problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
What is the smallest prime whose digits sum to $23$?
|
599
| 0.083333 |
Milton's equation on his homework paper got a coffee stain, making one of the coefficients unreadable. He remembers that the quadratic had two distinct negative, integer solutions. If the constant term of the quadratic is 48 instead of 36, what is the sum of all distinct possible integers that could be under the coffee stain?
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,brown);
label("$x + 48 = 0$", (3,1.5),E);
label("$x^2 +$", (0,1.5),W);
[/asy]
|
124
| 0.666667 |
In a specific sequence, the first term is $a_1 = 2010$ and the second term is $a_2 = 2011$. The values of the remaining terms are chosen so that
\[a_n + a_{n + 1} + a_{n + 2} = 2n\]
for all $n \geq 1$. Determine $a_{1000}$.
|
2676
| 0.333333 |
Let \(a\) and \(b\) be positive real numbers such that \(a + b = 3\). Find the set of all possible values of \(\frac{1}{a} + \frac{1}{b}\).
|
\left[\frac{4}{3}, \infty\right)
| 0.916667 |
Let $O$ be the origin. There exists a scalar $k$ so that for any points $A,$ $B,$ $C,$ and $D$ on the surface of a sphere centered at $O$ with radius $r$, and satisfying
\[4 \overrightarrow{OA} - 3 \overrightarrow{OB} + 6 \overrightarrow{OC} + k \overrightarrow{OD} = \mathbf{0},\]
the four points $A,$ $B,$ $C,$ and $D$ must lie on a common plane. Determine the value of $k$.
|
-7
| 0.916667 |
In the equation $\frac{1}{j} + \frac{1}{k} = \frac{1}{4}$, both $j$ and $k$ are positive integers. What is the sum of all possible values for $k$?
|
51
| 0.916667 |
Thirty gremlins and twenty imps are at the Interregional Mischief Convention. This year, five of the imps, due to a magical disagreement, refuse to shake hands with ten specific gremlins. The rest of the imps have no issues shaking hands with all gremlins. All gremlins are still friendly and shake hands with each other and with any imp willing to shake hands with them. How many handshakes were at the convention?
|
985
| 0.583333 |
Suppose we want to divide 12 rabbits into three groups, one with 4 rabbits, one with 6 rabbits, and one with 2 rabbits. How many ways can we form the groups such that BunBun is in the 4-rabbit group and Thumper is in the 6-rabbit group?
|
2520
| 0.833333 |
Let \( p, q, r, \) and \( s \) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2&=&r^2+s^2&=&2512, \\
pr&=&qs&=&1225.
\end{array}
\]
If \( T = p+q+r+s \), compute the value of \( \lfloor T \rfloor \).
|
140
| 0.583333 |
If we let $f(n)$ denote the sum of all the positive divisors of the integer $n$, how many integers $i$ exist such that $1 \le i \le 2500$ and $f(i) = 1 + \sqrt{i} + i$?
|
15
| 0.916667 |
Find the matrix $\mathbf{Q}$ such that for any vector $\mathbf{u},$ $\mathbf{Q} \mathbf{u}$ is the projection of $\mathbf{u}$ onto the vector $\begin{pmatrix} 1 \\ 1 \\ -2 \end{pmatrix}.$
|
\begin{pmatrix} \frac{1}{6} & \frac{1}{6} & -\frac{1}{3} \\ \frac{1}{6} & \frac{1}{6} & -\frac{1}{3} \\ -\frac{1}{3} & -\frac{1}{3} & \frac{2}{3} \end{pmatrix}
| 0.583333 |
What is the area, in square units, of a triangle with sides of $5, 5,$ and $6$ units? Express your answer in simplest radical form.
|
12
| 0.916667 |
How many positive 3-digit numbers are multiples of 25, but not of 75?
|
24
| 0.333333 |
A point $(x,y)$ is randomly selected such that $0 \leq x \leq 4$ and $0 \leq y \leq 8$. What is the probability that $x+y \leq 5$? Express your answer as a common fraction.
