problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
If $c$ and $d$ are integers with $c^2 > d^2$, what is the smallest possible positive value of $\frac{c^2+d^2}{c^2-d^2} + \frac{c^2-d^2}{c^2+d^2}$?
|
2
| 0.416667 |
What is the sum of all the solutions of the equation $x = \left|2x - |100 - 2x|\right|$?
|
\frac{460}{3}
| 0.583333 |
Consider a 10x10 chessboard with 100 squares. A person labels each square from $1$ to $100$. On each square $k$, they place $3^k$ grains of rice. Calculate how many more grains of rice are placed on the $15^{th}$ square than on the first $10$ squares combined.
|
14260335
| 0.75 |
Let $f(x) = 18x + 4$. Find the product of all $x$ that satisfy the equation $f^{-1}(x) = f((2x)^{-1})$.
|
-162
| 0.916667 |
In triangle $XYZ$, angle $XZY$ is 60 degrees, and angle $YZX$ is 80 degrees. Let $D$ be the foot of the perpendicular from $X$ to $YZ$, $O$ the center of the circle circumscribed about triangle $XYZ$, and $E$ the other end of the diameter which goes through $X$. Find the angle $DXE$, in degrees.
|
20^\circ
| 0.083333 |
If $m$ and $n$ are positive integers such that $\gcd(m, n) = 15$, what is the smallest possible value of $\gcd(14m, 20n)$?
|
30
| 0.916667 |
The quadratic $4x^2 - 40x + 100$ can be written in the form $(ax+b)^2 + c$, where $a$, $b$, and $c$ are constants. What is $2b-3c$?
|
-20
| 0.583333 |
Let \( x, y, \) and \( z \) be nonnegative numbers such that \( x^2 + y^2 + z^2 = 1 \). Find the maximum value of
\[ 2xy \sqrt{2} + 6yz. \]
|
\sqrt{11}
| 0.166667 |
Let $D$ be the determinant of the matrix whose column vectors are $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}.$ Find the determinant of the matrix whose column vectors are $2\mathbf{a},$ $\mathbf{b} + \mathbf{c},$ and $\mathbf{c} + \mathbf{a},$ in terms of $D.$
|
2D
| 0.416667 |
Find the product of all positive integral values of $m$ such that $m^2 - 40m + 399 = q$ for some prime number $q$. Note that there is at least one such $m$.
|
396
| 0.583333 |
Compute the value of:
\[
\left\lfloor \frac{2021! + 2018!}{2020! + 2019!} \right\rfloor.
\]
|
2020
| 0.666667 |
How many of the first $30$ rows in Pascal's triangle, excluding row $0$ and row $1$, consist of numbers that are exclusively even, not counting the $1$ at each end of the row?
|
4
| 0.25 |
For a given positive integer $n > 3^3$, what is the greatest common divisor of $n^3 + 2^3$ and $n + 3$?
|
1
| 0.25 |
Find the point in the plane $2x + 3y - 6z = 18$ that is closest to the point $(2, -1, 1)$.
|
\left(\frac{144}{49}, \frac{20}{49}, -\frac{89}{49}\right)
| 0.916667 |
Let \( f(x) = 3x - 2 \). Find the sum of all \( x \) that satisfy the equation \( f^{-1}(x) = f(x^{-2}) \).
|
-8
| 0.666667 |
The square quilt block shown is made from nine unit squares, two of which are divided in half diagonally to form triangles, and one square is divided into four smaller equal squares, one of which is shaded. What fraction of the square quilt is shaded? Express your answer as a common fraction.
