prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Given the line $y = \frac{3}{4}x + 6$ and a line $L$ parallel to the given line and $4$ units from it. Find a possible equation for line $L$. The original answer is in the form $y = \frac{3}{4}x + c$, please give the value of c. | 1 |
What is the smallest possible area, in square centimeters, of a rectangle that can completely contain the shape formed by joining six squares of side length $8$ cm as shown below? \[ \begin{asy} size(5cm,0); draw((0,2)--(0,3)); draw((1,1)--(1,3)); draw((2,0)--(2,3)); draw((3,0)--(3,2)); draw((4,0)--(4,1)); draw((2,0)--... | 768 |
In unit square \(ABCD\), the inscribed circle \(\omega\) intersects \(\overline{CD}\) at \(M\), and \(\overline{AM}\) intersects \(\omega\) at a point \(P\) different from \(M\). Find the length of \(AP\). The original answer is in the form \(\frac{k\sqrt{m}}{n}\), where the fraction is in its simplest form. Please cal... | 16 |
If $w$ is one of the imaginary roots of the equation $x^3=1$, find the value of the product $(1-w+w^2)(1+w-w^2)$. | 4 |
A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds \(T_{max}\). One end of the cord is attached to a fixed point, the other is attached to an object of mass \(3m\). If a second, smaller object of mass \(m\) moving at an initial speed \(v_0\) strikes the larger mass and the tw... | 7 |
Yan is somewhere between his home and the stadium. To get to the stadium, he can either walk directly to the stadium or walk home first and then ride his bicycle to the stadium. He rides 7 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to h... | 7 |
The point $(-1, -2)$ is rotated $270^{\circ}$ counterclockwise about the point $(3, 1)$. What are the coordinates of its new position? If the new position is $(x, y)$, please give the coordinates in the form of $(a, b)$ and find the value of $a + b$. | 5 |
What is the sum of distinct remainders when $(2n-1)^{502}+(2n+1)^{502}+(2n+3)^{502}$ is divided by $2012$ where $n$ is a positive integer? | 1514 |
Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. Find the probability that the product is divisible by $4$. The original answer is in the form \(\frac{k}{m}\), where k and m are integers. Please calculate the value of k + m. | 123 |
My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, find the expected score of the losing team. The or... | 5 |
Real numbers \(a\) and \(b\) are chosen with \(1 < a < b\) such that no triangle with positive area has side lengths \(1, a, b\) or \(\frac{1}{b}, \frac{1}{a}, 1\). The original answer is in the form \(\frac{k+\sqrt{n}}{m}\). Find the value of \(k + m + n\). | 10 |
Two circles are externally tangent. Lines \(\overline{PAB}\) and \(\overline{PA'B'}\) are common tangents with \(A\) and \(A'\) on the smaller circle and \(B\) and \(B'\) on the larger circle. If \(PA = AB = 4\), then what is the value of \(x\) if the area of the smaller circle is \(x\pi\)? | 2 |
$AB$ is the diameter of a circle centered at $O$. $C$ is a point on the circle such that angle $\angle BOC$ is $60^\circ$. If the diameter of the circle is $5$ inches, find the length of chord $AC$, expressed in inches. The answer should be in the form \(\frac{k\sqrt{m}}{n}\), where \(k\), \(m\), and \(n\) are integers... | 10 |
Find the sum of the roots of $\tan^2 x - 9\tan x + 1 = 0$ that are between $x = 0$ and $x = 2\pi$ radians. If the answer is $k\pi$, what is the value of $k$? | 3 |
For how many integers $n$, with $2 \leq n \leq 80$, is $\frac{(n-1)n(n+1)}{8}$ equal to an integer? Provide the number of such integers. | 49 |
Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. Find the number of possible values of $k$. | 1 |
$A(-1,1)$, $B$, and $C$ are points on the hyperbola $x^2 - y^2 = 1$. If $\triangle ABC$ is a regular triangle, find the area of $\triangle ABC$. The original answer is in the form of k\sqrt{m}, please give the value of k + m. | 6 |
On a large piece of paper, Dana creates a "rectangular spiral" by drawing line segments of lengths, in cm, of 1, 1, 2, 2, 3, 3, 4, 4, ... as shown. Dana's pen runs out of ink after the total of all the lengths he has drawn is 3000 cm. What is the length of the longest line segment that Dana draws, in cm? | 54 |