|
\frac{3}{8}
| 0.083333 |
Find the value of $x$ where $(2015 + x)^2 = x^2$.
|
-\frac{2015}{2}
| 0.833333 |
Given that 15 is the arithmetic mean of the set $\{8, 12, 23, 17, y\}$, what is the value of $y$?
|
15
| 0.916667 |
Determine the tens digit of $13^{2021}$.
|
1
| 0.583333 |
How many lattice points lie on the graph of the equation $x^2 - y^2 = 45$?
|
12
| 0.666667 |
Consider the geometric sequence starting with $5$, and each subsequent term is multiplied by $5/3$. Calculate the tenth term of this sequence and express your answer as a common fraction.
|
\frac{9765625}{19683}
| 0.833333 |
Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4\}$. How many such polynomials satisfy $P(-1) = 1$?
|
80
| 0.083333 |
Let $a$ and $b$ be positive real numbers such that $a > b$. Consider the following five statements:
1. $\frac{1}{a} > \frac{1}{b}$
2. $a^2 > b^2$
3. $a > b$
4. $|a| > 1$
5. $b < 1$
What is the maximum number of these statements that can be true for any values of $a$ and $b$?
|
4
| 0.583333 |
What is the smallest prime whose digits sum to $23$?
|
599
| 0.333333 |
Determine the degree measure of the smallest angle in a convex 15-sided polygon if the degree measures of the angles form an increasing arithmetic sequence with integer values.
|
135^\circ
| 0.833333 |
Find the smallest positive integer $x$ that satisfies the congruence $52x + 14 \equiv 6 \pmod {24}$.
|
4
| 0.666667 |
Place each of the digits 1, 2, 3 and 4 in exactly one square to make the smallest possible sum. Use the following structure: two digits form one number on the left, and the other two digits form another number on the right. The sum of these two numbers should be minimal. [Diagram explanation: two 2-digit numbers side by side]
|
37
| 0.916667 |
The entry fee for a theme park is $20 per adult and $15 per child. Recently, the park collected exactly $1600 in entry fees from both adults and children. Determine the ratio of adults to children at the theme park that is closest to $2$.
|
\frac{59}{28}
| 0.5 |
Calculate the result of $\begin{pmatrix} 3 \\ -2 \end{pmatrix} - 5 \begin{pmatrix} 2 \\ -3 \end{pmatrix} + 2 \begin{pmatrix} -1 \\ 4 \end{pmatrix}.$
|
\begin{pmatrix} -9 \\ 21 \end{pmatrix}
| 0.916667 |
Rationalize the denominator of $\frac{3}{4\sqrt{7} + 3\sqrt{13}}$ and write your answer in the form $\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$?
|
22
| 0.416667 |
Consider the matrix $\mathbf{B} = \begin{pmatrix} 1 & 4 \\ 6 & d \end{pmatrix}$ such that $\mathbf{B}^{-1} = p \mathbf{B}$ for some constant $p$. Additionally, suppose $d \neq 3$. Determine the ordered pair $(d, p)$.
|
(-1, \frac{1}{25})
| 0.833333 |
If $a$, $b$, and $c$ are positive integers such that $\gcd(a,b) = 960$ and $\gcd(a,c) = 324$, determine the smallest possible value of $\gcd(b,c)$.
|
12
| 0.666667 |
Real numbers $a$ and $b$ satisfy the equation $a^2 + b^2 = 12a - 4b + 20$. What is $a-b$?
|
8
| 0.583333 |
A regular pentagon and a square share a common vertex. Let the shared vertex be point $A$, and the adjacent vertices of the pentagon be $B$ and $E$. The square extends outward from $A$ along the sides $AB$ and $AE$. Calculate the degree measure of $\angle BAE$.
|
108^\circ
| 0.083333 |
In right triangle \( GHI \), we have \( \angle G = 40^\circ \), \( \angle H = 90^\circ \), and \( IH = 12 \). Find \( GH \) to the nearest tenth. You may use a calculator for this problem.