[asy]size(75);
fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,gray(.6)); // shading one small square
fill((1,1)--(1,2)--(2,2)--cycle,gray(.6)); // shading the triangle
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,gray(.6)); // shading the square
fill((1,2)--(0,2)--(1,3)--cycle,gray(.6)); // shading the triangle
draw(scale(3)*unitsquare);
for (int i = 1; i <= 2; ++i) {
draw((i,0)--(i,3));
draw((0,i)--(3,i));
}
draw((0.5,2.5)--(1,3)--(1,2.5)--(0.5,2)--cycle);
[/asy]
|
\frac{5}{36}
| 0.833333 |
As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $15n+4$ and $9n+2$?
|
2
| 0.833333 |
Triangle $ABC$ has a perimeter of 3010 units. The sides have lengths that are all integer values with $AB < AC \leq BC$. What is the smallest possible value of $AC - AB$?
|
1
| 0.583333 |
Compute the sum of the squares of the roots of the equation \[x^{10} + 3x^7 + 5x^2 + 404 = 0.\]
|
0
| 0.833333 |
Find $q(x)$ if the graph of $\frac{x^3-2x^2-5x+6}{q(x)}$ has vertical asymptotes at $2$, $-2$, and $1$, and no horizontal asymptote, and $q(4) = 24$.
|
q(x) = \frac{2}{3}x^3 - \frac{2}{3}x^2 - \frac{8}{3}x + \frac{8}{3}
| 0.083333 |
Find the number of real solutions to the equation
\[\frac{x}{50} = \cos x.\]
|
31
| 0.083333 |
Define a $\textit{good word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$, where $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$, and $D$ is never immediately followed by $A$. How many eight-letter good words are there?
|
8748
| 0.416667 |
Let point $O$ be the origin of a three-dimensional coordinate system, and let points $A,$ $B,$ and $C$ be located on the positive $x,$ $y,$ and $z$ axes, respectively. If $OA = \sqrt{144}$ and $\angle BAC = 45^\circ,$ then compute the area of triangle $ABC.$
|
72
| 0.583333 |
Suppose Alice and Bob played a game where they both picked a complex number. If the product of their numbers was \(48 - 16i\), and Alice picked \(7 + 4i\), what number did Bob pick?
|
\frac{272}{65} - \frac{304}{65}i
| 0.5 |
Suppose the function $g$ is graphed below, where each small box has width and height 1.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {
TicksArrx.push(i);
}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {
TicksArry.push(i);
}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray(0.22),extend=true),p=invisible);//,above=true);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true), p=invisible);//,Arrows);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx , pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
rr_cartesian_axes(-1,9,-1,9);
dot((0,0),red+5bp);
dot((2,2),red+5bp);
dot((4,4),red+5bp);
dot((6,6),red+5bp);
dot((8,8),red+5bp);
dot((1,7),red+5bp);
dot((3,5),red+5bp);
dot((5,3),red+5bp);
dot((7,1),red+5bp);
dot((9,0),red+5bp);
[/asy]
Samantha writes the number 2 on her thumb. She then applies $g$ to 2 and writes the output on her index finger. If Samantha continues this process of applying $g$ and writing the output on a new finger, what number will Samantha write on her tenth finger?
|
2
| 0.666667 |
Let \( f(x) = \frac{1}{ax^2 + bx + c} \), where \(a\), \(b\), and \(c\) are nonzero constants. Find all solutions to \( f^{-1}(x) = 0 \). Express your answer in terms of \(a\), \(b\), and \(c\).
|
x = \frac{1}{c}
| 0.833333 |
Given that \( b \) is an even multiple of \( 5959 \), find the greatest common divisor of \( 4b^2 + 73b + 156 \) and \( 4b + 15 \).
|
1
| 0.25 |
A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located, and crawls along a side of the square to that vertex. Find the probability that the bug moves to its starting vertex on its eighth move, expressed as a reduced fraction.
