problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
What is the value of $\sqrt[3]{4^6 + 4^6 + 4^6 + 4^6}$?
|
16 \sqrt[3]{4}
| 0.083333 |
For how many integer Fahrenheit temperatures between 32 and 2000 inclusive does the original temperature equal the final temperature when only temperatures where $F-32 \equiv 2 \pmod{9}$ are converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the nearest integer?
|
219
| 0.916667 |
Given the volume of the parallelepiped determined by the vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 6, find the volume of the parallelepiped determined by the vectors $\mathbf{a} + 2\mathbf{b}$, $\mathbf{b} + 4\mathbf{c}$, and $\mathbf{c} - 3\mathbf{a}$.
|
138
| 0.333333 |
Let $x$ be a real number. Consider the following five statements:
1. $0 < x^2 < 4$
2. $x^2 > 6$
3. $-2 < x < 0$
4. $0 < x < 2$
5. $-2 < x - x^2 < 3$
What is the maximum number of these statements that can be true for any value of $x$?
|
3
| 0.75 |
Find the maximum \( y \)-coordinate of a point on the graph of \( r = \sin 3\theta \).
|
\frac{9}{16}
| 0.833333 |
On a 9x9 chessboard, each square is labeled with the reciprocal of the sum of its row and column indices, given by $\frac{1}{r+c-1}$, where $r$ and $c$ are the row and column indices starting from 1. Nine squares are to be chosen such that there is exactly one chosen square in each row and each column. Find the minimum sum of the labels of the nine chosen squares.
|
1
| 0.333333 |
How many ways are there to put 7 balls into 4 boxes if the balls are indistinguishable and the boxes are also indistinguishable?
|
11
| 0.833333 |
Alice throws five identical darts. Each hits one of four identical dartboards on the wall. After throwing the five darts, she lists the number of darts that hit each board, from greatest to least. How many different lists are possible?
|
6
| 0.666667 |
A regular dodecagon $Q_1 Q_2 \dotsb Q_{12}$ is drawn in the coordinate plane with $Q_1$ at $(1,0)$ and $Q_7$ at $(3,0)$. If $Q_n$ represents the point $(x_n,y_n)$, compute the numerical value of the product:
\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \dotsm (x_{12} + y_{12} i).\]
|
4095
| 0.25 |
Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz = 1,$ $x + \frac{1}{z} = 6,$ and $y + \frac{1}{x} = 30.$ Find $z + \frac{1}{y}.$
|
\frac{38}{179}
| 0.75 |
Determine the intersection points and the ratio of the segments formed when the curve $y = \cos x$ intersects the line $y = \cos 60^\circ$. Assume the successive ratios $\dots p : q : p : q \dots$ with $p < q$. Compute the ordered pair of relatively prime positive integers $(p, q)$.
|
(1, 2)
| 0.5 |
Let $x_1, x_2, x_3, x_4, x_5$ be the roots of the polynomial $f(x) = x^5 - x^3 + 2x^2 + 1$, and let $g(x) = x^2 - 3$. Find
\[
g(x_1) g(x_2) g(x_3) g(x_4) g(x_5).
\]
|
-59
| 0.166667 |
Given that $a$ is an odd multiple of $7767$, find the greatest common divisor of $6a^2 + 5a + 108$ and $3a + 9$.
|
9
| 0.333333 |
A cylinder has a height that is increased by $2$ units, resulting in a $72\pi$ cubic units increase in volume. Likewise, increasing the radius by $2$ units also increases the volume by $72\pi$ cubic units. If the original radius is $3$, find the original height.
|
4.5
| 0.5 |
Elmo makes $N$ sandwiches for a school event. For each sandwich he uses $C$ slices of cheese at $3$ cents per slice and $T$ slices of tomato at $4$ cents per slice. The total cost of the cheese and tomato to make all the sandwiches is $\$3.05$. Assume that $C$, $T$, and $N$ are positive integers with $N>1$. What is the cost, in dollars, of the tomatoes Elmo uses to make the sandwiches?
