question string | answer string |
|---|---|
Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 191 |
Positive real numbers $b \neq 1$ and $n$ satisfy the equations
\[\sqrt{\log_b n} = \log_b \sqrt{n} \qquad \text{and} \qquad b \cdot \log_b n = \log_b (bn).\]
The value of $n$ is $\frac{j}{k},$ where $j$ and $k$ are relatively prime positive integers. Find $j+k.$ | 881 |
A plane contains $40$ lines, no $2$ of which are parallel. Suppose there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Find the... | 607 |
The sum of all positive integers $m$ for which $\tfrac{13!}{m}$ is a perfect square can be written as $2^{a}3^{b}5^{c}7^{d}11^{e}13^{f}$, where $a, b, c, d, e,$ and $f$ are positive integers. Find $a+b+c+d+e+f$. | 12 |
Let $P$ be a point on the circumcircle of square $ABCD$ such that $PA \cdot PC = 56$ and $PB \cdot PD = 90.$ What is the area of square $ABCD?$ | 106 |
Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. ... | 51 |
Call a positive integer $n$ extra-distinct if the remainders when $n$ is divided by $2, 3, 4, 5,$ and $6$ are distinct. Find the number of extra-distinct positive integers less than $1000$. | 49 |
Rhombus $ABCD$ has $\angle BAD<90^{\circ}$. There is a point $P$ on the incircle of the rhombus such that the distances from $P$ to lines $DA$, $AB$, and $BC$ are $9$, $5$, and $16$, respectively. Find the perimeter of $ABCD$. | 125 |
Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are integers in $\{-20, -19,-18, \dots , 18, 19, 20\}$, such that there is a unique integer $m \neq 2$ with $p(m) = p(2)$. | 738 |
There exists a unique positive integer $a$ for which the sum\[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\]is an integer strictly between $-1000$ and $1000$. For that unique $a$, find $a+U$.
(Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.) | 944 |
Find the number of subsets of $\{1,2,3,...,10\}$ that contain exactly one pair of consecutive integers. Examples of such subsets are $\{1,2,5\}$ and $\{1,3,6,7,10\}$.
| 235 |
Let $\triangle ABC$ be an equilateral triangle with side length $55$. Points $D$, $E$, and $F$ lie on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively, with $BD=7$, $CE=30$, and $AF=40$. A unique point $P$ inside $\triangle ABC$ has the property that\[\measuredangle AEP=\measuredangle BFP=\meas... | 75 |
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. A parallelepiped is a solid wi... | 125 |
The following analog clock has two hands that can move independently of each other.
Initially, both hands point to the number 12. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock while the other hand does not move.
Let $N$ b... | 608 |
Find the largest prime number $p<1000$ for which there exists a complex number $z$ satisfying
\begin{itemize}
\item the real and imaginary part of $z$ are both integers;
\item $|z|=\sqrt{p}$, and
\item there exists a triangle whose three side lengths are $p$, the real part of $z^{3}$, and the imaginary part of $z^{3... | 349 |
The number of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is $990$. Find the greatest number of apples g... | 220 |
Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292=444_{\text{eight}}$. | 585 |
Let $\triangle{ABC}$ be an isoceles triangle with $\angle A=90^{\circ}$. There exists a point $P$ inside $\triangle{ABC}$ such that $\angle PAB=\angle PBC=\angle PCA$ and $AP=10$. Find the area of $\triangle{ABC}$. | 250 |
Let $x$, $y$, and $z$ be real numbers satisfying the system of equations
\begin{align*}
xy+4z&=60\\
yz+4x&=60\\
zx+4y&=60.
\end{align*}Let $S$ be the set of possible values of $x$. Find the sum of the squares of the elements of $S$. | 273 |
Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of $S$ can be... | 719 |
Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside this region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n}$, ... | 35 |
Each vertex of a regular dodecagon (12-gon) is to be colored either red or blue, and thus there are $2^{12}$ possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle. | 928 |
Let $\omega=\cos\frac{2\pi}{7}+i\cdot\sin\frac{2\pi}{7}$, where $i=\sqrt{-1}$. Find
$$\prod_{k=0}^{6}(\omega^{3k}+\omega^k+1).$$ | 24 |
Circles $\omega_1$ and $\omega_2$ intersect at two points $P$ and $Q$, and their common tangent line closer to $P$ intersects $\omega_1$ and $\omega_2$ at points $A$ and $B$, respectively. The line parallel to line $AB$ that passes through $P$ intersects $\omega_1$ and $\omega_2$ for the second time at points $X$ and $... | 33 |
Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3$. One way to do this is shown below. Find the number of positive integer divisors of $N$.
[asy]... | 144 |
Find the number of collections of $16$ distinct subsets of $\{1, 2, 3, 4, 5\}$ with the property that for any two subsets $X$ and $Y$ in the collection, $X\cap Y \neq \emptyset$. | 81 |
In $\triangle ABC$ with side lengths $AB=13$, $BC=14$, and $CA=15$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $M$ is on $\overline{AP}$. There exists a unique point $Q$ on segment $\overline{AM}$ such that $\angle PBQ = \angle PCQ$. Then $AQ$ can b... | 247 |
Let $A$ be an acute angle such that $\tan A = 2\cos A$. Find the number of positive integers $n$ less than or equal to $1000$ such that $\sec^n A + \tan^n A$ is a positive integer whose units digit is $9$. | 167 |
A cube-shaped container has vertices $A$, $B$, $C$, and $D$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of the faces of the cube. Vertex $A$ of the cube is set on a horizontal plane $\mathcal P$ so that the plane of the rectangle $ABCD$... | 751 |
For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n\equiv1\pmod{2^n}$. Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n=a_{n+1}$. | 363 |
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