|
\frac{547}{2187}
| 0.083333 |
Below is the graph of $y = 3 \csc(2x - \pi)$ for some positive constants. Find the new value of **a** in this adjusted equation considering the graph's behavior:
[asy]
import TrigMacros;
size(500);
real h(real x) {
return 3*csc(2*x - pi);
}
draw(graph(h, -pi + 0.1, pi - 0.1),blue+linewidth(1));
limits((-pi,-7),(pi,7),Crop);
draw((-pi/4,-7)--(-pi/4,7),dashed);
draw((pi/4,-7)--(pi/4,7),dashed);
trig_axes(-pi,pi,-7,7,pi/2,1);
layer();
rm_trig_labels(-5, 5, 2);
label("$3$", (0,3), E);
label("$-3$", (0,-3), E);
[/asy]
|
3
| 0.583333 |
I have 6 marbles numbered 1 through 6 in a bag. Suppose I take out two different marbles at random. What is the expected value of the product of the numbers on the marbles?
|
\frac{35}{3}
| 0.25 |
In a chess tournament, each player played exactly one game against each of the other players. A win granted the winner 1 point, the loser 0 points, and each player 1/2 point if there was a tie. It was found that precisely half of the points each player earned came against the twelve players with the lowest number of points. What was the total number of players in the tournament?
|
24
| 0.166667 |
Find all real numbers \( b \) such that the equation
\[ x^3 - 2bx^2 + bx + b^2 - 2 = 0 \]
has exactly one real solution in \( x \).
|
0 \text{ and } 2
| 0.5 |
A book with 73 pages numbered 1 to 73 has its pages renumbered in reverse, from 73 to 1. How many pages do the new page number and old page number share the same units digit?
|
15
| 0.5 |
Determine the smallest possible value of $xy$ for positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{3y} = \frac{1}{6}$.
|
48
| 0.916667 |
Let $ABCDE$ be a convex pentagon, and let $H_A,$ $H_B,$ $H_C,$ $H_D$ denote the centroids of triangles $BCD,$ $ACE,$ $ABD,$ and $ABC,$ respectively. Determine the ratio $\frac{[H_A H_B H_C H_D]}{[ABCDE]}.$
|
\frac{1}{9}
| 0.75 |
A convex polyhedron $Q$ has $30$ vertices, $72$ edges, and $44$ faces, $32$ of which are triangular and $12$ of which are hexagonal. Determine the number of space diagonals of $Q$.
|
255
| 0.666667 |
A point $P$ is chosen at random in the interior of equilateral triangle $DEF$. What is the probability that $\triangle DFP$ has a greater area than each of $\triangle DEP$ and $\triangle EFP$?
|
\frac{1}{3}
| 0.833333 |
Consider two lines parameterized as follows:
\[\begin{pmatrix} -2 + s \\ 4 - ks \\ 2 + ks \end{pmatrix},\]
and
\[\begin{pmatrix} 2t \\ 2 + 2t \\ 3 - 2t \end{pmatrix}.\]
Determine the value of $k$ if the lines are coplanar.
|
-1
| 0.833333 |
Twelve 6-sided dice are rolled. What is the probability that exactly five of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth.
|
0.028
| 0.583333 |
In a class of 150 students, all students took the same test. What is the maximum number of students who can receive a grade above the class average?
|
149
| 0.5 |
Find the quotient when \(x^6 - 8\) is divided by \(x - 2.\)
|
x^5 + 2x^4 + 4x^3 + 8x^2 + 16x + 32
| 0.916667 |
If $\mathbf{B} = \begin{pmatrix} x & y \\ z & w \end{pmatrix},$ then its transpose is given by
\[\mathbf{B}^T = \begin{pmatrix} x & z \\ y & w \end{pmatrix}.\]Given that $\mathbf{B}^T = 2\mathbf{B}^{-1},$ find $x^2 + y^2 + z^2 + w^2.$
|
4
| 0.666667 |
**New Problem Statement**: The lengths of two sides of a triangle are 35 units and 43.5 units. If the third side also has an integral length, what is the least possible number of units in the perimeter of the triangle?
|
87.5
| 0.083333 |
Let $D$ be the circle with equation $x^2 + 16y + 81 = -y^2 - 12x$. Determine the center $(c,d)$ and radius $s$ of $D$, and find the value of $c+d+s$.
