prompt listlengths 1 1 | label stringlengths 1 23 |
|---|---|
[
{
"content": "Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | 6 |
[
{
"content": "In Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points-per-game a... | 40 |
[
{
"content": "A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rect... | 2 |
[
{
"content": "Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? (Note that if no die is rolled, the sum is 0.) Provide your answer as a fraction \\(\\frac{k}{m}\\) in simplest form, and give the value of ... | 11 |
[
{
"content": "The numbers $\\log(a^3b^7)$, $\\log(a^5b^{12})$, and $\\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^{th}$ term of the sequence is $\\log{b^n}$. What is the value of $n$?\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role": "us... | 112 |
[
{
"content": "An equilateral triangle has side length \\(6\\). Find the area of the region containing all points that are outside the triangle but not more than \\(3\\) units from a point of the triangle. The original answer is in the form \\(k + m\\pi\\), where \\(k\\) and \\(m\\) are integers. Please provide ... | 63 |
[
{
"content": "Let $a_1,a_2,\\ldots$ be a sequence determined by the rule $a_n=\\frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n=3a_{n-1}+1$ if $a_{n-1}$ is odd. Find the number of positive integers $a_1 \\le 2008$ for which $a_1$ is less than each of $a_2$, $a_3$, and $a_4$.\n\nLet's think step by step and outp... | 502 |
[
{
"content": "Triangle \\(ABC\\), with sides of length \\(5\\), \\(6\\), and \\(7\\), has one vertex on the positive \\(x\\)-axis, one on the positive \\(y\\)-axis, and one on the positive \\(z\\)-axis. Let \\(O\\) be the origin. Find the volume of tetrahedron \\(OABC\\). The original answer is in the form \\(\... | 95 |
[
{
"content": "Rectangle \\(PQRS\\) lies in a plane with \\(PQ=RS=2\\) and \\(QR=SP=6\\). The rectangle is rotated \\(90^\\circ\\) clockwise about \\(R\\), then rotated \\(90^\\circ\\) clockwise about the point \\(S\\) moved to after the first rotation. If the length of the path traveled by point \\(P\\) is \\(x... | 13 |
[
{
"content": "A cylindrical tank with a radius of $4$ feet and a height of $9$ feet is lying on its side. The tank is filled with water to a depth of $2$ feet. Find the volume of water in cubic feet in the form $x\\pi - y\\sqrt{z}$, where $x$, $y$, and $z$ are integers. Please provide the value of $x + y + z$.\... | 87 |
[
{
"content": "In the expansion of\n\\[\\left(1 + x + x^2 + \\cdots + x^{27}\\right)\\left(1 + x + x^2 + \\cdots + x^{14}\\right)^2,\\]\nfind the coefficient of $x^{28}$.\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | 224 |
[
{
"content": "A function \\(f\\) is defined by \\(f(z) = (4 + i) z^2 + \\alpha z + \\gamma\\) for all complex numbers \\(z\\), where \\(\\alpha\\) and \\(\\gamma\\) are complex numbers and \\(i^2 = -1\\). Suppose that \\(f(1)\\) and \\(f(i)\\) are both real. The original answer is in the form \\(\\sqrt{m}\\). F... | 2 |
[
{
"content": "Trapezoid $ABCD$ has bases $\\overline{AB}$ and $\\overline{CD}$ and diagonals intersecting at $K.$ Suppose that $AB = 9$, $DC = 12$, and the area of $\\triangle AKD$ is $24.$ Find the area of trapezoid $ABCD$.\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role":... | 98 |
[
{
"content": "The faces of a cubical die are marked with the numbers $1$, $2$, $2$, $3$, $3$, and $4$. The faces of another die are marked with the numbers $1$, $3$, $4$, $5$, $6$, and $8$. Find the probability that the sum of the top two numbers will be $5$, $7$, or $9$. The original answer is in \\frac{k}{m} ... | 25 |
[
{
"content": "Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ... | 5 |
[
{
"content": "A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. Find the number of heavy-tailed permutations.\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | 48 |
[
{
"content": "Two circles of radius \\(1\\) are to be constructed as follows. The center of circle \\(A\\) is chosen uniformly and at random from the line segment joining \\((0,0)\\) and \\((2,0)\\). The center of circle \\(B\\) is chosen uniformly and at random, and independently of the first choice, from the ... | 14 |
[
{
"content": "Jacob uses the following procedure to write down a sequence of numbers. First, he chooses the first term to be \\(6\\). To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts \\(1\\). If it comes up tails, he takes half of the previ... | 13 |
[
{
"content": "Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color? Express your answer in the form \\(\\frac{k}{m}\\) where the fraction is in its simplest form, and find the value of \\(k + m\\).\n\nLet... | 7 |
[
{
"content": "A parking lot has $16$ spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires $2$ adjacent spaces. What is the probability that she is able to park? ... | 45 |
[
{
"content": "For how many positive integer values of $n$ are both $\\frac{n}{3}$ and $3n$ three-digit whole numbers?\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | 12 |
[
{
"content": "Two subsets of the set $S=\\lbrace a,b,c,d,e\\rbrace$ are to be chosen so that their union is $S$ and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter? Provide your answer as an integer.\n\n... | 40 |
[
{
