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[
{
"content": "It takes 15 men working steadily 4 days to dig the foundation for a new apartment. How many days would it have taken 25 men working at the same rate to dig the foundation? Express your answer as a decimal to the nearest tenth.",
"role": "user"
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{
"content": "If it takes 15 men work... |
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{
"content": "Isabel loves to run in the forest and she does it within a circuit that is 365 meters long. She runs the circuit 7 times in the morning and runs it 3 times in the afternoon. How many meters does Isabel run during a week?",
"role": "user"
},
{
"content": "In a day, Isabel runs the circu... |
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{
"content": "A cistern with a capacity of 12000 liters is 2/3 full. Landon drains the cistern by 35% of the total volume of water in the cistern to irrigate his orchard. He then fills the cistern with 25% of the volume of water remaining in the cistern. Calculate the final volume of water in the cistern.",
... |
[
{
"content": "There is a group of ten students consisting of seven girls and three boys. When they gather for a photo, they stand in a line - the girls wearing identical blue shirts and the boys wearing identical white shirts. How many distinct ways are there to arrange the seven blue shirts and three white shi... |
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{
"content": "Let $p$ and $q$ be the two distinct solutions to the equation $$(x-5)(2x+9) = x^2-13x+40.$$What is $(p + 3)(q + 3)$?",
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{
"content": "First expand the equation:\n\\[ (x-5)(2x+9) = x^2-13x+40 \\Rightarrow 2x^2 + 9x - 10x - 45 = x^2 - 13x + 40 \\Rightarrow 2x^2 - x - 45 ... |
[
{
"content": "Let $x$, $y$, and $z$ be positive real numbers such that $x^2+y^2+z^2=1$. Find the smallest possible value of $\\frac{x^2}{1-yz}+\\frac{y^2}{1-zx}+\\frac{z^2}{1-xy}$.",
"role": "user"
},
{
"content": "Since $x^2 + y^2 + z^2 = 1$, we can use the Cauchy-Schwarz inequality to find the sma... |
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{
"content": "Karen had the giant box of crayons, containing twice as many crayons as were in Beatrice's box. But Beatrice's box contained twice as many crayons as were in Gilbert's box of crayons. And Gilbert had four times as many crayons as were in Judah's box of crayons. If Karen's box contained 128 crayons... |
[
{
"content": "Compute the coefficient of $x^2$ in the expansion of $(1+x)^{1293}$.",
"role": "user"
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{
"content": "We can use the Binomial Theorem to find the coefficient of $x^2$ in the expansion of $(1+x)^{1293}$.\n\nAccording to the Binomial Theorem, the expansion of $(1+x)^{1293}$ is given by:... |
[
{
"content": "In the triangle $ABC$, the angle bisector of $\\angle A$ intersects the side $BC$ at the point $D$. Given that $BD:DC = 2:3$ and the coordinates of the points $B$ and $C$ are $(2, 3)$ and $(4, 7)$ respectively, find the equation of the angle bisector of $\\angle A$ in the form $y = mx + b$.",
... |
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{
"content": "Three congruent circles centered at points $A$, $B$, and $C$ have radius $r$ and intersect pairwise at points $D$, $E$, and $F$ such that lines $AD$, $BE$, and $CF$ pass through a common point $O$.\nGiven that $\\angle AOB = \\frac{\\pi}{2}$, find $\\cos\\angle BOC$.",
"role": "user"
},
{
... |
[
{
"content": "A rental car company charges a base fee for the first day of rental and then adds a fixed amount for each additional day. If it costs Sarah $\\$85$ to rent a car for 4 days and Rachel $\\$180$ to rent a car for 7 days, what is the base fee for the first day?",
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{
"cont... |
[
{
