ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Let \( x \) be the number \( 0.\underbrace{0000\ldots0000}_{1996\text{ zeros}}1, \) where there are \( 1996 \) zeros after the decimal point. Find the largest value of the following expressions: \( 3+x, \ 3-x, \ 3\cdot x, \ \frac{3}{x}, \ \frac{x}{3} \). The original answer is in the form \(\frac{k}{x}\). Please provide the value of \(k\).
Answer: 3
Problem: In the number $74982.1035$, determine how many times greater the value of the place occupied by the digit $9$ is compared to the value of the place occupied by the digit $3$. Please provide the answer as an integer value representing the multiple.
Answer: 100000
Problem: All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A= \angle B = 90^\circ.$ If the answer is $x^\circ$, what is the degree measure of $\angle E?$
Answer: 150
Problem: All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A = \angle B = 90^{\circ}$. If the answer is $x^\circ$, what is the value of $x$ for the degree measure of $\angle E$?
Answer: 150
Problem: On the trip home from the meeting where this $AMC~10$ was constructed, the Contest Chair noted that his airport parking receipt had digits of the form $bbcac,$ where $0 \le a < b < c \le 9,$ and $b$ was the average of $a$ and $c.$ How many different five-digit numbers satisfy all these properties?
Answer: 20
Problem: In trapezoid $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$,
$AD = AB = 3$, and $DC = 6$. In addition, $E$ is on $\overline{DC}$, and $\overline{BE}$ is parallel to $\overline{AD}$. If the area of $\triangle BEC$ is $x$, find the value of $10x$.
Answer: 45
Problem: To complete the grid below, each of the digits $1$ through $4$ must occur once in each row and once in each column. Determine the number that will occupy the lower right-hand square.    
$$
\begin{array}{|c|c|c|c|}\hline1&&2&\\\hline2&3&&\\\hline&&&4\\\hline\\\hline\end{array}
$$
Answer: 2
Problem: A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms $247, 475$, and $756$ and end with the term $824$. Let $S$ be the sum of all the terms in the sequence. Find the largest prime factor that always divides $S$.
Answer: 37
Problem: Integers $a, b, c,$ and $d$, not necessarily distinct, are chosen independently and at random from $0$ to $2007$, inclusive. Find the probability that $ad-bc$ is even. The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 13
Problem: Two circles of radius $2$ are centered at $(2,0)$ and at $(0,2).$ What is the area of the intersection of the interiors of the two circles? Express your answer in the form k(\pi - m), and find the value of k + m.
Answer: 4
Problem: A traffic light runs repeatedly through the following cycle: green for \(30\) seconds, then yellow for \(3\) seconds, and then red for \(30\) seconds. Leah picks a random three-second time interval to watch the light. What is the probability, expressed as a fraction in simplest form \(\frac{k}{m}\), that the color changes while she is watching? Please provide the value of k + m.
Answer: 8
Problem: Four circles of radius \(1\) are each tangent to two sides of a square and externally tangent to a circle of radius \(2\), as shown. The area of the square is expressed in the form \(k + m\sqrt{2}\), where \(k\) and \(m\) are integers. Find the value of \(k + m\).
Answer: 34
Problem: The geometric series \(a+ar+ar^2+\ldots\) has a sum of \(7\), and the terms involving odd powers of \(r\) have a sum of \(3\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 7
Problem: Suppose that $m$ and $n$ are positive integers such that $75m = n^{3}$. What is the minimum possible value of $m + n$?
Answer: 60
Problem: The polynomial $f(x) = x^{4} + ax^{3} + bx^{2} + cx + d$ has real coefficients, and $f(2i) = f(2 + i) = 0.$ Find the value of $a + b + c + d$.
Answer: 9
Problem: A circle of radius $1$ is surrounded by $4$ circles of radius $r$ as shown. The original answer is in the form $k+\sqrt{m}$, please give the value of $k + m$.
Answer: 3
Problem: Let $a$, $b$, and $c$ be digits with $a\ne 0$. The three-digit integer $abc$ lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer $acb$ lies two thirds of the way between the same two squares. Find the value of $a+b+c$.
Answer: 16
Problem: The product of the two $99$-digit numbers    
$303,030,303,...,030,303$ and $505,050,505,...,050,505$    
has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$?
Answer: 8
Problem: The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by $4,$ and the second number is divided by $5.$ The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. Find the probability that the pair of numbers designates a shaded square. The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 3
Problem: Rhombus $ABCD$, with side length $6$, is rolled to form a cylinder of volume $6$ by taping $\overline{AB}$ to $\overline{DC}$. What is the value of $\sin(\angle ABC)$? Provide your answer in the form \(\frac{k\pi}{m}\), where \(k\) and \(m\) are integers. Calculate the sum of \(k + m\).
