$.", "role": "user" }, { "content": " To determine the total number of four-digit numbers less than 2340 using the digits 0,...
[ { "content": "Let $a_n=6^{n}+8^{n}$. Determine the remainder upon dividing $a_ {83}$ by $49$.\n", "role": "user" }, { "content": " To solve the problem, I need to find the remainder when \\( a_{83} = 6^{83} + 8^{83} \\) is divided by 49. I'll start by examining each term separately modulo 49 and the...	[ { "content": "Let $a_n=6^{n}+8^{n}$. Determine the remainder upon dividing $a_ {83}$ by $49$.\n", "role": "user" }, { "content": " If you think the answer is 0, write \\boxed{0}.\n\nAlright, so I have this problem here: I need to find the remainder when \\( a_{83} = 6^{83} + 8^{83} \\) is divided by...
[ { "content": "If $2^{10} \\cdot 2^{15}$ is expressed as some integer to the fifth power, what is that integer?", "role": "user" }, { "content": " To solve the problem \\(2^{10} \\cdot 2^{15}\\) and express it as some integer to the fifth power, I start by combining the exponents since the bases are ...	[ { "content": "If $2^{10} \\cdot 2^{15}$ is expressed as some integer to the fifth power, what is that integer?", "role": "user" }, { "content": " Let's see, I have to figure out what 2 to the power of 10 multiplied by 2 to the power of 15 is, and then express that result as some integer raised to th...
[ { "content": "Walter wakes up at 6:30 a.m., catches the school bus at 7:30 a.m., has 7 classes that last 45 minutes each, enjoys a 30-minute lunch break, and spends an additional 3 hours at school for various activities. He takes the bus home and arrives back at 5:00 p.m. Calculate the total duration of his bus...	[ { "content": "Walter wakes up at 6:30 a.m., catches the school bus at 7:30 a.m., has 7 classes that last 45 minutes each, enjoys a 30-minute lunch break, and spends an additional 3 hours at school for various activities. He takes the bus home and arrives back at 5:00 p.m. Calculate the total duration of his bus...
[ { "content": "Simplify the expression \\[\\sqrt{37-20\\sqrt3}.\\]", "role": "user" }, { "content": " To simplify the expression \\(\\sqrt{37 - 20\\sqrt{3}}\\), I start by assuming it can be written in the form \\(\\sqrt{a} - \\sqrt{b}\\), where \\(a\\) and \\(b\\) are positive integers.\n\nNext, I s...	[ { "content": "Simplify the expression \\[\\sqrt{37-20\\sqrt3}.\\]", "role": "user" }, { "content": "**\nAlright, so I've got this expression to simplify: \\(\\sqrt{37 - 20\\sqrt{3}}\\). Hmm, it looks a bit intimidating at first glance, but I remember from my math classes that expressions like this c...
[ { "content": "(12 points) Using the six digits 0, 1, 2, 3, 4, 5, complete the following three questions:\n(1) If digits can be repeated, how many different five-digit even numbers can be formed?\n(2) If digits cannot be repeated, how many different five-digit numbers divisible by 5, with the hundredth digit not...	[ { "content": "(12 points) Using the six digits 0, 1, 2, 3, 4, 5, complete the following three questions:\n(1) If digits can be repeated, how many different five-digit even numbers can be formed?\n(2) If digits cannot be repeated, how many different five-digit numbers divisible by 5, with the hundredth digit not...
[ { "content": "Given that $\\tan{\\alpha} = 3$, calculate:\n\n1. $$\\frac{4\\sin{\\alpha} - \\cos{\\alpha}}{3\\sin{\\alpha} + 5\\cos{\\alpha}}$$\n2. $$\\sin{\\alpha} \\cdot \\cos{\\alpha}$$", "role": "user" }, { "content": " Let me think about how to approach this problem step by step.\n\nFirst, I kn...	[ { "content": "Given that $\\tan{\\alpha} = 3$, calculate:\n\n1. $$\\frac{4\\sin{\\alpha} - \\cos{\\alpha}}{3\\sin{\\alpha} + 5\\cos{\\alpha}}$$\n2. $$\\sin{\\alpha} \\cdot \\cos{\\alpha}$$", "role": "user" }, { "content": " If the answer cannot be determined from the given information, explain why.\...