"content": "Let $\\alpha$ and $\\beta$ be the roots of the equation $x^2-x-1=0$.\n\nDetermine the value of $a$ for which the equations\n$\\begin{array}{l}\nx_1+\\alpha y_1=2a^2,\\\\\nx_2+\\alpha y_2=2a,\\\\\nx_3+\\alpha y_3=a+1,\\\\\nx_4+\\alpha y_4=a-1\n\\end{array}$\nand\n$\\begin{array}{l}\nx_1+\\beta y_1=... |
[
{
"content": "What is the coefficient of $x^4$ in the expansion of $(2x+3)^5$?",
"role": "user"
},
{
"content": "To find the coefficient of $x^4$ in the expansion of $(2x+3)^5$, we can use the Binomial Theorem.\n\nThe general form of the Binomial Theorem is:\n\\[ (a + b)^n = \\sum_{k=0}^{n} \\binom{... |
[
{
"content": "Let $a,$ $b,$ $c$ be positive real numbers. Find the set of all possible values of\n\\[g(a,b,c) = \\frac{a}{a + 2b} + \\frac{b}{b + 2c} + \\frac{c}{c + 2a}.\\]",
"role": "user"
},
{
"content": "## Step 1: Start by examining the given expression for g(a, b, c)\nWe are given the express... |
[
{
"content": "A city has 8 different bus lines, each with a unique route. A tourist wants to take exactly 3 different buses during her stay. In how many ways can she choose the 3 buses if she can't take bus A without taking bus B, and can't take bus C without taking bus D?",
"role": "user"
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{
"con... |
[
{
"content": "What is the smallest positive integer $n$ such that there exist positive integers $a$ and $b$ satisfying $\\mathop{\\text{lcm}}[a,b]=n$ and $\\mathop{\\text{gcd}}[a,b]=4$?",
"role": "user"
},
{
"content": "We know that the product of the gcd and the lcm of two numbers is equal to the p... |
[
{
"content": "Compute the value of\n\\[\\sin (\\log \\sqrt[3]{2}) + \\cos (\\log 3^{\\sqrt{2}}).\\]",
"role": "user"
},
{
"content": "The problem asks us to find the value of\n\\[ \\sin (\\log \\sqrt[3]{2}) + \\cos (\\log 3^{\\sqrt{2}}) \\]\n\nUsing properties of logarithms, we can rewrite the expre... |
[
{
"content": "A student group consists of 30 members. During the last event, only 20 members participated: one-third of the freshmen attended but all of the seniors attended. How many freshmen are in the student group?",
"role": "user"
},
{
"content": "Let's denote the number of freshmen as $f$ and ... |
[
{
"content": "The sum of the first $n$ terms in the infinite geometric sequence $\\left\\{1,\\frac{1}{3},\\frac{1}{9},\\frac{1}{27},\\dots \\right\\}$ is $\\frac{121}{81}$. Find $n$.",
"role": "user"
},
{
"content": "The given sequence is a geometric sequence with the first term $a_1 = 1$ and the co... |
[
{
"content": "Consider the sequence of real numbers $b_0, b_1, \\dots, b_n$ such that $b_0 = 1$, $b_1 = 2$, and $$ b_{k+1} = \\frac{b_{k-1}}{b_k} + 1 $$ for $k = 1, 2, \\dots, n-1$. If the sequence converges to a finite limit, find the value of this limit.",
"role": "user"
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{
"content": "We are gi... |
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{
"content": "Find the product of the greatest common divisor and the least common multiple of $18$ and $42.$",
"role": "user"
},
{
"content": "To find the greatest common divisor (GCD) and the least common multiple (LCM) of 18 and 42, let's first factor each number into its prime factors.\n\nThe pr... |
[
{
"content": "The quadratic $8x^2 - 20x - 12$ can be rewritten in the form of $(ax + b)(cx + d),$ where $a,$ $b,$ $c,$ and $d$ are all integers. What is the product $abcd$?",
"role": "user"
},
{
"content": "We are given the quadratic $8x^2 - 20x - 12$.\n\nTo factor this quadratic, we need to find tw... |
[
{
"content": "In twenty years, Ramon will be twice as old as Loui today. If Loui is currently 23 years old, how old is Ramon now?",
"role": "user"
},
{
"content": "Let's break it down:\n\nRamon will be twice as old as Loui today in 20 years.\nLoui is currently 23 years old, so twice Loui's age today... |
[
{
"content": "What is the greatest common factor of the two expressions $x^2 + 5x + 6$ and $x^2 + 3x + 2$?",
"role": "user"