|
-14 + \sqrt{19}
| 0.083333 |
Solve for $P$ if $\sqrt{P^3} = 81\sqrt[3]{81}$.
|
3^{32/9}
| 0.916667 |
A palindrome is a number that reads the same forward and backward. What is the smallest 5-digit palindrome in base 3 that can be expressed as a 3-digit palindrome in a different base? Provide your response in base 3.
|
10001_3
| 0.416667 |
Each letter represents a non-zero digit. Determine the value of \( t \) given the following equations:
\begin{align*}
a + b &= x \\
x + d &= t \\
t + a &= y \\
b + d + y &= 16
\end{align*}
|
8
| 0.75 |
Sasha has $\$4.80$ in U.S. coins. She has the same number of quarters and nickels, but twice as many dimes as quarters. What is the greatest number of quarters she could have?
|
9
| 0.916667 |
Rectangle $ABCD$ is inscribed in triangle $EFG$ such that side $AD$ of the rectangle is on side $EG$ of the triangle, as shown. The triangle's altitude from $F$ to side $EG$ is 10 inches, and $EG = 15$ inches. The length of segment $AB$ is equal to one-third the length of segment $AD$. What is the area of rectangle $ABCD$?
|
\frac{100}{3}
| 0.166667 |
In trapezoid $EFGH$, the sides $EF$ and $GH$ are both equal and parallel. The distances between the ends of the parallel sides are 5 units and $EG = 20$ units. If the vertical height from $F$ to line $GH$ is 6 units, compute the perimeter of trapezoid $EFGH$.
```plaintext
[asy]
pen p = linetype("4 4");
draw((0,0)--(5,6)--(20,6)--(25,0)--cycle);
draw((5,0)--(5,6), p);
draw((4.5,0)--(4.5, .5)--(5.0,0.5));
label(scale(0.75)*"E", (0,0), W);
label(scale(0.75)*"F", (5,6), NW);
label(scale(0.75)*"G", (20, 6), NE);
label(scale(0.75)*"H", (25, 0), E);
label(scale(0.75)*"15", (12.5,6), N);
label(scale(0.75)*"25", (12.5,0), S);
label(scale(0.75)*"6", (5, 3), E);
[/asy]
```
|
40 + 2\sqrt{61}
| 0.083333 |
How many positive integers \(n\) satisfy \[\dfrac{n+900}{80} = \lfloor \sqrt{n} \rfloor?\]
|
4
| 0.333333 |
A triangle forms from sticks of lengths 7, 24, and 26 inches joined end-to-end. If you either add or subtract the same integral length $x$ to/from each of the sticks, find the smallest value of $x$ such that the three resulting lengths can no longer form a triangle.
|
5
| 0.75 |
Given that $a$ is an even multiple of $947$, find the greatest common divisor (GCD) of $3a^2 + 47a + 101$ and $a + 19$.
|
1
| 0.5 |
Triangle $PQR$ has vertices $P(-3, 9)$, $Q(4, -2)$, and $R(10, -2)$. A line through $Q$ cuts the area of $\triangle PQR$ in half; find the sum of the slope and $y$-intercept of this line.
|
31
| 0.75 |
Given the function \( y = f(x) \) defined as follows:
- \( f(x) = -x - 1 \) for \( -4 \leq x \leq 0 \)
- \( f(x) = \sqrt{9 - (x - 3)^2} - 3 \) for \( 0 \leq x \leq 6 \)
- \( f(x) = 3(x - 6) \) for \( 6 \leq x \leq 7 \)
The graph of \( y = h(x) \) is a transformation of \( y = f(x) \). Determine \( h(x) \) in terms of \( f(x) \) if \( h(x) \) is \( f(x) \) reflected across the y-axis and then shifted 5 units to the right.
|
h(x) = f(5 - x)
| 0.083333 |
Let $t$ be a parameter that varies over all real numbers. Any parabola of the form
\[ y = 4x^2 + tx - t^2 - 3t \]
passes through a fixed point. Find this fixed point.