|
\$2.00
| 0.166667 |
How many different positive three-digit integers can be formed using only the digits in the set $\{2, 3, 4, 5, 5, 5, 6, 6\}$ if no digit may be used more times than it appears in the given set of available digits?
|
85
| 0.25 |
The Elvish language consists of 4 words: "elara", "quen", "silva", and "nore". In a sentence, "elara" cannot come directly before "quen", and "silva" cannot come directly before "nore"; all other word combinations are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Elvish?
|
48
| 0.083333 |
Let $a$ and $b$ be angles such that $\cos a + \cos b = 1$ and $\sin a + \sin b = \frac{1}{2}$. Additionally, assume that $\tan(a - b) = 1$. Find \[\tan \left( \frac{a + b}{2} \right).\]
|
\frac{1}{2}
| 0.666667 |
A truck travels due west at $\frac{3}{4}$ mile per minute on a straight road. At the same time, a circular storm, whose radius is $60$ miles, moves southwest at $\frac{1}{2}\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $130$ miles due north of the truck. Determine the average time $\frac{1}{2}(t_1 + t_2)$ during which the truck is within the storm circle, where $t_1$ is the time the truck enters and $t_2$ is the time the truck exits the storm circle.
|
208
| 0.75 |
Let $p$ and $q$ be positive integers such that \[\frac{6}{11} < \frac{p}{q} < \frac{5}{9}\] and $q$ is as small as possible. What is $q-p$?
|
9
| 0.916667 |
Mark had a box of chocolates. He consumed $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of what remained to his friend Lucy. Mark and his father then each ate 20 chocolates from what Mark had left. Finally, Mark's sister took between five and ten chocolates, leaving Mark with four chocolates. How many chocolates did Mark start with?
|
104
| 0.166667 |
On a 24-hour clock, an elapsed time of five hours looks the same as an elapsed time of $25$ hours. Because of this, we can say that five hours is "clock equivalent'' to its square number of hours. What is the least whole number of hours that is greater than $5$ hours and is "clock equivalent'' to its square number of hours?
|
9
| 0.916667 |
What is the 24th digit after the decimal point of the sum of the decimal equivalents for the fractions $\frac{1}{13}$ and $\frac{1}{8}$?
|
3
| 0.916667 |
In a similar geoboard setup, points are evenly spaced vertically and horizontally. Segment $DE$ is drawn using two points, as shown. Point $F$ is to be chosen from the remaining $23$ points. How many of these $23$ points will result in triangle $DEF$ being isosceles if $DE$ is three units long? [asy]
draw((0,0)--(0,6)--(6,6)--(6,0)--cycle,linewidth(1));
for(int i=1;i<6;++i)
{for(int j=1;j<6;++j)
{dot((i,j));}
}
draw((2,2)--(5,2),linewidth(1));
label("D",(2,2),SW);
label("E",(5,2),SE);
[/asy]
|
2
| 0.166667 |
Both roots of the quadratic equation $x^2 - 65x + k = 0$ are prime numbers. Find the number of possible values of $k.$
|
0
| 0.5 |
If \(n\) is the smallest positive integer for which there exist positive real numbers \(a\) and \(b\) such that
\[(a + bi)^n = 2(a - bi)^n,\]
compute \(\frac{b}{a}.\)
|
\sqrt{3}
| 0.25 |
Let $\mathbf{A}$ and $\mathbf{B}$ be matrices such that
\[\mathbf{A} + \mathbf{B} = \mathbf{A} \mathbf{B}.\]
If $\mathbf{A} \mathbf{B} = \begin{pmatrix} 10 & 6 \\ -4 & 2 \end{pmatrix},$ find $\mathbf{B} \mathbf{A}.$
|
\begin{pmatrix} 10 & 6 \\ -4 & 2 \end{pmatrix}
| 0.916667 |
A truncated cone has horizontal bases with radii 20 and 5. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
|
10
| 0.083333 |
Define a function $A(m, n)$ with the same recursive definition as provided:
\[ A(m,n) = \left\{
\begin{aligned}
&n+1& \text{ if } m = 0 \\
&A(m-1, 1) & \text{ if } m > 0 \text{ and } n = 0 \\
&A(m-1, A(m, n-1))&\text{ if } m > 0 \text{ and } n > 0.