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 m... | 27 |
How many pairs of positive integers $(x,y)$ are there such that $\sqrt{xy} - 71\sqrt{x} + 30 = 0$? | 8 |
Arithmetic sequences $(a_n)$ and $(b_n)$ have integer terms with $a_1 = b_1 = 1 < a_2 \le b_2$ and $a_nb_n = 2010$ for some $n$. What is the largest possible value of $n$? | 8 |
Let $d_1$ and $d_2$ be parallel lines in the plane. We are marking $11$ black points on $d_1$, and $16$ white points on $d_2$. We are drawing the segments connecting black points with white points. What is the maximum number of points of intersection of these segments that lies between the parallel lines (excluding the... | 6600 |
How many different integers can be expressed as the sum of three distinct members of the set $\{1, 4, 7, 10, 13, 16, 19\}$? | 13 |
Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniform... | 23 |
Let $f:(0,\infty) \rightarrow (0,\infty)$ be a function such that \[ 10\cdot \frac{x+y}{xy}=f(x)\cdot f(y)-f(xy)-90 \] for every $x,y \in (0,\infty)$. Find the value of $f(\frac{1}{11})$. | 21 |
Let $ABC$ be a triangle with $|AB|=18$, $|AC|=24$, and $m(\widehat{BAC}) = 150^\circ$. Let $D$, $E$, $F$ be points on sides $[AB]$, $[AC]$, $[BC]$, respectively, such that $|BD|=6$, $|CE|=8$, and $|CF|=2|BF|$. Let $H_1$, $H_2$, $H_3$ be the reflections of the orthocenter of triangle $ABC$ over the points $D$, $E$, $F$,... | 96 |
Let \(D(n)\) denote the number of ways of writing the positive integer \(n\) as a product \(n = f_1 \cdot f_2 \cdots f_k,\) where \(k \ge 1\), the \(f_i\) are integers strictly greater than 1, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factor... | 112 |
In triangle \(ABC\), \(\measuredangle CBA=72^\circ\), \(E\) is the midpoint of side \(AC\), and \(D\) is a point on side \(BC\) such that \(2BD=DC\); \(AD\) and \(BE\) intersect at \(F\). Find the ratio of the area of triangle \(BDF\) to the area of quadrilateral \(FDCE\). The original answer is in \(\frac{k}{m}\) form... | 6 |
A *triangular number* is a positive integer that can be expressed in the form $t_n = 1 + 2 + 3 + \cdots + n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1 = 1 = 1^2$, $t_8 = 36 = 6^2$, and $t_{49} = 1225 = 35^2$. Find the sum of the digits of the fourth sma... | 18 |
Triangles $ABC$ and $ADE$ have areas $2007$ and $7002$, respectively, with $B = (0,0)$, $C = (223,0)$, $D = (680,380)$, and $E = (689,389)$. Find the sum of all possible x-coordinates of $A$. | 1200 |
Find the sum of distinct residues of the number $2012^n + m^2 \pmod{11}$ where $m$ and $n$ are positive integers. | 39 |
Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are $1+2i$, $-2+i$, and $-1-2i$. Find the fourth number. The original answer is in the form x + yi, where x and y are integers. Please give the value of x + y. | 1 |
Toothpicks of equal length are used to build a rectangular grid. If the grid is 20 toothpicks high and 10 toothpicks wide, find the total number of toothpicks used. | 430 |
Let $c = \frac{2\pi}{11}.$ What is the value of \[ \frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}? \] | 1 |
Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|AB| + |BD| = |AC|$ and $m(\widehat{BAD}) = m(\widehat{DAC}) = 30^\circ$. If the answer is $x^\circ$, what is the value of $x$ for $m(\widehat{ACB})$? Please provide the value of $x$ as an integer. | 40 |
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only $4$ years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha's age and the mean of my grandparents’ ages? | 2 |
The sides of a triangle have lengths $11$, $15$, and $k$, where $k$ is an integer. Find the number of integer values of $k$ for which the triangle is obtuse. | 13 |
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex \(W\)? The original answer is in \(\frac{k}{m}\) fo... | 13 |
Given $a_1 = 1$, $a_{n+1} = \frac{a_n}{\sqrt{1 + 4a_n^2}}$ for $n \ge 1$, find the least $k$ such that $a_k < 10^{-2}$. | 2501 |