},
{
"content": "We can factor the expressions as follows:\n\\[ x^2 + 5x + 6 = (x + 3)(x + 2) \\]\n\\[ x^2 + 3x + 2 = (x + 2)(x + 1) \\]\n\nSince both expressions have the comm... |
[
{
"content": "In a convex cyclic quadrilateral $ABCD$, $\\angle B$ is a right angle, and the length of the diameter $AC$ is $15$. Given that the lengths of sides $AB$ and $CD$ are integers, and the length of side $AD$ is greater than the length of side $BC$, how many different convex cyclic quadrilaterals $ABCD... |
[
{
"content": "Find the projection of the vector $\\begin{pmatrix} 3 \\\\ 0 \\\\ -2 \\end{pmatrix}$ onto the line\n\\[\\frac{x}{2} = y = \\frac{z}{-1}.\\]",
"role": "user"
},
{
"content": "To find the projection of the vector onto the line, we need to find the projection of the vector onto the direct... |
[
{
"content": "Let $g$ be a function defined on the positive integers, such that\n\\[g(x^2) = g(x) + g(x)\\]for all positive integers $x.$ Given $g(4) = 6$ and $g(9) = 8,$ find $g(16).$",
"role": "user"
},
{
"content": "The given equation is $g(x^2) = g(x) + g(x)$. Simplify it to get:\n\\[ g(x^2) = ... |
[
{
"content": "A magazine costs $3 each. Jewel bought 10 magazines to be sold at $3.50 each. How much will be Jewel gain from selling these?",
"role": "user"
},
{
"content": "Jewel bought 10 magazines at $3 each, which means she spent $3 * 10 = 30$.\nShe sold them for $3.50 each, so she earned $3.50 ... |
[
{
"content": "Emily's age is four times the age of her cousin, David. Emily's sister, Olivia, is three years younger than Emily. Emily and Olivia are triplets with their brother, Benjamin. How old is David?",
"role": "user"
},
{
"content": "Let's denote Emily's age as $E$, David's age as $D$, Olivia... |
[
{
"content": "If $x,y,z$ are integers from the set of positive integers less than $11$ such that \\begin{align*}\nxyz&\\equiv 3\\pmod {11},\\\\\n7z&\\equiv 5\\pmod {11},\\\\\n9y&\\equiv 8+y\\pmod {11},\n\\end{align*}then what is the remainder when $x+y+z$ is divided by $11$?",
"role": "user"
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{
"c... |
[
{
"content": "Consider a square calendar whose each cell represents a day. A day is considered \"magic\" if the row number (from top to bottom) and column number (from left to right) are both divisors of a fixed number m.\n\nHow many magic dates are there if m = 31?",
"role": "user"
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{
"content": ... |
[
{
"content": "A company has five employees who work different numbers of hours per week. The average number of hours worked by all employees is 30 hours. If we know that the number of hours worked by each employee can be expressed as follows: x + 2, 20, 3x - 1, 25, x + 5, what is the value of x?",
"role": "... |
[
{
"content": "Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes $1, 10, 11, 12, \\ldots$ but by the 1,000th digit he (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last thr... |
[
{
"content": "Mr. Mendez awards extra credit on quizzes to his students with quiz grades that exceed the class mean. Given that 107 students take the same quiz, what is the largest number of students who can be awarded extra credit?",
"role": "user"
},
{
"content": "To maximize the number of student... |
[
{
"content": "A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?\n$\\textbf{(A)}\\ \\sqrt{13}\\qquad \\textbf{(B)}\\ \\sqrt{14}\\qquad \\textbf{(C)}\\ \\sqrt... |
[
{
"content": "What is the $100$th digit after the decimal point when $\\frac{3}{26}$ is expressed as a decimal?",
"role": "user"
},
{
"content": "To find the 100th digit after the decimal point in the decimal representation of $\\frac{3}{26}$, we first express $\\frac{3}{26}$ as a decimal using long... |
[
{
"content": "How many terms of the arithmetic sequence 88, 85, 82, $\\dots$ appear before the number $-17$ appears?",
"role": "user"
},
{