\end{aligned} \right.\]
Compute $A(3, 2).$
|
29
| 0.5 |
What is the greatest common factor of all three-digit palindromes that are divisible by 3?
|
3
| 0.666667 |
Let
\[
\mathbf{B} = \begin{pmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix}.
\]
Compute $\mathbf{B}^{200}$.
|
\mathbf{B}^{200} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
| 0.916667 |
Find all real values of \( x \) that satisfy \( \frac{x(x+1)}{(x-3)^2} \ge 10 \). (Give your answer in interval notation.)
|
\left[\frac{61 - \sqrt{481}}{18}, 3\right) \cup \left(3, \frac{61 + \sqrt{481}}{18}\right]
| 0.5 |
Compute the smallest positive integer $n$ such that
\[\sum_{k = 0}^n \log_{10} \left( 1 + \frac{1}{2^{2^k}} \right) \ge 1 + \log_{10} \frac{511}{512}.\]
|
3
| 0.25 |
The exact amount of fencing that enclosed the six congruent equilateral triangular corrals is reused to form one large equilateral triangular corral. What is the ratio of the total area of the six small corrals to the area of the new large corral? Express your answer as a common fraction.
|
\frac{1}{6}
| 0.833333 |
Five dice are stacked in a pyramid as shown, with two on the bottom and three on top. Fourteen of their thirty faces are visible. The visible numbers are 1, 1, 2, 3, 1, 2, 3, 3, 4, 5, 1, 6, 6, and 5. What is the total number of dots NOT visible in this view?
|
62
| 0.083333 |
For how many integer values of $n$ between 1 and 500 inclusive does the decimal representation of $\frac{n}{525}$ terminate?
|
23
| 0.916667 |
The function $g$ is defined for all integers and satisfies the following conditions:
\[ g(n) = \begin{cases}
n-3 & \mbox{if } n \geq 500 \\
g(g(n+4)) & \mbox{if } n < 500
\end{cases} \]
Find $g(56)$.
|
497
| 0.583333 |
Find the largest three-digit integer that is divisible by each of its distinct, non-zero digits, starting with an 8.
|
864
| 0.416667 |
Compute the smallest positive integer $n$ such that $n + i,$ $(n + i)^2,$ and $(n + i)^3$ are the vertices of a triangle in the complex plane whose area is greater than 4030.
|
10
| 0.5 |
A solid rectangular block is constructed by sticking together $N$ identical 1-cm cubes face to face. When the block is viewed such that three of its faces are visible, exactly $378$ of the 1-cm cubes are not visible. Determine the smallest possible value of $N$.
|
560
| 0.166667 |
What is the coefficient of $x^3y^5$ in the expansion of $\left(\frac{4}{3}x - \frac{2y}{5}\right)^8$?
|
-\frac{114688}{84375}
| 0.333333 |
What is the ones digit of $1^{2021} + 2^{2021} + 3^{2021} + \cdots + 2021^{2021}?$
|
1
| 0.75 |
The cost of three pencils and four pens is \$3.20, and the cost of two pencils and three pens is \$2.50. What is the cost of one pencil and two pens?
|
1.80
| 0.333333 |
The polynomial equation \[x^4 + ax^2 + bx + c = 0,\] where \(a\), \(b\), and \(c\) are rational numbers, has \(3-\sqrt{5}\) as a root. It also has a sum of its roots equal to zero. What is the integer root of this polynomial?
|
-3
| 0.416667 |
How many integers \( n \) are there such that \( 200 < n < 300 \) and \( n \) leaves the same remainder when divided by \( 7 \) and by \( 9 \)?