How many solutions does the equation $\sin \left( \frac{\pi}{2} \cos x\right)=\cos \left( \frac{\pi}{2} \sin x\right)$ have in the closed interval $[0,\pi]$? Provide your answer as an integer. | 2 |
How many different solutions does the congruence $x^3 + 3x^2 + x + 3 \equiv 0 \pmod{25}$ have? Provide your answer as an integer. | 6 |
Four positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to the fourth integer. Thus, the numbers $29$, $23$, $21$, and $17$ are obtained. Find one of the original integers. | 21 |
The minimum value of the quotient of a (base ten) number of three different nonzero digits divided by the sum of its digits is x. If the answer is x, what is the value of 10x? | 105 |
The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments. \[\text{[asy]}\] \text{size(300);} \text{defaultpen(linewidth(0.8));} \text{real r = 0.35;} \text{path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r);} \text{path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-... | 2 |
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $8:00 \text{ AM}$, and all three always take the same amount of time to eat lunch. On Monday, the three of them painted $50\%$ of a house, quitting at $4:00 \text{ PM}$. On Tuesday, when Paula wasn't there, the two ... | 48 |
The largest prime factor of $16384$ is $2$, because $16384 = 2^{14}$. What is the sum of the digits of the largest prime factor of $16383$? | 10 |
Given the linear fractional transformation of \(x\) into \(f_1(x) = \frac{2x-1}{x+1}\). Define \(f_{n+1}(x) = f_1(f_n(x))\) for \(n=1,2,3,\ldots\). Assuming that \(f_{35}(x) = f_5(x)\), find the expression for \(f_{28}(x)\). The original answer is in the form \(\frac{k}{m-nx}\). Please find the value of k + m + n. | 3 |
The expression \((x + y + z)^{2006} + (x - y - z)^{2006}\) is simplified by expanding it and combining like terms. How many terms are in the simplified expression? Please provide your answer as an integer. | 1008016 |
For a geometric series $(a_n)$ with all terms real, if $S_{10} = 10$ and $S_{30} = 70$, find the value of $S_{40}$. | 150 |
The base-nine representation of the number $N$ is $27,006,000,052_{\text{nine}}$. Find the remainder when $N$ is divided by $5$. | 3 |
A simplified model of a bicycle of mass \(M\) has two tires that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is \(w\), and the center of mass of the bicycle is located midway between the tires and a height \(h\) above the... | 2 |
A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds \(T_{max}\). One end of the cord is attached to a fixed point, the other is attached to an object of mass \(3m\). If a second, smaller object of mass \(m\) moving at an initial speed \(v_0\) strikes the larger mass and the tw... | 7 |
Let the bisectors of the exterior angles at \(B\) and \(C\) of triangle \(ABC\) meet at \(D\). Then, if all measurements are in degrees, find the value of angle \(BDC\). The original answer is in the form \(\frac{k}{m}(n - A)\). Please provide the value of k + m + n. | 183 |
The parabola $P$ has its focus at $(0,0)$ and passes through the points $(4,3)$ and $(-4,-3)$. Determine the number of points $(x,y) \in P$ with integer coefficients such that $|4x + 3y| \leq 1000$. | 40 |
At most, how many elements can a set have such that all elements are less than $102$ and the set does not contain the sum of any two of its elements? Provide your answer as an integer. | 51 |
In a bag of marbles, \(\frac{3}{5}\) of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? The original answer is in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please provide t... | 11 |
Find the number of noncongruent integer-sided triangles with positive area and perimeter less than $15$ that are neither equilateral, isosceles, nor right triangles. | 5 |
For what value of $k$ does the pair of equations $y = x^2$ and $y = 3x + k$ have two identical solutions? Provide your answer as a fraction in the form $-\frac{m}{n}$, where m and n are integers. Please find the value of m + n. | 13 |
Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other. | 20 |