[ { "content": "In the diagram, the circle is inscribed in the square. This means that the circle and the square share points \\(S, T, U,\\) and \\(V\\), and the width of the square is exactly equal to the diameter of the circle. Rounded to the nearest tenth, what percentage of line segment \\(XY\\) is outside th...	[ { "content": "In the diagram, the circle is inscribed in the square. This means that the circle and the square share points \\(S, T, U,\\) and \\(V\\), and the width of the square is exactly equal to the diameter of the circle. Rounded to the nearest tenth, what percentage of line segment \\(XY\\) is outside th...
[ { "content": "Place the numbers 1, 2, 3, 4, 5, 6, 7, and 8 on the eight vertices of a cube such that the sum of any three numbers on a face is at least 10. Find the minimum sum of the four numbers on any face.", "role": "user" }, { "content": " To solve this problem, I need to place the numbers 1 th...	[ { "content": "Place the numbers 1, 2, 3, 4, 5, 6, 7, and 8 on the eight vertices of a cube such that the sum of any three numbers on a face is at least 10. Find the minimum sum of the four numbers on any face.", "role": "user" }, { "content": " \n\nAlright, so I have this problem where I need to pla...
[ { "content": "A dark drawer contains $90$ red socks, $70$ green socks, $50$ blue socks, and $30$ purple socks. Someone randomly selects socks without seeing their color. What is the smallest number of socks that they must pick to guarantee at least $12$ pairs? A pair of socks consists of two socks of the same ...	[ { "content": "A dark drawer contains $90$ red socks, $70$ green socks, $50$ blue socks, and $30$ purple socks. Someone randomly selects socks without seeing their color. What is the smallest number of socks that they must pick to guarantee at least $12$ pairs? A pair of socks consists of two socks of the same ...
[ { "content": "In a new sequence, the first term is \\(a_1 = 5000\\) and the second term is \\(a_2 = 5001\\). Furthermore, the values of the remaining terms are designed so that \\(a_n + a_{n+1} + a_{n+2} = 2n\\) for all \\( n \\geq 1 \\). Determine \\(a_{1000}\\).", "role": "user" }, { "content": " ...	[ { "content": "In a new sequence, the first term is \\(a_1 = 5000\\) and the second term is \\(a_2 = 5001\\). Furthermore, the values of the remaining terms are designed so that \\(a_n + a_{n+1} + a_{n+2} = 2n\\) for all \\( n \\geq 1 \\). Determine \\(a_{1000}\\).", "role": "user" }, { "content": " ...
[ { "content": "Given that the right focus of ellipse $I$: $\\frac{{x}^{2}}{{a}^{2}}+ \\frac{{y}^{2}}{{b}^{2}}=1 (a > 0,b > 0)$ is $(2 \\sqrt{2},0)$, and ellipse $I$ passes through point $(3,1)$.\n(1) Find the equation of ellipse $I$;\n(2) Let line $l$ with slope $1$ intersect ellipse $I$ at two distinct points $...	[ { "content": "Given that the right focus of ellipse $I$: $\\frac{{x}^{2}}{{a}^{2}}+ \\frac{{y}^{2}}{{b}^{2}}=1 (a > 0,b > 0)$ is $(2 \\sqrt{2},0)$, and ellipse $I$ passes through point $(3,1)$.\n(1) Find the equation of ellipse $I$;\n(2) Let line $l$ with slope $1$ intersect ellipse $I$ at two distinct points $...
[ { "content": "What number can be added to both the numerator and denominator of $\\frac{3}{5}$ so that the resulting fraction will be equivalent to $\\frac{5}{6}$?", "role": "user" }, { "content": " To find the number \\( x \\) that can be added to both the numerator and denominator of \\( \\frac{3}...	[ { "content": "What number can be added to both the numerator and denominator of $\\frac{3}{5}$ so that the resulting fraction will be equivalent to $\\frac{5}{6}$?", "role": "user" }, { "content": " Let me think about how to approach this problem step by step.\n\nOkay, so I have this fraction, 3/5, ...
[ { "content": "John is cycling east at a speed of 8 miles per hour, while Bob is also cycling east at a speed of 12 miles per hour. If Bob starts 3 miles west of John, determine the time it will take for Bob to catch up to John.", "role": "user" }, { "content": " To solve the problem of when Bob catc...	[ { "content": "John is cycling east at a speed of 8 miles per hour, while Bob is also cycling east at a speed of 12 miles per hour. If Bob starts 3 miles west of John, determine the time it will take for Bob to catch up to John.", "role": "user" }, { "content": " Let me think about how to approach th...