|
7
| 0.833333 |
Given the points $P = (2, -3, 1)$, $Q = (4, -7, 4)$, $R = (3, -2, -1)$, and $S = (5, -6, 2)$ in space, determine if they form a flat quadrilateral and find its area.
|
\sqrt{110}
| 0.583333 |
Let \(x\), \(y\), and \(z\) be real numbers such that
\[x^3 + y^3 + z^3 - 3xyz = 8.\]
Find the minimum value of \(x^2 + y^2 + z^2.\)
|
4
| 0.75 |
Let \(w, x, y, z\) be distinct real numbers that sum to \(0\). Compute \[ \dfrac{wy+xz}{w^2+x^2+y^2+z^2}. \]
|
-\frac{1}{2}
| 0.583333 |
Let $g(x) = x^4 + 16x^3 + 80x^2 + 128x + 64$. Let $z_1, z_2, z_3, z_4$ be the roots of $g$. Find the smallest possible value of $|z_{a}z_{b} + z_{c}z_{d}|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$.
|
16
| 0.25 |
In the sequence of natural numbers listed in ascending order, what is the smallest prime number that occurs after a sequence of six consecutive positive integers, all of which are nonprime?
|
97
| 0.583333 |
Compute the number of real solutions $(x, y, z, w)$ to the system of equations:
\begin{align*}
x + 1 &= z + w + zwx, \\
y - 1 &= w + x + wxy, \\
z + 2 &= x + y + xyz, \\
w - 2 &= y + z + yzw.
\end{align*}
|
1
| 0.666667 |
How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \[\left\{ \begin{aligned} x+2y&=2 \\ \left| |x| - |y| \right| &= 2 \end{aligned}\right.\]
|
2
| 0.416667 |
Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that
\[P(1) = P(3) = P(5) = P(7) = a\] and
\[P(2) = P(4) = P(6) = P(8) = -a.\] Additionally,
\[P(0) = 0 \] and
\[P(9) = a.\]
What is the smallest possible value of $a$?
|
945
| 0.083333 |
The real numbers $a,$ $b,$ and $c$ satisfy the equation
\[a^3 + b^3 + c^3 + a^2 + b^2 + 1 = d^2 + d + \sqrt{a + b + c - 2d}.\] Find $d$.
|
1 \text{ or } -\frac{4}{3}
| 0.666667 |
How many integers $n$ satisfy the condition $150 < n < 250$ and have the same remainder when divided by $7$ or by $9$?
|
7
| 0.25 |
Consider a triangle $DEF$ where the angles of the triangle satisfy
\[ \cos 3D + \cos 3E + \cos 3F = 1. \]
Two sides of this triangle have lengths 12 and 14. Find the maximum possible length of the third side.
|
2\sqrt{127}
| 0.166667 |
Six couples are at a gathering. Each person shakes hands exactly once with everyone else except for their spouse and one additional person of their choice. How many handshakes were exchanged?
|
54
| 0.666667 |
An annulus is formed by two concentric circles with radii $R$ and $r$, where $R > r$. Let $\overline{OA}$ be a radius of the outer circle, $\overline{AB}$ be a tangent to the inner circle at point $B$, and $\overline{OC}$ be the radius of the outer circle that passes through $B$. Let $x = AB$, $y = BC$, and $z = AC$. Determine the area of the annulus. Express your answer in terms of $\pi$ and at most one of the variables $x, R, r, y, z$.
|
\pi x^2
| 0.416667 |
Sides $\overline{AM}$ and $\overline{CD}$ of regular dodecagon $ABCDEFGHIJKL$ are extended to meet at point $P$. What is the degree measure of angle $P$?
|
90^\circ
| 0.083333 |
Define the function $f(n)$ to return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point, given that $k$ is of the form $3^n * 2^n$. What is the number of positive integer divisors of $f(2010)$?