"content": "This is an arithmetic sequence with first term $a_1 = 88$ and common difference $d = -3$. The general formula for the nth term of an arithmetic seque... |
[
{
"content": "In a set of eight objects arranged in a row, find the number of subsets that contain at least two consecutive objects.",
"role": "user"
},
{
"content": "We can count the number of subsets with at least two consecutive objects by first counting the total number of subsets and then subtr... |
[
{
"content": "A bookstore is having a sale. For every novel purchased, the customer receives 3 bookmarks as a gift. If the store sells 15 novels in the morning and 32 novels in the evening, how many bookmarks has the store given away in total?",
"role": "user"
},
{
"content": "For every novel purcha... |
[
{
"content": "Maria surveyed a group of people about their knowledge of bees. To the nearest tenth of a percent, she found that $73.2\\%$ of the people surveyed thought bees made honey. Of the people who thought bees made honey, $56.1\\%$ said that bees were native to North America. Since bees are actually nati... |
[
{
"content": "The solution to the inequality\n\\[y = -x^2 + ax + b \\le 0\\]is $(-\\infty,-3] \\cup [5,\\infty).$ Find the vertex of the parabola $y = -x^2 + ax + b.$",
"role": "user"
},
{
"content": "Since the inequality $y \\le 0$ holds for $(-\\infty, -3] \\cup [5, \\infty)$, the quadratic funct... |
[
{
"content": "A set of 11 elements is given, and it is required to determine the number of all possible partitions of the set into two nonempty subsets.\n\n (I) The first subset consists of 2 elements, and the second of 9 elements.\n(II) The first subset consists of 9 elements, and the second of 2 elements.\n\n... |
[
{
"content": "Let $m=\\angle CYX$ and $n=\\angle BXY.$ Since $\\angle AXE$ is $108^\\circ$ less than $3m$ and $\\angle CYX$ is $m$, then $$3m-108^\\circ=m+n$$ $$2m=n+108^\\circ$$ Since $\\overline{AB}\\parallel \\overline{CD}$, then we know that the corresponding angles of $\\overline{AB}$ and $\\overline{CD}$ ... |
[
{
"content": "Maya knits sweaters during the six weekdays and sells them on Sundays. She makes 3 sweaters a day and sells each sweater for $12. In 6 weeks she managed to sell all the sweaters. How much money did she collect during that time?",
"role": "user"
},
{
"content": "Maya makes 3 sweaters a ... |
[
{
"content": "In a classroom, there are blue chairs, green chairs, and white chairs. There are 10 blue chairs. The green chairs are 3 times as many as the blue chairs, and there are 13 fewer white chairs than the green and blue chairs combined. How many chairs are there in a classroom?",
"role": "user"
},... |
[
{
"content": "A regular 12-gon is inscribed in a circle of radius 12. The sum of the lengths of all sides and diagonals of the 12-gon can be written in the form\n\\[a + b \\sqrt{2} + c \\sqrt{3} + d \\sqrt{6},\\]where $a$, $b$, $c$, and $d$ are positive integers. Find $a+b+c+d$.",
"role": "user"
},
{
... |
[
{
"content": "At a birthday party, there are 20 guests. A total of 50 cupcakes are prepared for them. One-third of the guests eat 2 cupcakes each, and the remaining guests eat 1 cupcake each. How many cupcakes are leftover?",
"role": "user"
},
{
"content": "There are 20 guests at the party.\nOne-thi... |
[
{
"content": "A magazine contains 1005 different codes, each consisting of a 5-digit number with the digits 0, 1, 2, 3, or 4. How many of these codes contain exactly three digits that are the same?",
"role": "user"
},
{
"content": "We need to count the number of codes that contain exactly three digi... |
[
{
"content": "In a university, 40% of the students are seniors, 25% are freshmen, and the rest are graduate students. If there are 800 students, how many more graduate students are there than freshmen?",
"role": "user"
},
{
"content": "Let's denote the total number of students as $S$, the number of ... |
[
{
