Question bank Math Olympiad Vault 2022

Compute $\sqrt{2022^{2}-12^{6}}$.

Alice is thinking of a positive real number $x$, and Bob is thinking of a positive real number $y$. Given that $x^{\sqrt{y}}=27$ and $(\sqrt{x})^{y}=9$, compute $x y$.

A regular $2022$-gon has perimeter $6.28$. To the nearest positive integer, compute the area of the $2022$-gon.

Three distinct vertices are randomly selected among the five vertices of a regular pentagon. Let $p$ be the probability that the triangle formed by the chosen vertices is acute. Compute $10 p$.

What is the smallest $r$ such that three disks of radius $r$ can completely cover up a unit disk?

Herbert rolls $6$ fair standard dice and computes the product of all of his rolls. If the probability that the product is prime can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.

Sets $A$, $B$, and $C$ satisfy $|A| = 92$, $|B| = 35$, $|C| = 63$, $|A \cap B| = 16$, $|A \cap C| = 51$, $|B \cap C| = 19$. Compute the number of possible values of $|A \cap B \cap C|$.

Compute the number of ways to color 3 cells in a $3 \times 3$ grid so that no two colored cells share an edge.

Let $ABC$ be a triangle with $\angle A = 60^\circ$. Line $\ell$ intersects segments $AB$ and $AC$ and splits triangle $ABC$ into an equilateral triangle and a quadrilateral. Let $X$ and $Y$ be on $\ell$ such that lines $BX$ and $CY$ are perpendicular to $\ell$. Given that $AB = 20$ and $AC = 22$, compute $XY$.

Positive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $a x^{2} - b x + c = 0$ has exactly one real solution $r$, and this value $r$ is less than $100$. Compute the maximum possible value of $r$.

Two linear functions $f(x)$ and $g(x)$ satisfy the properties that for all $x$,
- $f(x) + g(x) = 2$
- $f(f(x)) = g(g(x))$
and $f(0) = 2022$. Compute $f(1)$.

Compute the number of positive integers that divide at least two of the integers in the set $\{1^{1}, 2^{2}, 3^{3}, 4^{4}, 5^{5}, 6^{6}, 7^{7}, 8^{8}, 9^{9}, 10^{10}\}$.

Emily's broken clock runs backwards at five times the speed of a regular clock. Right now, it is displaying the wrong time. How many times will it display the correct time in the next 24 hours? It is an analog clock (i.e. a clock with hands), so it only displays the numerical time, not AM or PM. Emily's clock also does not tick, but rather updates continuously.

The English alphabet, which has 26 letters, is randomly permuted. Let $p_{1}$ be the probability that $\mathrm{AB}$, $\mathrm{CD}$, and $EF$ all appear as contiguous substrings. Let $p_{2}$ be the probability that $ABC$ and $DEF$ both appear as contiguous substrings. Compute $\frac{p_{1}}{p_{2}}$.

Let triangle $A B C$ be an acute triangle with circumcircle $\Gamma$. Let $X$ and $Y$ be the midpoints of minor arcs $\widehat{A B}$ and $\widehat{A C}$ of $\Gamma$, respectively. If line $X Y$ is tangent to the incircle of triangle $A B C$ and the radius of $\Gamma$ is $R$, find, with proof, the value of $X Y$ in terms of $R$.

How many ways are there to arrange the numbers $1,2,3,4,5,6$ on the vertices of a regular hexagon such that exactly 3 of the numbers are larger than both of their neighbors? Rotations and reflections are considered the same.

Find the number of ordered pairs $(A, B)$ such that the following conditions hold:
- $A$ and $B$ are disjoint subsets of $\{1,2, \ldots, 50\}$.
- $|A|=|B|=25$
- The median of $B$ is 1 more than the median of $A$.

Alice is bored in class, so she thinks of a positive integer. Every second after that, she subtracts from her current number its smallest prime divisor, possibly itself. After 2022 seconds, she realizes that her number is prime. Find the sum of all possible values of her initial number.

Let $ABCD$ and $AEFG$ be unit squares such that the area of their intersection is $\frac{20}{21}$. Given that $\angle BAE \lt 45^{\circ}$, $\tan \angle BAE$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.

Let $(a_{1}, a_{2}, \ldots, a_{8})$ be a permutation of $(1,2, \ldots, 8)$. Find, with proof, the maximum possible number of elements of the set
$$
\left\{a_{1}, a_{1}+a_{2}, \ldots, a_{1}+a_{2}+\cdots+a_{8}\right\}
$$
that can be perfect squares.