|
14.3
| 0.583333 |
Let $g(x)$ satisfy the equation $g(x - y) = g(x) g(y)$ for all real numbers $x$ and $y$. Also, assume $g(x) \neq 0$ for all real numbers $x$. Determine the value of $g(5)$.
|
g(5) = 1
| 0.916667 |
Compute $i^{-200} + i^{-199} + i^{-198} + \cdots + i^{-1} + i^0 + i^1 + \cdots + i^{199} + i^{200}$.
|
1
| 0.666667 |
What is the sum of the non-zero digits of the base $8$ representation of $999_{10}$?
|
19
| 0.75 |
Suppose that $b$ is a positive integer greater than or equal to $2.$ When $315$ is converted to base $b$, the resulting representation has $5$ digits. What is the number of possible values for $b$?
|
1
| 0.833333 |
Let's consider a seven-digit number $C\,985\,F\,72$, where $C$ and $F$ are unknown digits. If this number is divisible by $9$, find the sum of all possible values of $C+F$.
|
19
| 0.166667 |
Each of the seven letters in "ALGEBRA" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "ANGLE"? Express your answer as a common fraction.
|
\frac{5}{7}
| 0.25 |
Triangle $PQR$ has side lengths $PQ=7$, $QR=8$, and $PR=9$. Two ants start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is $QS$?
|
5
| 0.25 |
Find the coefficient of the $x^2$ term in the expansion of the product $(2ax^3 + 5x^2 - 3x)(3bx^2 - 8x - 5)$.
|
-1
| 0.75 |
A right pyramid has a square base, each side measuring 8 cm. Its peak is positioned 15 cm above the center of the square base. Calculate the sum of the lengths of the pyramid's eight edges, rounded to the nearest tenth.
|
96.1 \text{ cm}
| 0.916667 |
Determine how many base-2 digits (bits) are present when the base-16 number $A987B_{16}$ is written in base 2.
|
20
| 0.916667 |
John has 8 green marbles and 7 purple marbles. He chooses a marble at random, records its color, and then does not put the marble back. He repeats this process 6 times. What is the probability that he chooses exactly three green marbles?
|
\frac{392}{1001}
| 0.666667 |
Find the number of cubic centimeters in the volume of the cylinder formed by rotating a square with side length 10 centimeters about its vertical line of symmetry. Assume the height of the cylinder is now 20 centimeters. Express your answer in terms of $\pi$.
|
500\pi
| 0.833333 |
Let $\star (x)$ be the sum of the digits of a positive integer $x$. Define $\mathcal{S}$ as the set of positive integers such that for all elements $n$ in $\mathcal{S}$, $\star (n)=9$ and $0\le n< 10^{5}$. Compute $\star(m)$ where $m$ is the number of elements in $\mathcal{S}$.
|
13
| 0.916667 |
Evaluate the expression \[(5^{1001} + 6^{1002})^2 - (5^{1001} - 6^{1002})^2\] and express it in the form \(k \cdot 30^{1001}\) for some integer \(k\).
|
24
| 0.083333 |
What is the base 4 representation of the base 2 number $10111100_2$?
|
2330_4
| 0.75 |
At constant temperature, the pressure of a sample of gas is inversely proportional to its volume. A sample of gas is initially in a 3.5-liter container at a pressure of 8 kPa. If all of the gas is transferred to a 10.5-liter container at the same temperature, what will the new pressure be?
|
\frac{8}{3} \text{ kPa}
| 0.5 |
A stock investment increased by $15\%$ in the first year. At the start of the next year, by what percent must the stock now decrease to return to its original price at the beginning of the first year?
|
13.04\%
| 0.333333 |
In convex quadrilateral $ABCD, \angle B \cong \angle D, AB = CD = 200,$ and $AD \neq BC$. The perimeter of $ABCD$ is $720$. Find $\lfloor 1000 \cos B \rfloor.$
|
800
| 0.916667 |
For each positive integer $n$, let $n!$ denote the product $1\cdot 2\cdot 3\cdot\,\cdots\,\cdot (n-1)\cdot n$.