"content": "Given the vertices of a regular hexagon inscribed in a circle of radius $r$ (which means the circle passes through all vertices), calculate the side length of the hexagon that is tangent to the circle at its midpoint. Consider the distance between the center of the circle and the point of tangency... |
[
{
"content": "The function \nf(z) = (z - 1)^4 + z^4 + i (2 z + 1)\nmaps some complex number a + bi to a complex number lying on the real axis.\nFind the smallest positive value of |a| + |b|.",
"role": "user"
},
{
"content": "## Step 1: Understand the problem\nWe are given a function f(z) that maps a... |
[
{
"content": "A polynomial $P(x)$ with real coefficients has two complex roots $\\frac{-3\\pm i\\sqrt{299}}{14}$. Find the product of all the real roots of $P(x)$.",
"role": "user"
},
{
"content": "Since $P(x)$ has real coefficients, the complex roots must come in conjugate pairs. The product of a c... |
[
{
"content": "Let $\\mathbf{A} = \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix}$. Find the matrix $\\mathbf{B}$ such that for any matrix $\\mathbf{X}$, $\\mathbf{A}\\mathbf{X} + \\mathbf{B}\\mathbf{X} = \\mathbf{X}\\mathbf{A} + \\mathbf{X}\\mathbf{B}$. If no such matrix $\\mathbf{B}$ exists, then enter the ze... |
[
{
"content": "What is the range of the function $y = \\frac{x^2 - 4x + 3}{x-3}$? (Express your answer using interval notation.)",
"role": "user"
},
{
"content": "## Step 1: Factor the numerator\nFirst, let's factor the numerator of the function to simplify it. We can factor the quadratic expression ... |
[
{
"content": "A box contains 10 different colored balls: 4 red, 3 blue, 2 green, and 1 yellow. If 3 balls are drawn at random from the box, what is the probability that at least 2 of them are of the same color?",
"role": "user"
},
{
"content": "To solve this problem, we can calculate the probability... |
[
{
"content": "A quadratic function $f(x)$ can be expressed in the form $f(x) = ax^2 + bx + c$ and in the vertex form $f(x) = a(x - h)^2 + k$. Given the standard form of the quadratic function $f(x) = 3x^2 - 8x + 5$, determine the vertex $(h, k)$.",
"role": "user"
},
{
"content": "To find the vertex ... |
[
{
"content": "What is the largest positive integer $n$ for which there exists a positive integer $k$ such that $7^n$ leaves a remainder of 1 when divided by $k$, and $7^{n+1}$ leaves a remainder of 3 when divided by $k$?",
"role": "user"
},
{
"content": "We know that $7^{n+1} = 7^n \\cdot 7$. Since ... |
[
{
"content": "Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$.",
"role": "user"
},
{
"content": "Let's first solve for the conditions on $x$ given by each interval separately.\n\nCondition for $2x$ being in $(1, 2)$:\n\\[ 1 < 2x < 2 \\Rightarrow \\frac{1}{2} < x... |
[
{
"content": "Compute the value of $x$ that satisfies the equation $\\cos(2x) = \\frac{1}{2}$.",
"role": "user"
},
{
"content": "The equation $\\cos(2x) = \\frac{1}{2}$ can be solved using the unit circle or knowledge of common cosine values.\n\nFrom the unit circle, we know that $\\cos(\\theta) = \... |
[
{
"content": "Let\n$$x^3 + ax^2 + bx + c = (x + p)(x + q)(x + r),$$\n$$x^3 - 7x^2 + 16x - 12 = (x - p)(x - q)(x - s),$$\nand $p$, $q$, $r$, and $s$ are positive integers. What is the value of $p + q + r + s$?",
"role": "user"
},
{
"content": "Let's use the Vieta's formulas to relate the coefficients... |
[
{
"content": "There are three kinds of rabbits on the farm: white, brown, and gray. A quarter of the rabbits are white, and there are twice as many gray rabbits as brown rabbits. If there are 44 rabbits total on the farm, how many of those rabbits are gray?",
"role": "user"
},
{
"content": "Let $w$,... |
[
{
"content": "Determine the domain of the function $y=\\sqrt{|\\sin(\\log_3 x)|}$, where $x$ is a positive real number.",
"role": "user"
},
{
"content": "The domain of the function involves several steps.\n\nStep 1: $\\log_3 x$ must be defined.\nSince $x$ is a positive real number, $\\log_3 x$ is de... |
[
{