|
4044121
| 0.416667 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 3$ and
\[f(x^2 + y^2) = (x + y)(f(x) + f(y))\] for all real numbers $x$ and $y.$
Determine the value of $f(5)$.
|
f(5) = 45
| 0.75 |
A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly $300$ of the 1-cm cubes cannot be seen. Find the smallest possible value of $N$.
|
462
| 0.166667 |
Given the right triangles ABC and ABD, what is the length of segment BC, in units? [asy]
size(150);
pair A, B, C, D, X;
A=(0,0);
B=(0,15);
C=(-20,0);
D=(-45,0);
draw(A--B--D--A);
draw(B--C);
draw((0,2)--(-2,2)--(-2,0));
label("$50$", (B+D)/2, NW);
label("$25$", (C+D)/2, S);
label("$20$", (A+C)/2, S);
label("A", A, SE);
label("B", B, NE);
label("D", D, SW);
label("C", C, S);
[/asy]
|
25
| 0.666667 |
An isosceles triangle $ABC$ with $AB = AC$ is given. A point $P$ is chosen at random within the triangle. Determine the probability that $\triangle ABP$ has a smaller area than both $\triangle ACP$ and $\triangle BCP$.
|
\frac{1}{3}
| 0.75 |
Quadrilateral $EFGH$ is a parallelogram. A line through point $G$ makes a $30^\circ$ angle with side $GH$. Determine the degree measure of angle $E$.
[asy]
size(100);
draw((0,0)--(5,2)--(6,7)--(1,5)--cycle);
draw((5,2)--(7.5,3)); // transversal line
draw(Arc((5,2),1,-60,-20)); // transversal angle
label("$H$",(0,0),SW); label("$G$",(5,2),SE); label("$F$",(6,7),NE); label("$E$",(1,5),NW);
label("$30^\circ$",(6.3,2.8), E);
[/asy]
|
150
| 0.75 |
In a triangle $ABC$, $BP$ and $BQ$ trisect $\angle ABC$. A line $BM$ bisects $\angle PBQ$ and another line $CN$ bisects $\angle QBC$. Find the ratio of the measure of $\angle MBQ$ to the measure of $\angle NCQ$.
|
1
| 0.833333 |
Find the number of solutions to
\[
\sin x = \left( \frac{1}{3} \right)^x
\]
on the interval $(0, 50\pi)$.
|
50
| 0.583333 |
Let $g(x) = \left\lceil \frac{\cos(x)}{x + 3} \right\rceil$ for $x > -3$, and $g(x) = \left\lfloor \frac{\cos(x)}{x + 3} \right\rfloor$ for $x < -3$. ($g(x)$ is not defined at $x = -3$.) Determine if there is an integer that is not in the range of $g(x)$.
|
0
| 0.333333 |
Calculate the number of roots of unity that are also roots of the quadratic equation $z^2 + az - 1 = 0$ for some integer $a$.
|
2
| 0.75 |
For some positive integer \( n \), the number \( 150n^3 \) has \( 150 \) positive integer divisors, including \( 1 \) and the number \( 150n^3 \). How many positive integer divisors does the number \( 108n^5 \) have?
|
432
| 0.25 |
Let \( m \) be the smallest integer whose cube root is of the form \( n + r \), where \( n \) is a positive integer and \( r \) is a positive real number less than \( 1/500 \). Find \( n \).
|
13
| 0.333333 |
The cost of three notebooks and two pens is $\$7.40$, and the cost of two notebooks and five pens is $\$9.75$. What is the cost of one notebook and three pens?
|
5.53
| 0.166667 |
Suppose
\[
\frac{1}{x^3 - x^2 - 17x + 45} = \frac{A}{x + 5} + \frac{B}{x - 3} + \frac{C}{x + 3}
\]
where $A$, $B$, and $C$ are real constants. Determine the value of $A$.
|
A = \frac{1}{16}
| 0.583333 |
Find the minimum value of
\[
y^2 + 9y + \frac{81}{y^3}
\]
for \(y > 0\).
|
39
| 0.916667 |
Let \( c \) be a complex number. Suppose there exist distinct complex numbers \( r \), \( s \), and \( t \) such that for every complex number \( z \), we have
\[
(z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct)
\]
where \( r \), \( s \), \( t \) are the third roots of unity. Compute the number of distinct possible values of \( c \).
|
3
| 0.666667 |
Find the number of real solutions to the equation
\[
\frac{x}{50} = \cos x.