"content": "The solutions of the equation $z^4+4z^3i-6z^2-4zi-i=0$ are the vertices of a convex polygon in the complex plane. What is the area of the polygon? Provide your answer in the form \\frac{k}{m}, and calculate the value of k + m.\n\nLet's think step by step and output the final answer within \\boxed{... | 9 |
[
{
"content": "In square \\(ABCE\\), \\(AF=2FE\\) and \\(CD=2DE\\). What is the ratio of the area of \\(\\triangle BFD\\) to the area of square \\(ABCE\\)? The answer is in the form \\(\\frac{k}{m}\\), please find the value of \\(k + m\\).\n\nLet's think step by step and output the final answer within \\boxed{}.... | 23 |
[
{
"content": "Quadrilateral $ABCD$ has $AB = BC = CD$, $m\\angle ABC = 70^\\circ$ and $m\\angle BCD = 170^\\circ$. If the answer is $x^\\circ$, what is the value of $x$ for the degree measure of $\\angle BAD$?\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | 85 |
[
{
"content": "Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square? The original answer is in the form of \\(\\frac{k}{m}\\) where the fraction is in simple... | 71 |
[
{
"content": "A sequence \\((a_1,b_1), (a_2,b_2), (a_3,b_3), \\ldots\\) of points in the coordinate plane satisfies\n\\((a_{n + 1}, b_{n + 1}) = (\\sqrt {3}a_n - b_n, \\sqrt {3}b_n + a_n)\\) for \\(n = 1,2,3,\\cdots\\).\nSuppose that \\((a_{100},b_{100}) = (2,4)\\). Find the value of \\(a_1 + b_1\\). The origin... | 100 |
[
{
"content": "Let $ABCD$ be a trapezoid with $AB||CD, AB=11, BC=5, CD=19,$ and $DA=7$. Bisectors of $\\angle A$ and $\\angle D$ meet at $P$, and bisectors of $\\angle B$ and $\\angle C$ meet at $Q$. If the area of hexagon $ABQCDP$ is $x\\sqrt{3}$, what is the value of $x$?\n\nLet's think step by step and output... | 30 |
[
{
"content": "What is the sum of the digits of the square of $111111111$? Provide your answer as a single integer.\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | 81 |
[
{
"content": "Kiana has two older twin brothers. The product of their three ages is $128$. What is the sum of their three ages?\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | 18 |
[
{
"content": "By inserting parentheses, it is possible to give the expression\n\\[2\\times3 + 4\\times5\\]\nseveral values. How many different values can be obtained?\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | 4 |
[
{
"content": "A carton contains milk that is \\(2\\%\\) fat, an amount that is \\(40\\%\\) less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk? Provide your answer as a fraction in the form \\(\\frac{k}{m}\\), where the fraction is in its simplest form. Plea... | 13 |
[
{
"content": "A flagpole is originally $5$ meters tall. A hurricane snaps the flagpole at a point $x$ meters above the ground so that the upper part, still attached to the stump, touches the ground $1$ meter away from the base. If the answer is $x$ meters, what is the value of $10x$?\n\nLet's think step by step... | 24 |
[
{
"content": "A particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$, it mistakenly displays a $9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time? The... | 3 |
[
{
"content": "On a checkerboard composed of \\(64\\) unit squares, find the probability that a randomly chosen unit square does not touch the outer edge of the board. Express your answer in the form \\(\\frac{k}{m}\\), where the fraction is in its simplest form. Please give the value of \\(k + m\\).\n\nLet's th... | 25 |
[
{
"content": "How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$?\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | 6 |
[
{
"content": "The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43$ dollars. Some of the $30$ sixth graders each bought a pencil, and they paid a total of $1.95$ dollars. How many more sixth graders than seventh grade... | 4 |
[
{
"content": "Distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=BC=CD=1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?\n\nLet's... | 3 |
[
{
"content": "On Monday, Millie puts a quart of seeds, $25\\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\\%$ of the millet in the feeder, but they eat all of the other ... | 5 |
[
{
"content": "For what value of $n$ is $i + 2i^2 + 3i^3 + \\cdots + ni^n = 48 + 49i$? Find the value of $n$.\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | 97 |
[
{
"content": "A recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\\frac{1}{4}$ cup sugar, $1$ cup water, and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredients, what is the g... | 39 |
[
{
"content": "Let $a$, $b$, $c$, and $d$ be real numbers with $|a-b|=2$, $|b-c|=3$, and $|c-d|=4$. Find the sum of all possible values of $|a-d|$.\n\nLet's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | 18 |
[
{
"content": "Points \\(A\\) and \\(C\\) lie on a circle centered at \\(O\\), each of \\(\\overline{BA}\\) and \\(\\overline{BC}\\) are tangent to the circle, and \\(\\triangle ABC\\) is equilateral. The circle intersects \\(\\overline{BO}\\) at \\(D\\). The original answer is in the form \\(\\frac{k}{m}\\). Pl... | 3 |
[
{
"content": "Trapezoid \\(ABCD\\) has \\(AD \\parallel BC\\), \\(BD = 1\\), \\(\\angle DBA = 23^\\circ\\), and \\(\\angle BDC = 46^\\circ\\). The ratio \\(BC: AD\\) is \\(9: 5\\). What is the value of \\(CD\\)? The original answer is in the form \\(\\frac{k}{m}\\), where \\(\\frac{k}{m}\\) is a fully simplifie... | 9 |
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