"content": "Let $a$, $b$, and $c$ be the side lengths of a triangle, and let $s$ be the semiperimeter. Let\n\n\\begin{align*}\nP &= \\frac{\\sqrt{s-a}+\\sqrt{s-b}+\\sqrt{s-c}}{\\sqrt{s}},\\\\\nQ &= \\frac{\\sqrt{s+a}+\\sqrt{s+b}+\\sqrt{s+c}}{\\sqrt{s}}.\n\\end{align*}Find the maximum value of $PQ$.",
"rol... |
[
{
"content": "Emily is going on a road trip. She has to drive 15 miles to the first rest stop, twice that distance to the second rest stop, then 2 times the total distance driven so far to get to her final destination. How many miles does Emily drive between the second rest stop and her final destination?",
... |
[
{
"content": "A triangle in a Cartesian coordinate plane has vertices (5, -2), (10, 5) and (5, 5). How many square units are in the area of the triangle? Express your answer as a decimal to the nearest tenth.",
"role": "user"
},
{
"content": "To find the area of the triangle with vertices (5, -2), (... |
[
{
"content": "Let $x$ and $y$ be real numbers such that\n\\[xy - \\frac{x}{y^2} - \\frac{y}{x^2} = 3.\\]Find the sum of all possible values of $(x - 1)(y - 1).$",
"role": "user"
},
{
"content": "We want to find the sum of all possible values of $(x - 1)(y - 1)$, given that $xy - \\frac{x}{y^2} - \\f... |
[
{
"content": "The square of an integer is 182 greater than the integer itself. What is the sum of all integers for which this is true?",
"role": "user"
},
{
"content": "Let $x$ be the integer in question. The problem tells us that $x^2 = x + 182$.\n\nThis equation can be rearranged into the quadrati... |
[
{
"content": "Calculate the result of the subtraction problem: $4.19-2.8$.",
"role": "user"
},
{
"content": "To solve this problem, we simply need to subtract 2.8 from 4.19.\n\nFirst, let's line up the decimal points and perform the subtraction:\n```\n4.19\n- 2.80\n------\n1.39\n```\nSo, the result ... |
[
{
"content": "Find the value of m such that\n\\[\\frac{9}{x + y} = \\frac{m}{x + z} = \\frac{13}{z - y}.\\]",
"role": "user"
},
{
"content": "We can rewrite the given equation as:\n\\[\\frac{9}{x + y} = \\frac{m}{x + z} = \\frac{13}{z - y} = k,\\]\nwhere $k$ is a constant.\n\nFrom the first and seco... |
[
{
"content": "Let $f(x)$ be a quadratic function of the form $ax^2 + bx + c.$ Find the maximum value of $f(x)$ in terms of $a, b,$ and $c.$",
"role": "user"
},
{
"content": "The maximum value of a quadratic function occurs at its vertex. To find the vertex, we can use the formula:\n\\[ x = \\frac{-... |
[
{
"content": "Alex is a graphic designer who receives 8 design project requests daily from various clients. She can only work on 5 projects each day due to time constraints. After 7 days, how many design project requests will she have remaining to work on?",
"role": "user"
},
{
"content": "Alex rece... |
[
{
"content": "Seven congruent isosceles triangles are arranged to form a larger isosceles triangle $ABC$, as shown. The area of $ABC$ is $900$. What is the length, $y$, rounded off to the nearest integer?\n\n[asy]\nreal y = 1; real z = 2/3;\n\n// Draw outer triangle and labels\npair a = (0, 0); pair b = (4 * ... |
[
{
"content": "Let $p(x)=x^3+ax^2+bx+c$ be a polynomial with integer coefficients. Suppose that $p(x)$ has three real roots, one of which is $x=r^2$, where $r$ is a positive integer. If $q(x)=x^3+ax^2+bx+c$ has only integer roots, what is the smallest possible value of $r$ such that $p(x)$ and $q(x)$ have no com... |
[
{
"content": "In a regular octahedron, there are several segments that can be formed by connecting two vertices that are not adjacent to each other. How many such segments exist in a regular octahedron?",
"role": "user"
},
{
"content": "In a regular octahedron, there are 6 vertices.\n\nTo find the n... |
[
{
"content": "In quadrilateral $ABCD$, diagonal $\\overline{AC}$ is a diameter of a semicircle that passes through points $B$ and $D$. Given that $\\angle BAC=60^\\circ$, $AD=10$ and $AB=8$, find the length of side $\\overline{BC}$.",
"role": "user"