\]
|
31
| 0.166667 |
A $4 \times 2$ rectangle and a $5 \times 3$ rectangle are contained within a square. One of the rectangles must be rotated 90 degrees relative to the other, and the sides of the square are parallel to one of the rectangle's sides at a time. What is the smallest possible area of the square?
|
25
| 0.75 |
In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 30 square units, calculate the area of trapezoid $PQRS$.
|
147
| 0.083333 |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 48.$ Find the minimum value of
\[x^2 + 4xy + 4y^2 + 3z^2.\]
|
144
| 0.416667 |
In an acute triangle \( \triangle ABC \), altitudes \( \overline{AD} \) and \( \overline{BE} \) intersect at point \( H \). If \( HD = 6 \) and \( HE = 3 \), calculate \( (BD)(DC) - (AE)(EC) \).
|
27
| 0.333333 |
What is the largest possible median for the seven number set $\{x, 2x, y, 3, 2, 5, 7\}$ if $x$ and $y$ can be any integers?
|
7
| 0.833333 |
The least common multiple of two positive integers is divided by their greatest common divisor, yielding a result of 24. If one of these integers is 36, what is the smallest possible value of the other integer?
|
96
| 0.166667 |
Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 3$, $\|\mathbf{b}\| = 4$, and $\|\mathbf{c}\| = 5$. Find all possible values of $\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c}$.
|
[-32, 32]
| 0.916667 |
A basketball team has 12 members, of which only 4 are tall enough to play as centers, while the rest can play any position. Determine the number of ways to choose a starting lineup consisting of one point guard, one shooting guard, one small forward, one center, and one power forward, given that the shooting guard and small forward can't be the same person who plays as the point guard.
|
31680
| 0.166667 |
The graph of the function $f(x)$ is shown. How many values of $x$ satisfy $f(f(x)) = 5$? [asy]
import graph; size(7.4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.4,xmax=5.66,ymin=-1.05,ymax=6.16;
for(int i = -4; i <= 5; ++i) {
draw((i,-1)--(i,6), dashed+mediumgrey);
}
for(int i = 1; i <= 6; ++i) {
draw((-4,i)--(5,i), dashed+mediumgrey);
}
Label laxis; laxis.p=fontsize(10);
xaxis("$x$",-4.36,5.56,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); yaxis("$y$",-0.92,6.12,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); draw((xmin,(-(0)-(-2)*xmin)/-2)--(-1,(-(0)-(-2)*-1)/-2),linewidth(1.2)); draw((-1,1)--(3,5),linewidth(1.2)); draw((3,(-(-16)-(2)*3)/2)--(xmax,(-(-16)-(2)*xmax)/2),linewidth(1.2));
label("$f(x)$",(-3.52,4.6),SE*lsf);
dot((-4.32,4.32),ds); dot((5.56,2.44),ds);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]
|
3
| 0.333333 |
In the diagram, $EF$ and $GH$ are straight lines. What is the value of $y?$ [asy]
draw((0,0)--(12,0));
draw((0,5)--(12,5));
draw((2,0)--(4,5)--(10,0));
label("$70^\circ$",(4,4.5),W);
label("$40^\circ$",(5,4.5),E);
label("$E$",(0,5),W);
label("$G$",(0,0),W);
label("$F$",(12,5),E);
label("$H$",(12,0),E);
label("$110^\circ$",(2,0),NW);
label("$y^\circ$",(8.5,0),N);
[/asy]
|
40
| 0.333333 |
Each of \(b_1, b_2, \dots, b_{150}\) is equal to \(1\) or \(-1\). Find the minimum positive value of
\[
\sum_{1 \le i < j \le 150} b_i b_j.