Answer: 10
Problem: A set of 25 square blocks is arranged into a \(5 \times 5\) square. Find the number of different combinations of 3 blocks that can be selected from that set so that no two are in the same row or column.
Answer: 600
Problem: Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? Express your answer in the form \(\frac{k-m\sqrt{2}}{n}\), where the fraction is in simplest form. Please find the value of \(k + m + n\).
Answer: 20
Problem: The parallelogram bounded by the lines $y=ax+c$, $y=ax+d$, $y=bx+c$, and $y=bx+d$ has area $18$. The parallelogram bounded by the lines $y=ax+c$, $y=ax-d$, $y=bx+c$, and $y=bx-d$ has area $72$. Given that $a$, $b$, $c$, and $d$ are positive integers, find the smallest possible value of $a+b+c+d$.
Answer: 16
Problem: The first $2007$ positive integers are each written in base $3$. How many of these base-$3$ representations are palindromes?
Answer: 100
Problem: Two cards are dealt from a deck of four red cards labeled $A$, $B$, $C$, $D$ and four green cards labeled $A$, $B$, $C$, $D$. A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 11
Problem: For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n.$ Find the number of values of $n$ such that $n + S(n) + S(S(n)) = 2007.$
Answer: 4
Problem: A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90^{\circ}$ right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?
Answer: 5
Problem: A pyramid with a square base is cut by a plane that is parallel to its base and $2$ units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. The original answer is in the form k + m\sqrt{2}. Please provide the value of k + m, where k and m are integers.
Answer: 6
Problem: Square \(ABCD\) has area \(36,\) and \(\overline{AB}\) is parallel to the \(x\)-axis. Vertices \(A,\) \(B\), and \(C\) are on the graphs of \(y = \log_{a}x,\) \(y = 2\log_{a}x,\) and \(y = 3\log_{a}x,\) respectively. The original answer is in the form \(\sqrt[k]{3}\). Please determine the value of \(k\).
Answer: 6
Problem: What is the area of the shaded pinwheel shown in the $5 \times 5$ grid? Provide your answer as an integer.
Answer: 6
Problem: Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base-$10$ representation consists of only $4$'s and $9$'s, with at least one of each. What are the last four digits of $n?$
Answer: 4944
Problem: How many pairs of positive integers $(a,b)$ are there such that ${gcd}(a,b)=1$ and $\frac{a}{b} + \frac{14b}{9a}$ is an integer?
Answer: 4
Problem: How many pairs of positive integers $(a,b)$ are there such that $a$ and $b$ have no common factors greater than $1$ and:  
$\frac{a}{b} + \frac{14b}{9a}$  
is an integer? Provide the number of such pairs.
Answer: 4
Problem: Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of $\{1,2,3,\cdots,12\},$ including the empty set, are spacy?
Answer: 129
Problem: A bakery owner turns on his doughnut machine at $8:30$ AM. At $11:10$ AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the entire job? Write your answer in the format "HH:MM AM/PM", and then provide the sum of HH and MM.
Answer: 34
Problem: A $4\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?

$\begin{array}{|c|c|c|c|}\hline1&2&3&4\\\hline8&9&10&11\\\hline15&16&17&18\\\hline22&23&24&25\\\hline\end{array}$

Find the positive difference between the two diagonal sums.
Answer: 4
Problem: A $4\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?
Answer: 4
Problem: The ten-letter code $BEST ~OF ~LUCK$ represents the ten digits $0-9$, in order. What $4$-digit number is represented by the code word $CLUE$?
Answer: 8671
Problem: On circle \(O\), points \(C\) and \(D\) are on the same side of diameter \(\overline{AB}\), \(\angle AOC = 30^\circ\), and \(\angle DOB = 45^\circ\). Find the ratio of the area of the smaller sector \(COD\) to the area of the circle. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 31
Problem: In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids?
Answer: 5
Problem: What is the volume of a cube whose surface area is twice that of a cube with volume $1$? Provide your answer in the form of $a\sqrt{b}$ where $a$ and $b$ are integers, and give the value of $a + b$.
Answer: 4
Problem: Older television screens have an aspect ratio of $4: 3$. That is, the ratio of the width to the height is $4: 3$. The aspect ratio of many movies is not $4: 3$, so they are sometimes shown on a television screen by "letterboxing" - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $2: 1$ and is shown on an older television screen with a $27$-inch diagonal. If the height of each darkened strip is $x$ inches, what is the value of $10x$?
Answer: 27
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: What is the area of the region defined by the inequality $|3x-18|+|2y+7|\le3$?
Answer: 3
Problem: 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
Problem: 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
Problem: 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
Problem: Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?
Answer: 6
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3n$ three-digit whole numbers?
Answer: 12
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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