},
{
"content": "Given that diagonal $\\overline{A... |
[
{
"content": "The fifth term of a geometric sequence of positive numbers is $11$ and the eleventh term is $5$. What is the eighth term of the sequence? Express your answer in simplest radical form. [asy]\nsize(150); defaultpen(linewidth(2));\nreal loc = 0;\nfor(int i = 0; i < 11; ++i) {\n\nif(i == 4)\n\nlabel(... |
[
{
"content": "A horse gallops through a meadow and is joined by a group of 5 deer. 3 birds land on each deer, and 2 squirrels scamper alongside each bird. How many animals are galloping or moving through the meadow?",
"role": "user"
},
{
"content": "Let's break down the problem step by step:\n\n* 1 ... |
[
{
"content": "In a class of 30 students, 18 students speak English, 15 students speak Spanish, and 7 students speak both English and Spanish. If each student speaks at least one of these two languages, and the number of students who speak only English is 3 more than the number of students who speak only Spanish... |
[
{
"content": "Find the value of $k$ for which the polynomial $2x^4 - 3x^3 + kx^2 - 5x + 1$ has a sum of coefficients equal to the product of its coefficients.",
"role": "user"
},
{
"content": "The sum of the coefficients of the polynomial is\n\\[ 2 - 3 + k - 5 + 1 = k - 5 \\]\n\nThe product of the c... |
[
{
"content": "Determine the remainder when $3x^2 + 2x - 5$ is divided by $x + 2$.",
"role": "user"
},
{
"content": "To find the remainder when $3x^2 + 2x - 5$ is divided by $x + 2$, we can use polynomial long division or synthetic division. However, in this case, we can also use the Remainder Theore... |
[
{
"content": "A function $f(x)$ is defined as follows:\n\n$f(x) = \\begin{cases}\nx^2+3x+2 & \\text{if } x < 0 \\\\\n2x+1 & \\text{if } x = 0 \\\\\n3x-2 & \\text{if } x > 0\n\\end{cases}$\n\nFind the value of $f(f(-2))$.",
"role": "user"
},
{
"content": "To find $f(f(-2))$, we first need to calculat... |
[
{
"content": "Alexis needs to exercise for 1.5 hours every day to stay healthy. After she joined a sports team, she needs to exercise for 4 more hours every week. How many hours will Alexis exercise in three weeks?",
"role": "user"
},
{
"content": "Let's break down Alexis's exercise time per week. S... |
[
{
"content": "What value of $x$ will give the maximum value for $-x^2- 6x + 12$?",
"role": "user"
},
{
"content": "To find the value of $x$ that will give the maximum value for the quadratic function $-x^2 - 6x + 12$, we need to find the vertex of the parabola it represents.\n\nThe $x$-coordinate of... |
[
{
"content": "Tom writes the squares of all positive integers from 1 to n, inclusive, on the blackboard, then erases 10n of these numbers. If the average of the remaining numbers is 23, find n.",
"role": "user"
},
{
"content": "## Step 1: The problem requires us to find the value of n, where Tom wr... |
[
{
"content": "Consider the line parameterized by\n\\begin{align*}\nx&= 2s + 3,\\\\\ny& = 3s-1.\n\\end{align*}Find a vector $\\begin{pmatrix}c \\\\ d \\end{pmatrix}$ pointing from the origin to this line that is parallel to $\\begin{pmatrix}5 \\\\-2 \\end{pmatrix}$.",
"role": "user"
},
{
"content": "... |
[
{
"content": "A car is traveling around a circular track at a constant speed. At 3:45 PM, the car is at a distance of 2.5 km from the starting point, and at 4:00 PM, it is at a distance of 6.8 km from the starting point. If the track is 10 km long, what is the speed of the car in kilometers per hour?",
"rol... |
[
{
"content": "On a circular street, there are 5 houses, each painted either blue or red. Let b be the number of houses with at least one blue neighbor and r be the number of houses with at least one red neighbor. How many different ordered pairs (b,r) are possible?",
"role": "user"
},
{
"content": "... |
[
{
"content": "Simplify: $2x(4x^2+5)-3(x^2-7x+2)$. Express your answer in the form $Ax^3+Bx^2+Cx+D$.",
"role": "user"
},
{