What is the remainder when $10!$ is divided by $13$?
|
6
| 0.583333 |
The sides of a triangle have lengths $13, 17,$ and $k,$ where $k$ is a positive integer. For how many values of $k$ is the triangle obtuse?
|
14
| 0.833333 |
If $x$ and $y$ are positive real numbers such that $5x^2 + 15xy = x^3 + 2x^2y + 3xy^2,$ what is the value of $x$?
|
5
| 0.333333 |
Given that $b$ is an odd multiple of $7769$, find the greatest common divisor of $4b^2+81b+144$ and $2b+7$.
|
1
| 0.75 |
A right triangle has an area of 180 square units and a leg length of 30 units. What is the perimeter of the triangle, in units?
|
P = 42 + 2\sqrt{261}
| 0.083333 |
From this infinite list of numbers, how many are integers? $$\sqrt{6561},\sqrt[3]{6561},\sqrt[4]{6561},\sqrt[5]{6561},\sqrt[6]{6561},\ldots$$
|
3
| 0.083333 |
If $x$, $y$, and $z$ are positive with $xy=30$, $xz = 60$, and $yz=90$, what is the value of $x+y+z$?
|
x+y+z = 11\sqrt{5}
| 0.333333 |
The sequence $b_1,$ $b_2,$ $b_3,$ $\dots$ satisfies $b_1 = 23,$ $b_7 = 83,$ and for all $n \ge 3,$ $b_n$ is the arithmetic mean of the first $n - 1$ terms. Find $b_2.$
|
143
| 0.333333 |
In the diagram, what is the perimeter of polygon $PQRSU$? [asy]
import olympiad;
size(6cm); // ADJUST
pair p = (0, 8);
pair q = (4, 8);
pair r = (4, 4);
pair s = (9, 0);
pair u = (0, 0);
draw(p--q--r--s--u--cycle);
label("$P$", p, NW);
label("$Q$", q, NE);
label("$R$", r, E + NE);
label("$S$", s, SE);
label("$U$", u, SW);
label("$8$", p / 2, W);
label("$4$", p + (q - p) / 2, 2 * N);
label("$9$", s / 2, S);
draw(rightanglemark(p, u, s));
draw(rightanglemark(u, p, q));
draw(rightanglemark(p, q, r));
add(pathticks(p--q, s=6));
add(pathticks(q--r, s=6));
[/asy]
|
25+\sqrt{41}
| 0.416667 |
Let $g(x) = \frac{x + 8}{x - 1}$. Define a sequence $(g_n)$ of functions where $g_1 = g$ and
\[ g_n = g \circ g_{n - 1} \]
for all $n \geq 2$. Find the number of distinct real numbers $x$ such that
\[ g_n(x) = x \]
for some positive integer $n$.
|
2
| 0.416667 |
Chandra has five bowls, each a different color (red, blue, yellow, green, purple). She also has four glasses, one of each color except purple. If she chooses a bowl and a glass from the cupboard, under the condition that the bowl and glass cannot be the same color, how many valid pairings are possible?
|
16
| 0.25 |
Thirteen girls are standing in a circle. A ball is passed clockwise. The first girl, Bella, starts with the ball, skips the next four girls and throws to the sixth girl. The girl receiving the ball then skips the next four girls and continues the pattern. How many throws are needed for the ball to come back to Bella?
|
13
| 0.833333 |
A triangle has an inscribed circle with a radius of 8 cm. Calculate the length of $\overline{AB}$ if the triangle is a right triangle with one angle measuring $30^\circ$. Express your answer in simplest radical form.
|
\overline{AB} = 16(\sqrt{3} + 1)
| 0.5 |
The quadratic $x^2 + 500x + 2500$ can be written in the form $(x + a)^2 + d$, where $a$ and $d$ are constants. Find the value of $\frac{d}{a}$.
|
-240
| 0.916667 |
A sphere is divided into eight equal parts (octants). The circumference of the sphere is $16\pi$ inches. Find the number of cubic inches in the volume of one of these parts. Express your answer in terms of $\pi$.