|
(3, 36)
| 0.833333 |
Allen and Christie each arrive at a conference at a random time between 1:00 and 2:00. Each stays for 20 minutes, then leaves. What is the probability that Allen and Christie see each other at the conference?
|
\frac{5}{9}
| 0.75 |
Let $S$ be a subset of $\{1,2,3,...,100\}$ such that no pair of distinct elements in $S$ has a product divisible by $5$. What is the maximum number of elements in $S$?
|
80
| 0.5 |
The number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_{12} \le 2005$ such that $a_i - i$ is odd for $1 \le i \le 12$ can be expressed as ${m \choose n}$ for some positive integers $m > n$. Compute the remainder when $m$ is divided by 1000.
|
8
| 0.083333 |
Both $a$ and $b$ are positive integers and $b > 1$. When $a^b$ is the greatest possible value less than 600, what is the sum of $a$ and $b$?
|
26
| 0.833333 |
A school estimates its student population to be between $1000$ and $1200$ students. Each student studies either German or Russian, and some study both. The number studying German is between $70\%$ and $75\%$ of the student population, while those studying Russian are between $35\%$ and $45\%$. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Calculate $M - m$.
|
190
| 0.666667 |
What is the units digit of the sum $1! + 1^2 + 2! + 2^2 + 3! + 3^2 + 4! + 4^2 + \cdots + 10! + 10^2$?
|
8
| 0.25 |
Let \( x, y, z \) be complex numbers such that:
\[
xy + 3y = -9,
yz + 3z = -9,
zx + 3x = -9.
\]
Determine all possible values of \( xyz \).
|
27
| 0.583333 |
Let \(a, b, c, d, e\) be real numbers such that
\[
a^2 + b^2 + c^2 + d^2 + e^2 = 5.
\]
Find the maximum value of \(a^3 + b^3 + c^3 + d^3 + e^3.\)
|
5\sqrt{5}
| 0.916667 |
Let \( a, b, c \) be real numbers such that \( 2a + 3b + c = 6 \). Find the maximum value of
\[ ab + ac + bc. \]
|
\frac{9}{2}
| 0.833333 |
How many rows of Pascal's Triangle contain the number $41$?
|
1
| 0.75 |
Find the smallest constant $N$ such that
\[\frac{a^2 + b^2 + c^2}{ab+bc+ca} \geq N\]
whenever $a, b, c$ are the sides of a triangle and are in arithmetic progression.
|
1
| 0.583333 |
Triangle $PQR$ has vertices $P(0, 10)$, $Q(2, 0)$, $R(10, 0)$. A horizontal line with equation $y=s$ intersects line segment $\overline{PQ}$ at $V$ and line segment $\overline{PR}$ at $W$, forming $\triangle PVW$ with area 20. Compute $s$.
|
10 - 5\sqrt{2}
| 0.416667 |
For polynomial $P(x)=1-\dfrac{1}{2}x+\dfrac{1}{4}x^{2}$, define
\[Q(x)=P(x)P(x^{4})P(x^{6})P(x^{8})P(x^{10})P(x^{12})=\sum_{i=0}^{100} b_ix^{i}.\] Find $\sum_{i=0}^{100} |b_i|$.
|
\frac{117649}{4096}
| 0.166667 |
Both \(a\) and \(b\) are positive integers and \(b > 2\). When \(a^b\) is the greatest possible value less than 500, what is the sum of \(a\) and \(b\)?
|
10
| 0.833333 |
In a circle with center $O$, $AD$ is a diameter, $PQR$ is a chord, $AP = 8$, and $\angle PAO = \text{arc } QR = 45^\circ$. Find the length of $PQ$.
|
8
| 0.166667 |
How many 0's are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\frac{1}{2^3\cdot5^5}$?
|
4
| 0.416667 |
Find the sum of the coefficients of the $x^3$ and $x^5$ terms in the expansion of $(x+1)^{50}$.