The number $2013$ has the property that it includes four consecutive digits ($0$, $1$, $2$, and $3$). Find how many $4$-digit numbers include $4$ consecutive digits. | 162 |
If the sum of digits of only $m$ and $m+n$ from the numbers $m$, $m+1$, \ldots, $m+n$ are divisible by $8$ where $m$ and $n$ are positive integers, what is the largest possible value of $n$? | 15 |
In triangle $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the area of $\triangle ABC$ is $96$, then what is the area of $\triangle AEF$? | 36 |
Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt{2}$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt{5}$ and $\angle AEC = 45^{\circ}$. Find the value of $CE^2+DE^2$. | 100 |
Find the product of uncommon real roots of the two polynomials $x^4 + 2x^3 - 8x^2 - 6x + 15$ and $x^3 + 4x^2 - x - 10$. | 6 |
Find the sum of the $2$ smallest prime factors of $2^{1024} - 1$. | 8 |
What is the least positive integer $k$ such that $n+k \in S$ for every $n \in S$, where $S = \{ n : n3^n + (2n+1)5^n \equiv 0 \pmod{7} \}$? | 21 |
If $x_1=5, x_2=401$, and \[ x_n=x_{n-2}-\frac{1}{x_{n-1}} \] for every $3\leq n \leq m$, find the largest value of $m$. | 2007 |
A sequence of complex numbers $z_0, z_1, z_2, \ldots$ is defined by the rule \[ z_{n+1} = \frac{i z_n}{\overline{z_n}} \] where $\overline{z_n}$ is the complex conjugate of $z_n$ and $i^2 = -1$. Suppose that $|z_0| = 1$ and $z_{2005} = 1$. Find the number of possible values for $z_0$. The original answer is in the form... | 2006 |
Let $N$ be the second smallest positive integer that is divisible by every positive integer less than 7. What is the sum of the digits of $N$? | 3 |
A painting $18'' \times 24''$ is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, find the ratio of the smaller to the larger dimension of the framed painting in the form $... | 5 |
A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are \((3,17)\) and \((48,281)\)? Include both endpoints of the segment in your count. | 4 |
Two men at points $R$ and $S$, 76 miles apart, set out at the same time to walk towards each other. The man at $R$ walks uniformly at the rate of $\frac{9}{2}$ miles per hour; the man at $S$ walks at the constant rate of $\frac{13}{4}$ miles per hour for the first hour, $\frac{15}{4}$ miles per hour for the second hour... | 4 |
Calculate the product \(\frac{1}{3} \times \frac{2}{4} \times \frac{3}{5} \times \cdots \times \frac{18}{20} \times \frac{19}{21}\). What is the value of this product in its simplest form? The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 211 |
The parallelogram bounded by the lines $y = ax + c$, $y = ax + d$, $y = bx + c$, and $y = bx + d$ has an area of 18. The parallelogram bounded by the lines $y = ax + c$, $y = ax - d$, $y = bx + c$, and $y = bx - d$ has an area of 72. Given that $a$, $b$, $c$, and $d$ are positive integers, find the smallest possible va... | 16 |
Let $D$ be a point on side $[AB]$ of triangle $ABC$ with $|AB| = |AC|$, such that $[CD]$ is an angle bisector and $m(\widehat{ABC}) = 40^\circ$. Let $F$ be a point on the extension of $[AB]$ beyond $B$ such that $|BC| = |AF|$. Let $E$ be the midpoint of $[CF]$. If $G$ is the intersection of lines $ED$ and $AC$, what is... | 150 |
Find the number of prime numbers $p$ such that $2p^4 - 7p^2 + 1$ is equal to the square of an integer. | 1 |
If the perimeter of a rectangle is \(p\) and its diagonal is \(d\), find the difference between the length and width of the rectangle. The original answer is in the form \(\frac{\sqrt{n \cdot d^2 - p^2}}{m}\). Please calculate the value of \(k + m + n\). | 11 |
A list of integers has mode $32$ and mean $22$. The smallest number in the list is $10$. The median $m$ of the list is a member of the list. If the list member $m$ were replaced by $m + 10$, the mean and median of the new list would be $24$ and $m + 10$, respectively. If $m$ were instead replaced by $m - 8$, the median... | 20 |
Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\overline{DE} \perp \overline{BC}$. Find the ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 4 |
How many functions $f : A \rightarrow A$ are there satisfying $f(f(a)) = a$ for every $a \in A=\{1,2,3,4,5,6,7\}$? | 232 |
