meetforever/GRPO_CL huggingface dataset:
user: Points \(A\) and \(B\) are on a circle of radius \(5\) and \(AB = 6\). Point \(C\) is the midpoint of the minor arc \(AB\). What is the length of the line segment \(AC\)? The answer is in the form \(\sqrt{k}\). Please find the value of \(k\).
Answer: 10
user: Each of the sides of a square $S_1$ with area $16$ is bisected, and a smaller square $S_2$ is constructed using the bisection points as vertices. The same process is carried out on $S_2$ to construct an even smaller square $S_3$. What is the area of $S_3$?
Answer: 4
user: Points \(A\) and \(B\) are on a circle of radius \(5\) and \(AB=6\). Point \(C\) is the midpoint of the minor arc \(AB\). Find the length of the line segment \(AC\). The answer is in the form \(\sqrt{m}\), please give the value of m.
Answer: 10
user: A cone-shaped mountain has its base on the ocean floor and has a height of $8000$ feet. The top $\frac{1}{8}$ of the volume of the mountain is above water. What is the depth of the ocean at the base of the mountain in feet?
Answer: 4000
user: A ball is dropped from a height of $3$ meters. On its first bounce, it rises to a height of $2$ meters. It keeps falling and bouncing to $\frac{2}{3}$ of the height it reached in the previous bounce. On which bounce will it not rise to a height of $0.5$ meters? Provide your answer as an integer.
Answer: 5
user: Points \(A\) and \(B\) lie on a circle centered at \(O\), and \(\angle AOB = 60^\circ\). A second circle is internally tangent to the first and tangent to both \(\overline{OA}\) and \(\overline{OB}\). The original answer for the ratio of the area of the smaller circle to that of the larger circle is in \(\frac{k}{m}\) format, where k and m are integers. Please find the value of k + m.
Answer: 10
user: Vertex \(E\) of equilateral \(\triangle{ABE}\) is in the interior of unit square \(ABCD\). Let \(R\) be the region consisting of all points inside \(ABCD\) and outside \(\triangle{ABE}\) whose distance from \(AD\) is between \(\frac{1}{3}\) and \(\frac{2}{3}\). The area of \(R\) is given in the form \(\frac{k-m\sqrt{n}}{p}\), where \(n\) is a positive integer. Find the value of \(k + m + n + p\).
Answer: 56
user: What is the area of the region defined by the inequality $|3x-18|+|2y+7|\le3$?
Answer: 3
user: Triangle $OAB$ has $O=(0,0)$, $B=(5,0)$, and $A$ in the first quadrant. In addition, $\angle ABO=90^\circ$ and $\angle AOB=30^\circ$. Suppose that $OA$ is rotated $90^\circ$ counterclockwise about $O$. The coordinates of the image of $A$ are expressed in the form \((- \frac{k}{3}\sqrt{m}, n)\). Please find the value of \(k + m + n\).
Answer: 13
user: A circle has a radius of \(\log_{10}{(a^2)}\) and a circumference of \(\log_{10}{(b^4)}\). Find the value of \(\log_{a}{b}\). The original answer is in the form of \(k\pi\), please provide the value of \(k\).
Answer: 1
user: How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$, where $b<100$? Provide your answer as an integer.
Answer: 6
user: Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?
Answer: 6
user: 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 average for the $10$ games was also an integer. Find the product of the number of points she scored in the ninth and tenth games.
Answer: 40
user: 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 rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $(a,b)$?
Answer: 2
user: 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 \(k + m\).
Answer: 11
user: 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 $n$?
Answer: 112
user: A shape is created by joining seven unit cubes, as shown. Find the ratio of the volume in cubic units to the surface area in square units. Express your answer as a ratio in the form \(a : b\). Please provide the sum of a and b.
Answer: 37
user: 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 the value of \(k + m\).
Answer: 63
user: 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 $a_4$.
Answer: 502
user: Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\triangle ABC$ is a right triangle with area $2008$. What is the sum of the digits of the $y$-coordinate of $C$?
Answer: 18
user: 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 \(\sqrt{m}\). Please provide the value of \(m\).
Answer: 95
user: Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run? Express your answer in the form of $a\pi + b$, where $a$ and $b$ are integers. Please provide the value of a + b.
Answer: 60
user: 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 \(x\pi\), the original answer is in the form of \(k+\sqrt{m}\). Please find the value of \(k+m\).
Answer: 13
user: 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 $x + y + z$.
Answer: 87
user: In the expansion of
\[\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2,\]
find the coefficient of $x^{28}$.
Answer: 224
user: 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 \(\sqrt{m}\). Find the smallest possible value of \(m\).
Answer: 2
user: Eight points are spaced at intervals of one unit around a \(2 \times 2\) square, as shown. Two of the \(8\) points are chosen at random. What is the probability that the two points are one unit apart? The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 9
user: 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 $ABCD$.