|
2138360
| 0.833333 |
In an isosceles triangle $ABC$ with $AB=AC$, point $D$ is the midpoint of $\overline{AC}$ and $\overline{BE}$, where $\overline{BE}$ is 13 units long. Determine the length of $\overline{BD}$. Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--(3,13^.5)--(6,0)--cycle);
draw((0,0)--(-3,13^.5)--(6,0));
label("$A$", (6,0), SE);
label("$B$", (0,0), SW);
label("$C$", (3,13^.5), N);
label("$D$", (1.5,6.5), N);
label("$E$", (-3,13^.5), NW);
[/asy]
|
6.5
| 0.333333 |
Robin decided to trek along Alpine Ridge Trail, which she completed in five days. The first three days she hiked a total of 30 miles. The second and fourth days she averaged 15 miles per day. On the last two days, she covered a total of 28 miles. The total distance she traveled on the first and fourth days was combined to 34 miles. How long in total was the Alpine Ridge Trail?
|
58 \text{ miles}
| 0.5 |
Suppose $C$ is a point not on line $AF$, and $D$ is a point on line $AF$ such that $CD \perp AF$. Additionally, let $B$ be a point on line $CF$ but not midpoint such that $AB \perp CF$. Given $AB = 6$, $CD = 9$, and $AF = 15$, determine the length of $CF$.
|
22.5
| 0.166667 |
Let $a,$ $b,$ and $c$ be constants, and suppose the inequality \[\frac{(x-a)(x-b)}{x-c} \geq 0\] is true if and only if either $x < -6$ or $20 \leq x \leq 23.$ Given that $a < b,$ find the value of $a + 2b + 3c.$
|
48
| 0.25 |
Let $\mathbf{u}$, $\mathbf{v}_1$, and $\mathbf{v}_2$ be unit vectors, all mutually orthogonal. Let $\mathbf{w}$ be a vector such that $\mathbf{u} \times (\mathbf{v}_1 + \mathbf{v}_2) + \mathbf{u} = \mathbf{w}$ and $\mathbf{w} \times \mathbf{u} = \mathbf{v}_1 + \mathbf{v}_2$. Compute $\mathbf{u} \cdot (\mathbf{v}_1 + \mathbf{v}_2 \times \mathbf{w})$.
|
1
| 0.5 |
If $x+\frac1x = -5$, what is $x^5+\frac1{x^5}$?
|
-2525
| 0.916667 |
Consider the equation
\[
(x - \sqrt[3]{23})(x - \sqrt[3]{63})(x - \sqrt[3]{113}) = \frac{1}{3}
\]
which has three distinct solutions \( r, s, \) and \( t \). Calculate the value of \( r^3 + s^3 + t^3 \).
|
200
| 0.166667 |
Let $c$ and $d$ be nonzero complex numbers such that $c^2 - cd + d^2 = 0$. Evaluate
\[
\frac{c^6 + d^6}{(c - d)^6}.
\]
|
2
| 0.75 |
Given that $\binom{24}{5}=42504$, $\binom{24}{6}=134596$, and $\binom{23}{5}=33649$, find $\binom{26}{6}$.
|
230230
| 0.916667 |
Ella is attempting to calculate the sum of distances from a point $Q = (3,3)$ to the vertices of $\triangle DEF$, where $D$ is at the origin, $E$ is at $(7,2)$, and $F$ is at $(4,5)$. Solve for the sum of the distances from point $Q$ to each vertex of the triangle $\triangle DEF$, and write your answer in the form $p + q\sqrt{r}$, where $p$ and $q$ are integers. What is $p + q$?
|
5
| 0.416667 |
Find the angle $\theta$ in degrees for the expression
\[
\text{cis } 55^\circ + \text{cis } 65^\circ + \text{cis } 75^\circ + \dots + \text{cis } 145^\circ
\]
expressed in the form $r \, \text{cis } \theta$, where $r > 0$ and $0^\circ \le \theta < 360^\circ$.