|
\frac{256}{3}\pi
| 0.583333 |
A circle centered at $O$ is circumscribed about $\triangle ABC$ as shown. Point $A$ is such that $\angle BOA = 120^\circ$ and $\angle BOC = 140^\circ$. What is the measure of $\angle BAC$, in degrees?
|
70^\circ
| 0.75 |
Let $d$ be a positive integer such that when $143$ is divided by $d$, the remainder is $3.$ Compute the sum of all possible values of $d$ between 20 and 100.
|
153
| 0.916667 |
Our club has 24 members, 12 boys and 12 girls. In how many ways can we choose a president, a vice-president, and a secretary if the president and the vice-president must be of the same gender, but the secretary can be of any gender and no one can hold more than one office?
|
5808
| 0.25 |
Compute \[
\left\lfloor \frac{2010! + 2008!}{2011! + 2009!}\right\rfloor.
\]
|
0
| 0.916667 |
Find the smallest solution to the equation \[\frac{1}{x-1} + \frac{1}{x-5} = \frac{4}{x-4}.\]
|
\frac{5 - \sqrt{33}}{2}
| 0.916667 |
Simplify $1-(2-(3-(4-(5-(6-x)))))$.
|
x-3
| 0.833333 |
Let $p$ and $q$ be the roots of the equation $x^2 - 7x + 12 = 0$. Compute the value of:
\[ p^3 + p^4 q^2 + p^2 q^4 + q^3. \]
|
3691
| 0.666667 |
Let $g(n)$ return the number of distinct ordered pairs of positive integers $(a, b)$ such that for each ordered pair, $a^2 + b^2 = n$. Note that when $a \neq b$, $(a, b)$ and $(b, a)$ are distinct. What is the smallest positive integer $n$ for which $g(n) = 4$?
|
65
| 0.75 |
The sum of 81 consecutive integers is $9^5$. What is their median?
|
729
| 0.666667 |
Simplify the product \[\frac{6}{3}\cdot\frac{9}{6}\cdot\frac{12}{9} \dotsm \frac{3n+3}{3n} \dotsm \frac{3003}{3000}.\]
|
1001
| 0.666667 |
Let $a_1, a_2, \dots$ be a sequence where $a_1 = 5$, $a_2 = 3$, and $a_n = \frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$?
|
3
| 0.916667 |
A 10 by 10 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 7 black squares, can be drawn on the checkerboard?
|
140
| 0.166667 |
For all real numbers $x$ and $y$, define the mathematical operation $\diamond$ such that $x \diamond 0 = 2x, x \diamond y = y \diamond x$, and $(x + 1) \diamond y = (x \diamond y) \cdot (y + 2)$. What is the value of $6 \diamond 3$?
|
93750
| 0.25 |
Find the integer $n$, $0 \le n \le 15$, such that \[n \equiv 14567 \pmod{16}.\]
|
7
| 0.833333 |
If 5 daps are equivalent to 4 dops, and 3 dops are equivalent to 8 dips, how many daps are equivalent to 48 dips?
|
22.5 \text{ daps}
| 0.916667 |
Bag A has 5 white marbles and 6 black marbles. Bag B has 8 yellow marbles and 6 blue marbles. Bag C has 3 yellow marbles and 9 blue marbles. A marble is drawn at random from Bag A. If it is white, a marble is drawn at random from Bag B, otherwise, if it is black, a marble is drawn at random from Bag C. What is the probability that the second marble drawn is yellow?
|
\frac{61}{154}
| 0.916667 |
Find the quadratic polynomial, with real coefficients, which has $-3 - 4i$ as a root, and where the coefficient of $x$ is $-10$.
|
-\frac{5}{3}x^2 - 10x - \frac{125}{3}
| 0.75 |
Quadrilateral $EFGH$ is 12 cm by 6 cm. $P$ is the midpoint of $\overline{FG}$, and $Q$ is the midpoint of $\overline{GH}$. A diagonal $\overline{EQ}$ is drawn. Calculate the number of square centimeters in the area of the region $EPQH$.
