Question bank Math Olympiad Vault 2023

Suppose $a$ and $b$ are positive integers such that $a^{b} = 2^{2023}$. Compute the smallest possible value of $b^{a}$.

Compute the number of dates in the year 2023 such that when put in MM/DD/YY form, the three numbers are in strictly increasing order.
For example, $06 / 18 / 23$ is such a date since $6\lt 18\lt 23$, while today, $11 / 11 / 23$, is not.

Let $ABCD$ be a convex quadrilateral such that $\angle ABD = \angle BCD = 90^{\circ}$, and let $M$ be the midpoint of segment $BD$. Suppose that $CM = 2$ and $AM = 3$. Compute $AD$.

For any positive integer $a$, let $\tau(a)$ be the number of positive divisors of $a$. Find, with proof, the largest possible value of $4 \tau(n)-n$ over all positive integers $n$.

A regular $n$-gon $P_{1} P_{2} \ldots P_{n}$ satisfies $\angle P_{1} P_{7} P_{8} = 178^{\circ}$. Compute $n$.

The formula to convert Celsius to Fahrenheit is
$$
F^{\circ} = 1.8 \cdot C^{\circ} + 32
$$
In Celsius, it is $10^{\circ}$ warmer in New York right now than in Boston. In Fahrenheit, how much warmer is it in New York than in Boston?

If $a$ and $b$ are positive real numbers such that $a \cdot 2^{b} = 8$ and $a^{b} = 2$, compute $a^{\log_{2} a} 2^{b^{2}}$.

The number $5.6$ may be expressed uniquely (ignoring order) as a product $\underline{a} \cdot \underline{b} \times \underline{c} . \underline{d}$ for digits $a, b, c, d$ all nonzero. Compute $\underline{a} \cdot \underline{b}+\underline{c} \cdot \underline{d}$.

A real number $x$ satisfies $9^{x} + 3^{x} = 6$. Compute the value of $16^{1 / x} + 4^{1 / x}$.

Points $X$, $Y$, and $Z$ lie on a circle with center $O$ such that $XY = 12$. Points $A$ and $B$ lie on segment $XY$ such that $OA = AZ = ZB = BO = 5$. Compute $AB$.

Tyler has an infinite geometric series with sum $10$. He increases the first term of his sequence by $4$ and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum $15$. Compute the common ratio of Tyler's series.

Compute the number of positive four-digit multiples of $11$ whose sum of digits (in base ten) is divisible by $11$.

In a convex quadrilateral $ABCD$, the diagonals intersect at $O$, $M$ and $N$ are points on the segments $OA$ and $OD$ respectively. Suppose $MN$ is parallel to $AD$ and $NC$ is parallel to $AB$. Prove that $\angle ABM = \angle NCD$.

Let $L O V E R$ be a convex pentagon such that $L O V E$ is a rectangle. Given that $O V = 20$ and $L O = V E = R E = R L = 23$, compute the radius of the circle passing through $R$, $O$, and $V$.

Let $ABCDEF$ be a regular hexagon, and let $P$ be a point inside quadrilateral $ABCD$. If the area of triangle $PBC$ is $20$, and the area of triangle $PAD$ is $23$, compute the area of hexagon $ABCDEF$.

Let $ABCDE$ be a convex pentagon such that
$$
\begin{aligned}
& AB + BC + CD + DE + EA = 64 \text{ and } \\
& AC + CE + EB + BD + DA = 72.
\end{aligned}
$$
Compute the perimeter of the convex pentagon whose vertices are the midpoints of the sides of $ABCDE$.

Suppose $P(x)$ is a cubic polynomial with integer coefficients such that $P(\sqrt{5})=5$ and $P(\sqrt[3]{5})=5 \sqrt[3]{5}$. Compute $P(5)$.

Let $n$ be a positive integer, and let $s$ be the sum of the digits of the base-four representation of $2^{n}-1$. If $s=2023$ (in base ten), compute $n$ (in base ten).

Two distinct 2-digit prime numbers $p$, $q$ can be written one after the other in 2 different ways to form two 4-digit numbers. For example, $11$ and $13$ yield $1113$ and $1311$. If the two 4-digit numbers formed are both divisible by the average value of $p$ and $q$, find all possible pairs \{$p, q$\}.

Richard starts with the string $HHMMMMTT$. A move consists of replacing an instance of $HM$ with $MH$, replacing an instance of $MT$ with $TM$, or replacing an instance of $TH$ with $HT$. Compute the number of possible strings he can end up with after performing zero or more moves.