"content": "To simplify the expression, we need to distribute the terms and then combine like terms.\n\n1. Distribute the terms:\n \\begin{align*}\n 2x(4x^2+5) &= 8x^3 + 10x ... |
[
{
"content": "Emily wants to wallpaper her office and her home gym. The office is a rectangle 15 feet by 20 feet, and she needs to cover all four walls. The home gym is a square 25 feet by 25 feet, and she only needs to cover two walls. All the walls in Emily's house are 12 feet tall. How many total square feet... |
[
{
"content": "On a busy straight stretch of two-way double-lane highway, trucks all travel at the same speed and all follow the safety rule: the distance from the back of the truck ahead to the front of the truck behind is exactly one truck length for each 20 kilometers per hour of speed or fraction thereof. A ... |
[
{
"content": "What is the smallest positive integer n such that n^2 is congruent to 4 (mod 9) and n^3 is congruent to 8 (mod 27)?",
"role": "user"
},
{
"content": "For $n^2 \\equiv 4 \\pmod{9}$:\n\\[ n^2 - 4 \\equiv 0 \\pmod{9} \\Rightarrow (n + 2)(n - 2) \\equiv 0 \\pmod{9} \\]\nThis implies that $... |
[
{
"content": "A bag contains 4 red marbles, 6 blue marbles, and 2 green marbles. A marble is drawn at random from the bag and then replaced. A second marble is then drawn. What is the probability that the first marble is red and the second marble is not green?",
"role": "user"
},
{
"content": "The t... |
[
{
"content": "A right cone with a base radius of 4 units is inscribed in a hemisphere of radius 6 units. The volume, in cubic units, of the space inside the hemisphere and outside the cone is $V\\pi$. Find $V$, as a common fraction, given that the height of the cone is 2 units less than the radius of the hemisp... |
[
{
"content": "Ava attended a music festival with a 2-hour concert by her favorite band. If there were two 10-minute breaks and all the songs were 4 minutes long, except for two songs that lasted 8 minutes each, how many songs did the band play?",
"role": "user"
},
{
"content": "The concert was 2 hou... |
[
{
"content": "At a restaurant, Emily worked 35 hours a week, Michael worked 28 hours, and Olivia worked 22 hours. If the employees are paid $18 per hour, how much money does their manager spend paying all of them every month?",
"role": "user"
},
{
"content": "To find the total amount of money the ma... |
[
{
"content": "Calculate $\\frac{1}{4} \\cdot \\frac{2}{5} \\cdot \\frac{3}{6} \\cdot \\frac{4}{7} \\cdots \\frac{49}{52} \\cdot \\frac{50}{53}$. Express your answer as a common fraction.",
"role": "user"
},
{
"content": "Looking at the terms, we notice that we can cancel each numerator with the deno... |
[
{
"content": "In a rectangular pyramid with base dimensions 12 and 18, and slant height 15, find the length of the shortest segment from the apex to the base that is perpendicular to one of the base's diagonals.",
"role": "user"
},
{
"content": "To find the length of the shortest segment from the ap... |
[
{
"content": "How many positive 4-digit numbers are divisible by 13 and have a sum of digits equal to 15?",
"role": "user"
},
{
"content": "We can use the divisibility rule for 13, which states that a number is divisible by 13 if and only if the difference between the sum of the digits in the even p... |
[
{
"content": "A sector of a circle with radius 8 units is divided into 4 equal parts. Three of the parts are shaded as shown in the diagram below. If the area of the sector is 24π square units, what percentage of the sector's area is shaded?\n\n[asy]\nimport graph;\ndefaultpen(linewidth(0.7));\n\ndraw((0,0)--(8... |
[
{
"content": "Twenty-four hours ago, Olivia and her team took 120 snapshots of their product. Today, Olivia and her team will take 25% fewer snapshots of the same product. If they were to take 400 snapshots of the product, how many snapshots would they take to reach the target?",
"role": "user"
},
{
... |
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