[ { "content": "To assess the shooting level of a university shooting club, an analysis group used stratified sampling to select the shooting scores of $6$ senior members and $2$ new members for analysis. After calculation, the sample mean of the shooting scores of the $6$ senior members is $8$ (unit: rings), wit...	[ { "content": "To assess the shooting level of a university shooting club, an analysis group used stratified sampling to select the shooting scores of $6$ senior members and $2$ new members for analysis. After calculation, the sample mean of the shooting scores of the $6$ senior members is $8$ (unit: rings), wit...
[ { "content": "The function $f: N \\to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \\in\\{0,1\\}$ for all $m,n \\in N$ . Determine $f (2014)$ . $N_0=\\{0,1,2,...\\}$ ", "role": "user" }, { "content": " \n</think>\n\nTo determine \\( f(2014) \\),...	[ { "content": "The function $f: N \\to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \\in\\{0,1\\}$ for all $m,n \\in N$ . Determine $f (2014)$ . $N_0=\\{0,1,2,...\\}$ ", "role": "user" }, { "content": " \nAlright, let's try to figure out this pro...
[ { "content": "Find the units digit of $9^{8^7}$.", "role": "user" }, { "content": " To find the units digit of \\(9^{8^7}\\), we can follow these steps:\n\n1. Understand the Problem: We need to determine the units digit of a very large exponentiation: \\(9^{8^7}\\). Direct computation is infeasi...	[ { "content": "Find the units digit of $9^{8^7}$.", "role": "user" }, { "content": " Let me think about how to find the units digit of \\(9^{8^7}\\). Hmm, units digit problems usually involve looking for patterns in the last digit of powers. I remember that for many numbers, the units digits repeat i...
[ { "content": "Two points are drawn on each side of a square with an area of 81 square units, dividing the side into 3 congruent parts. Quarter-circle arcs connect the points on adjacent sides to create the figure shown. What is the length of the boundary of the bolded figure? Express your answer as a decimal...	[ { "content": "Two points are drawn on each side of a square with an area of 81 square units, dividing the side into 3 congruent parts. Quarter-circle arcs connect the points on adjacent sides to create the figure shown. What is the length of the boundary of the bolded figure? Express your answer as a decimal...
[ { "content": "The number \\( N \\) is of the form \\( p^{\\alpha} q^{\\beta} r^{\\gamma} \\), where \\( p, q, r \\) are prime numbers, and \\( p q - r = 3, p r - q = 9 \\). Additionally, the numbers \\( \\frac{N}{p}, \\frac{N}{q}, \\frac{N}{r} \\) respectively have 20, 12, and 15 fewer divisors than the number ...	[ { "content": "The number \\( N \\) is of the form \\( p^{\\alpha} q^{\\beta} r^{\\gamma} \\), where \\( p, q, r \\) are prime numbers, and \\( p q - r = 3, p r - q = 9 \\). Additionally, the numbers \\( \\frac{N}{p}, \\frac{N}{q}, \\frac{N}{r} \\) respectively have 20, 12, and 15 fewer divisors than the number ...
[ { "content": "In triangle $ABC,$ $a = 7,$ $b = 9,$ and $c = 4.$ Let $I$ be the incenter.\n\n[asy]\nunitsize(0.8 cm);\n\npair A, B, C, D, E, F, I;\n\nB = (0,0);\nC = (7,0);\nA = intersectionpoint(arc(B,4,0,180),arc(C,9,0,180));\nI = incenter(A,B,C);\n\ndraw(A--B--C--cycle);\ndraw(incircle(A,B,C));\n\nlabel(\"$A...	[ { "content": "In triangle $ABC,$ $a = 7,$ $b = 9,$ and $c = 4.$ Let $I$ be the incenter.\n\n[asy]\nunitsize(0.8 cm);\n\npair A, B, C, D, E, F, I;\n\nB = (0,0);\nC = (7,0);\nA = intersectionpoint(arc(B,4,0,180),arc(C,9,0,180));\nI = incenter(A,B,C);\n\ndraw(A--B--C--cycle);\ndraw(incircle(A,B,C));\n\nlabel(\"$A...
[ { "content": "Define the polynomials $P_0, P_1, P_2 \\cdots$ by:\n\\[ P_0(x)=x^3+213x^2-67x-2000 \\]\n\\[ P_n(x)=P_{n-1}(x-n), n \\in N \\]\nFind the coefficient of $x$ in $P_{21}(x)$.", "role": "user" }, { "content": " \n\nOkay, so I have this problem about polynomials defined recursively. It start...	[ { "content": "Define the polynomials $P_0, P_1, P_2 \\cdots$ by:\n\\[ P_0(x)=x^3+213x^2-67x-2000 \\]\n\\[ P_n(x)=P_{n-1}(x-n), n \\in N \\]\nFind the coefficient of $x$ in $P_{21}(x)$.", "role": "user" }, { "content": " \nOkay, so I've got this problem where I need to find the coefficient of \\( x \...