|
100^\circ
| 0.833333 |
The integer $z$ has 18 positive factors. The numbers 16 and 18 are factors of $z$. What is $z$?
|
2^5 \cdot 3^2 = 288
| 0.75 |
Consider a bug starting at vertex $A$ of a cube, where each edge of the cube is 1 meter long. At each vertex, the bug can move along any of the three edges emanating from that vertex, with each edge equally likely to be chosen. Let $p = \frac{n}{6561}$ represent the probability that the bug returns to vertex $A$ after exactly 8 meters of travel. Find the value of $n$.
|
1641
| 0.166667 |
The number $6\,21H\,408\,3G5$ is divisible by $6$. If $H$ and $G$ each represent a single digit, what is the sum of all distinct possible values of the product $HG$?
|
0
| 0.333333 |
How many ways are there to distribute 4 distinguishable balls into 3 indistinguishable boxes?
|
14
| 0.75 |
The product of two consecutive page numbers is $20{,}412$. What is the sum of these two page numbers?
|
285
| 0.833333 |
A point $(x, y)$ is randomly selected such that $0 \leq x \leq 4$ and $0 \leq y \leq 5$. What is the probability that $x + y \leq 5$? Express your answer as a common fraction.
|
\frac{3}{5}
| 0.166667 |
Let $a<b<c$ be three integers such that $a,b,c$ is an arithmetic progression and $c,a,b$ is a geometric progression. What is the smallest possible value of $c$?
|
4
| 0.833333 |
A region \( S \) in the complex plane is defined by
\[
S = \{x + iy : -2 \leq x \leq 2, -2 \leq y \leq 2\}.
\]
A complex number \( z = x + iy \) is chosen uniformly at random from \( S \). What is the probability that \( \left(\frac{1}{2} + \frac{1}{2}i\right)z \) is also in \( S \)?
|
1
| 0.833333 |
For certain real values of $a, b, c,$ and $d_{}$, the polynomial $x^4 + ax^3 + bx^2 + cx + d = 0$ has four non-real roots. The sum of two of these roots is $5 + 2i$ and the product of the other two roots is $10 - i$, where $i^2 = -1$. Find $b$.
|
49
| 0.083333 |
Simplify the following expression:
\[(2 + 2\cot (A+\alpha) - 2\csc (A+\alpha))(2 + 2\tan (A+\alpha) + 2\sec (A+\alpha)).\]
|
8
| 0.833333 |
The infinite sequence $T=\{t_1, t_2, t_3, \ldots\}$ is defined by $t_1 = 3$ and $t_n = 3^{t_{n-1}}$ for each integer $n > 1$. What is the remainder when $t_{50}$ is divided by $7$?
|
6
| 0.833333 |
A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 63. If $m$ is also divisible by 11, what is the greatest possible value of $m$?
|
9702
| 0.083333 |
For real numbers $x,$ $y,$ and $z,$ consider the matrix
\[\begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}\]
Determine if the matrix is invertible and, if not, list all possible values of
\[\frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y}.\]
|
\frac{3}{2}
| 0.083333 |
The graph of the function \( g(x) \) is defined on the interval from \( -3 \) to \( 5 \), inclusive, on \( x \). Let \( g(x) \) be defined as follows: \( g(x) = -\frac{1}{2}x^2 + x + 3 \) when \( -3 \leq x \leq 1 \), and \( g(x) = \frac{1}{3}x^2 - 3x + 11 \) when \( 1 < x \leq 5 \). Determine how many values of \( x \) satisfy \( g(g(x)) = 3 \).
|
1
| 0.166667 |
Find all values of $x$ such that $0 \le x < 2\pi$ and $\sin x - \cos x = \sqrt{2}$.
|
\frac{3\pi}{4}
| 0.916667 |
A solid rectangular block is created by gluing together \(N\) 1-cm cube units. When this block is situated such that three faces are visible, \(462\) of the 1-cm cubes cannot be seen. Determine the smallest possible value of \(N\).
|
672
| 0.166667 |
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