\]
|
23
| 0.75 |
Let $x,$ $y,$ $z$ be real numbers, all greater than 4, such that
\[\frac{(x + 3)^2}{y + z - 3} + \frac{(y + 5)^2}{z + x - 5} + \frac{(z + 7)^2}{x + y - 7} = 45.\]
Enter the ordered triple $(x,y,z).$
|
(12, 10, 8)
| 0.083333 |
Consider the sequence of numbers defined recursively by $t_1=2$ and for $n>1$ by $t_n=n+t_{n-1}$ when $n$ is even, and by $t_n=\frac{t_{n-1}}{n}$ when $n$ is odd. Given that $t_n=\frac{1}{221}$, find $n$.
|
221
| 0.833333 |
How many numbers are in the list $ -22, -15, -8, \ldots, 43, 50?$
|
11
| 0.583333 |
The lengths of the sides of a triangle are $\log_{10}15$, $\log_{10}90$, and $\log_{10}m$, where $m$ is a positive integer. Determine how many possible values there are for $m$.
|
1343
| 0.916667 |
Given that $y$ is a multiple of $3456$, what is the greatest common divisor of $g(y) = (5y+4)(9y+1)(12y+6)(3y+9)$ and $y$?
|
216
| 0.416667 |
Let $l, m, n,$ and $p$ be real numbers, and let $A, B, C$ be points such that the midpoint of $\overline{BC}$ is $(l,p,0),$ the midpoint of $\overline{AC}$ is $(0,m,p),$ and the midpoint of $\overline{AB}$ is $(p,0,n).$ Find the value of:
\[
\frac{AB^2 + AC^2 + BC^2}{l^2 + m^2 + n^2 + p^2}.
\]
|
8
| 0.583333 |
Find the units digit of $n$ given that $mn = 34^5$ and $m$ has a units digit of 6.
|
4
| 0.916667 |
Determine how many ordered pairs of positive integers $(x, y)$, where $x < y$, have a harmonic mean of $5^{20}$.
|
20
| 0.5 |
A recently released album has a cover price of \$30. The music store offers a \$5 discount and two percentage discounts: 10% and 25%. A savvy shopper wants to maximize savings by choosing the optimal order of applying these discounts. Calculate how much more she will save by choosing the best order over the worst one. Express your answer in cents.
|
162.5\text{ cents}
| 0.083333 |
Find the reflection of the vector $\begin{pmatrix} 2 \\ 5 \end{pmatrix}$ over the normalized vector of $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$.
|
\begin{pmatrix} 4.6 \\ -2.8 \end{pmatrix}
| 0.5 |
Three mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 9:30 a.m., and each stays for exactly $n$ minutes. The probability that any one arrives while another is in the cafeteria is $50 \%,$ and $n = d - e\sqrt {f},$ where $d, e,$ and $f$ are positive integers, and $f$ is not divisible by the square of any prime. Find $d + e + f.$
|
47
| 0.166667 |
Consider a sequence $y_1,$ $y_2,$ $y_3,$ $\dots$ defined by
\begin{align*}
y_1 &= \sqrt[4]{4}, \\
y_2 &= (\sqrt[4]{4})^{\sqrt[3]{4}},
\end{align*}
and in general,
\[y_n = (y_{n - 1})^{\sqrt[3]{4}}\] for $n > 1.$ What is the smallest value of $n$ for which $y_n$ is an integer?
|
4
| 0.416667 |
The operation $\oplus$ is defined by
\[a \oplus b = \frac{a + b}{1 + ab}.\]
Compute
\[1 \oplus (2 \oplus (3 \oplus (\dotsb \oplus (999 \oplus 1000) \dotsb))).\]
|
1
| 0.833333 |
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