[ { "content": "Given that the polynomial $x^2-kx+16$ has only positive integer roots, find the average of all distinct possibilities for $k$.", "role": "user" }, { "content": " To solve the problem, we start with the polynomial \\( x^2 - kx + 16 \\). It is given that this polynomial has only positive...	[ { "content": "Given that the polynomial $x^2-kx+16$ has only positive integer roots, find the average of all distinct possibilities for $k$.", "role": "user" }, { "content": " \nOkay, so I have this problem here: the polynomial \\(x^2 - kx + 16\\) has only positive integer roots. I need to find the ...
[ { "content": "Compute $95^2$ in your head.", "role": "user" }, { "content": " To compute \\(95^2\\) mentally, I'll use the algebraic identity for squaring a number close to 100. The formula I remember is \\((100 - a)^2 = 10000 - 200a + a^2\\), where \\(a = 5\\) in this case.\n\nFirst, I'll calculate...	[ { "content": "Compute $95^2$ in your head.", "role": "user" }, { "content": " To calculate \\(95^2\\) mentally, I can use the algebraic identity for squaring a binomial. Specifically, I'll express 95 as \\(100 - 5\\). This gives me:\n\n\\[\n95^2 = (100 - 5)^2\n\\]\n\nExpanding the square using the f...
[ { "content": "For what positive value of $t$ is $\|{-4+ti}\| = 2\\sqrt{13}$?", "role": "user" }, { "content": " To find the positive value of \\( t \\) such that \\( \|{-4 + ti}\| = 2\\sqrt{13} \\), we start by recognizing that the modulus of a complex number \\( a + bi \\) is given by \\( \\sqrt{a^2 + ...	[ { "content": "For what positive value of $t$ is $\|{-4+ti}\| = 2\\sqrt{13}$?", "role": "user" }, { "content": " Please think step by step, as if I'm just learning this.\n\nOkay, so I have this problem here: For what positive value of \\( t \\) is \\( \|{-4 + ti}\| = 2\\sqrt{13} \\)? Hmm, I need to figur...
[ { "content": "Given functions \\(f(x)\\) and \\(g(x)\\) are defined on \\(\\mathbb{R}\\) with \\(g(x)\\neq 0\\), and \\(f(x)g'(x) > f'(x)g(x)\\). It is also given that \\(f(x)=a^{x}\\cdot g(x)\\) where \\(a > 0, a\\neq 1\\), and \\(\\frac{f(1)}{g(1)} + \\frac{f(-1)}{g(-1)} = \\frac{5}{2}\\). Considering the fin...	[ { "content": "Given functions \\(f(x)\\) and \\(g(x)\\) are defined on \\(\\mathbb{R}\\) with \\(g(x)\\neq 0\\), and \\(f(x)g'(x) > f'(x)g(x)\\). It is also given that \\(f(x)=a^{x}\\cdot g(x)\\) where \\(a > 0, a\\neq 1\\), and \\(\\frac{f(1)}{g(1)} + \\frac{f(-1)}{g(-1)} = \\frac{5}{2}\\). Considering the fin...
[ { "content": "Let $N$ be the number of functions $f:\\{1,2,3,4,5,6,7,8,9,10\\} \\rightarrow \\{1,2,3,4,5\\}$ that have the property that for $1\\leq x\\leq 5$ it is true that $f(f(x))=x$ . Given that $N$ can be written in the form $5^a\\cdot b$ for positive integers $a$ and $b$ with $b$ not div...	[ { "content": "Let $N$ be the number of functions $f:\\{1,2,3,4,5,6,7,8,9,10\\} \\rightarrow \\{1,2,3,4,5\\}$ that have the property that for $1\\leq x\\leq 5$ it is true that $f(f(x))=x$ . Given that $N$ can be written in the form $5^a\\cdot b$ for positive integers $a$ and $b$ with $b$ not div...
[ { "content": "The hundreds digit of a three-digit number is $2$ more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?", "role": "user" }, { "content": " Let's denote the ...	[ { "content": "The hundreds digit of a three-digit number is $2$ more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?", "role": "user" }, { "content": " Hmm, okay, so I n...