|
45\text{ cm}^2
| 0.25 |
I want to design a 4-character license plate. The requirements are:
1. The first character must be a letter.
2. The second and third characters can each be either a letter or a digit.
3. The fourth character must be a digit.
4. The first character and the third character should be the same. How many ways can I design such a license plate?
|
9360
| 0.916667 |
Factor $x^6 - 64$ as far as possible, where the factors are monic polynomials with real coefficients.
|
(x - 2)(x + 2)(x^2 + 2x + 4)(x^2 - 2x + 4)
| 0.083333 |
Solve for $x$: \[\frac{x-75}{4} = \frac{5-3x}{7}.\]
|
\frac{545}{19}
| 0.916667 |
Find the units digit of $7 \cdot 13 \cdot 1957 - 7^4$.
|
6
| 0.916667 |
If \(x > 0\) and \(y > 0\), define a new operation \(\Delta\) as follows: $$x \Delta y = \dfrac{x + y}{1 + xy + x^2y^2}.$$
Calculate \(3 \Delta 4.\)
|
\frac{7}{157}
| 0.75 |
During a discussion about polynomial roots, Sarah considers a quadratic polynomial \[x^2 - tx + q,\] with roots $\alpha$ and $\beta$. She notes that \[\alpha + \beta = \alpha^2 + \beta^2 = \alpha^3 + \beta^3 = \cdots = \alpha^{2010} + \beta^{2010}.\] She aims to find the maximum possible value of \[\dfrac{1}{\alpha^{2011}} + \dfrac{1}{\beta^{2011}}.\]
|
2
| 0.583333 |
The area of a triangle is 450 square feet and its perimeter is required. If the length of one side (considered as the base) is 25 feet, find the altitude corresponding to this base and the perimeter, assuming the triangle is equilateral.
|
75 \text{ feet}
| 0.666667 |
Calculate \(3^{18} \div 27^2\) and multiply the result by 7. Write your answer as an integer.
|
3720087
| 0.916667 |
Find the product of all positive integer values of $c$ such that $10x^2 + 24x + c = 0$ has two real roots.
|
87,178,291,200
| 0.916667 |
There is an unlimited supply of congruent equilateral triangles made of colored paper, each triangle being a solid color with the same color on both sides. A large equilateral triangle is constructed from four smaller triangles as shown. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles match in color. Given that there are eight different colors of triangles and the center triangle can only be one of three specific colors (red, blue, green) chosen from these eight, how many distinguishable large equilateral triangles can be constructed?
|
360
| 0.333333 |
Rose wants to fill her new rectangular flower bed, which has been divided into five differently sized rectangular regions, with a unique type of flower per region. The new dimensions, in feet, of the rectangular regions are:
- Region 1: 2 ft by 3 ft
- Region 2: 3 ft by 4 ft
- Region 3: 2 ft by 5 ft
- Region 4: 4 ft by 4 ft
- Region 5: 3 ft by 6 ft
She plants one flower per square foot in each region. The costs of the flowers are as follows: Asters cost $1 each, Begonias $1.75 each, Cannas $2 each, Dahlias $2.25 each, and Easter lilies $3.50 each. What is the least possible cost, in dollars, for planting her garden?
|
113.5
| 0.083333 |
What is the remainder when $1491 \cdot 2001$ is divided by $250$?
|
241
| 0.5 |
Simplify and rationalize the denominator: $$\frac{1}{1+ \frac{1}{\sqrt{5}+2}}.$$
|
\frac{\sqrt{5} + 1}{4}
| 0.833333 |
In a survey conducted in Utah, 540 people were asked about their preferred term for carbonated beverages. The results are depicted in a pie chart. The central angle of the "Soda" sector of the graph is $162^\circ$. Determine the number of respondents who favored the term "Soda".
|
243
| 0.833333 |
What is the smallest prime whose digits sum to $20$?
|
389
| 0.416667 |
The measures of the angles of a pentagon are in the ratio 2:2:3:4:5. Find the measure of the smallest angle.
|
67.5^\circ
| 0.916667 |
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