The two-digit integers from $19$ to $92$ are written consecutively to form the larger integer $N = 19202122\ldots909192$. If $3^{k}$ is the highest power of $3$ that is a factor of $N$, what is the value of $k$? | 1 |
In order to pass $B$ going $40$ mph on a two-lane highway, $A$, going $50$ mph, must gain $30$ feet. Meanwhile, $C$, who is $210$ feet from $A$, is headed toward him at $50$ mph. If $B$ and $C$ maintain their speeds, then, in order to pass safely, $A$ must increase his speed by how many mph? | 5 |
The probability that event $A$ occurs is \(\frac{3}{4}\); the probability that event $B$ occurs is \(\frac{2}{3}\). Let \(p\) be the probability that both \(A\) and \(B\) occur. Find the smallest interval necessarily containing \(p\) and express it in the form \(\left[\frac{x}{y}, \frac{c}{d}\right]\), where \(x, y, c,... | 22 |
A truck is initially moving at velocity \(v\). The driver presses the brake to slow the truck to a stop. The brake applies a constant force \(F\) to the truck. The truck rolls a distance \(x\) before coming to a stop, and the time it takes to stop is \(t\). Find the expression that represents the initial momentum of th... | 3 |
How many ways are there to partition $7$ students into the groups of $2$ or $3$? Provide your answer as an integer. | 105 |
Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. Find the probability that the four chords are the sides of a convex quadrilateral. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\). | 92 |
Asli will distribute $100$ candies among her brother and $18$ friends of his. Asli splits the friends of her brother into several groups and distributes all the candies into these groups. In each group, the candies are shared in a fair way such that each child in a group takes the same number of candies, and this numbe... | 16 |
What is the largest real number $C$ that satisfies the inequality $x^2 \geq C \lfloor x \rfloor (x-\lfloor x \rfloor)$ for every real $x$? | 4 |
The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is called the power of the triple. Find the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$. | 7290 |
Three mutually tangent spheres of radius \(1\) rest on a horizontal plane. A sphere of radius \(2\) rests on them. What is the distance from the plane to the top of the larger sphere? Express your answer in the form \(3 + \frac{\sqrt{n}}{m}\) where \(n\) and \(m\) are integers. Please give the value of \(n + m\). | 72 |
Find the integer that divides $3^{3n+1} + 5^{3n+2} + 7^{3n+3}$ for every positive integer $n$. | 7 |
Find the radius of the inscribed circle of a triangle with sides of length \(13\), \(30\), and \(37\). Provide your answer in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form. Calculate the value of \(k + m\). | 11 |
In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. What is the common ratio? Provide your answer in the form of \(\frac{\sqrt{k} - m}{n}\) and find the value of \(k + m + n\). | 8 |
Call a number "prime-looking" if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000? Provide your answer as an integer. | 100 |
Calculate the value of \(\cot 10 + \tan 5\) and express your answer in terms of a trigonometric function. If the answer is \(\csc x\), what is the value of \(x\)? | 10 |
A positive number is mistakenly divided by $6$ instead of being multiplied by $6$. Calculate the error committed, and provide the answer to the nearest percent. | 97 |
Find the sum of all prime numbers between $1$ and $100$ that are simultaneously $1$ greater than a multiple of $4$ and $1$ less than a multiple of $5$. What is the sum? | 118 |
Let $P$ equal the product of $3,659,893,456,789,325,678$ and $342,973,489,379,256$. Find the number of digits in $P$. | 34 |
Find the number of real values of $x$ satisfying the equation $2^{2x^2 - 7x + 5} = 1$. | 2 |
The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a 9-digit number $M$, all of whose digits are distinct. Find the digit that is not present in the number $M$. | 0 |
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