Answer: 98
user: 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} format, please give the value of k + m.
Answer: 25
user: 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 ahead of him just leaving the next pail. How many times will Michael and the truck meet?
Answer: 5
user: 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.
Answer: 48
user: 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 line segment joining \((0,1)\) to \((2,1)\). Find the probability that circles \(A\) and \(B\) intersect. The original answer is in the form \(\frac{k \sqrt{m} - n}{p}\). Please find the value of \(k + m + n + p\).
Answer: 14
user: 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 previous term and subtracts \(1\). What is the probability that the fourth term in Jacob's sequence is an integer? The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 13
user: 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 \(k + m\).
Answer: 7
user: A round table has radius \(4\). Six rectangular place mats are placed on the table. Each place mat has width \(1\) and length \(x\) as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length \(x\). Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. Find the value of \(x\). The original answer is in the form \(\frac{k\sqrt{n}-\sqrt{p}}{m}\). Please give the value of \(k+m+n+p\).
Answer: 15
user: 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? The original answer is in the form of a simplified fraction \frac{k}{m}, please give the value of k + m.
Answer: 45
user: For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3n$ three-digit whole numbers?
Answer: 12
user: 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.
Answer: 40
user: 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.
Answer: 9
user: 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 \(k + m\).
Answer: 23
user: 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 $\angle BAD$?
Answer: 85
user: Let $A_0=(0,0)$. Distinct points $A_1,A_2,\cdots$ lie on the $x$-axis, and distinct points $B_1,B_2,\cdots$ lie on the graph of $y=\sqrt{x}$. For every positive integer $n$, $A_{n-1}B_nA_n$ is an equilateral triangle. Find the least $n$ for which the length $A_0A_n\geq100$.
Answer: 17
user: 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 simplest form. Please find the value of k + m.
Answer: 71
user: A sequence \((a_1,b_1), (a_2,b_2), (a_3,b_3), \ldots\) of points in the coordinate plane satisfies
\((a_{n + 1}, b_{n + 1}) = (\sqrt {3}a_n - b_n, \sqrt {3}b_n + a_n)\) for \(n = 1,2,3,\cdots\).
Suppose that \((a_{100},b_{100}) = (2,4)\). Find the value of \(a_1 + b_1\). The original answer is in the form \(\frac{x}{y \cdot 2^{z}}\), where y is not divisible by 2. Please provide the value of \(x + y + z\).
Answer: 100
user: 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 $x$?
Answer: 30
user: The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden? Provide your answer as an integer.
Answer: 6
user: A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths \(15\) and \(25\) meters. What fraction of the yard is occupied by the flower beds? The answer should be in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is in its simplest form. Please find the value of \(k + m\).
Answer: 6
user: A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds? The original answer is in the form \frac{k}{m}, where k and m are integers, and the fraction is in its simplest form. Please give the value of k + m.
Answer: 6
user: What is the sum of the digits of the square of $111111111$? Provide your answer as a single integer.
Answer: 81
user: Kiana has two older twin brothers. The product of their three ages is $128$. What is the sum of their three ages?
Answer: 18
user: By inserting parentheses, it is possible to give the expression
\[2\times3 + 4\times5\]
several values. How many different values can be obtained?
Answer: 4
user: 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. Please give the value of k + m.
Answer: 13
user: Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
Answer: 3
user: Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac{5}{2} \angle B$. What is the value of $\angle D$ in degrees? If the answer is $x^\circ$, what is the value of $10x$?
Answer: 525
user: Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have? Fill in the blank with the number of sides.
Answer: 23
user: 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 $10x$?
Answer: 24
user: 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 original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 3
user: 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 \(k + m\).
Answer: 25
user: How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$?
Answer: 6
user: The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For example, figure $F_3$ has $13$ diamonds. How many diamonds are there in figure $F_{20}$?
Answer: 761
user: 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 graders bought a pencil?
Answer: 4
user: 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?
Answer: 3
user: 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 seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet? If Monday is day 1, what is the number of the day?
Answer: 5
user: Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. The original answer is in $\frac{k}{m}$ format, where $\frac{k}{m}$ is in simplest form. Please find the value of $k + m$.
Answer: 5
user: For what value of $n$ is $i + 2i^2 + 3i^3 + \cdots + ni^n = 48 + 49i$? Find the value of $n$.
Answer: 97
user: 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 greatest number of servings of hot chocolate he can make? Express your answer in the form of an improper fraction \frac{k}{m}, and find the value of k + m.
Answer: 39
user: 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 $|a-d|$.
Answer: 18
user: 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 \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 3
user: 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 simplified fraction. Please provide the value of \(k + m\).
Answer: 9