[ { "content": "In the diagram below, $AB = 30$ and $\\angle ADB = 90^\\circ$. If $\\sin A = \\frac{3}{5}$ and $\\sin C = \\frac{1}{4}$, what is the length of $DC$?", "role": "user" }, { "content": " \n\nAlright, let's see. I've got this geometry problem here, and I need to find the length of DC. The ...	[ { "content": "In the diagram below, $AB = 30$ and $\\angle ADB = 90^\\circ$. If $\\sin A = \\frac{3}{5}$ and $\\sin C = \\frac{1}{4}$, what is the length of $DC$?", "role": "user" }, { "content": " I think it's 15.\n\nWait, let me try to figure this out step by step. Maybe I made a mistake earlier.\...
[ { "content": "Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\\frac mn$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.\n", "role": "user" }, { "c...	[ { "content": "Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\\frac mn$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.\n", "role": "user" }, { "c...
[ { "content": "A four-digit number with digits in the thousands, hundreds, tens, and units places respectively denoted as \\(a, b, c, d\\) is formed by \\(10 \\cdot 23\\). The sum of these digits is 26. The tens digit of the product of \\(b\\) and \\(d\\) equals \\((a+c)\\). Additionally, \\(( b d - c^2 )\\) is ...	[ { "content": "A four-digit number with digits in the thousands, hundreds, tens, and units places respectively denoted as \\(a, b, c, d\\) is formed by \\(10 \\cdot 23\\). The sum of these digits is 26. The tens digit of the product of \\(b\\) and \\(d\\) equals \\((a+c)\\). Additionally, \\(( b d - c^2 )\\) is ...
[ { "content": "How many integers between 1000 and 3000 have all three of the numbers 18, 24, and 36 as factors?", "role": "user" }, { "content": " To determine how many integers between 1000 and 3000 have all three numbers 18, 24, and 36 as factors, I first need to find the least common multiple (LCM...	[ { "content": "How many integers between 1000 and 3000 have all three of the numbers 18, 24, and 36 as factors?", "role": "user" }, { "content": " \n\nAlright, so I've got this problem here: \"How many integers between 1000 and 3000 have all three of the numbers 18, 24, and 36 as factors?\" Hmm, okay...
[ { "content": "Let the function \\( f(x) \\) be defined on \\([0,1]\\), satisfying \\( f(0) = f(1) \\) and for any \\( x, y \\in [0,1] \\) having \\( \|f(x) - f(y)\| < \|x - y\| \\). Find the smallest real number \\( m \\) such that for any function \\( f(x) \\) meeting the above conditions and for any \\( x, y \\in...	[ { "content": "Let the function \\( f(x) \\) be defined on \\([0,1]\\), satisfying \\( f(0) = f(1) \\) and for any \\( x, y \\in [0,1] \\) having \\( \|f(x) - f(y)\| < \|x - y\| \\). Find the smallest real number \\( m \\) such that for any function \\( f(x) \\) meeting the above conditions and for any \\( x, y \\in...
[ { "content": "A set consists of 120 distinct blocks. Each block is one of 3 materials (plastic, wood, metal), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 5 shapes (circle, hexagon, square, triangle, rectangle). How many blocks in the set differ from the 'wood small blue hexagon' in ...	[ { "content": "A set consists of 120 distinct blocks. Each block is one of 3 materials (plastic, wood, metal), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 5 shapes (circle, hexagon, square, triangle, rectangle). How many blocks in the set differ from the 'wood small blue hexagon' in ...
[ { "content": "Consider an $8 \\times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e. it cannot stay in the same square). What is the expected number of minutes until the rook reaches t...	[ { "content": "Consider an $8 \\times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e. it cannot stay in the same square). What is the expected number of minutes until the rook reaches t...
[ { "content": "Given the line $y=x+1$ intersects with the ellipse $mx^2+my^2=1(m > n > 0)$ at points $A$ and $B$, where the x-coordinate of the midpoint of the chord $AB$ is equal to $-\\frac{1}{3}$, find the eccentricity of the hyperbola $\\frac{y^2}{m^2}-\\frac{x^2}{n^2}=1$.", "role": "user" }, { "...	[ { "content": "Given the line $y=x+1$ intersects with the ellipse $mx^2+my^2=1(m > n > 0)$ at points $A$ and $B$, where the x-coordinate of the midpoint of the chord $AB$ is equal to $-\\frac{1}{3}$, find the eccentricity of the hyperbola $\\frac{y^2}{m^2}-\\frac{x^2}{n^2}=1$.", "role": "user" }, { "...
[ { "content": "On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning.", "role": "user" }, ...	[ { "content": "On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is 1088 feet per second and one mile is 5280 feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning.", "role": "user" }, ...