Olympiad problems

2024 Harvard-MIT Mathematics Tournament, 19

Tags: guts

let $A_1A_2\ldots A_{19}$ be a regular nonadecagon. Lines $A_1A_5$ and $A_3A_4$ meet at $X.$ Compute $\angle A_7 X A_5.$

2024 Harvard-MIT Mathematics Tournament, 29

Tags: guts

For each prime $p,$ a polynomial $P(x)$ with rational coefficients is called $p$-[i]good[/i] if and only if there exist three integers $a,b,$ and $c$ such that $0 \le a < b < c < \tfrac{p}{3}$ and $p$ divides all the numerators of $P(a), P(b),$ and $P(c),$ when written in simplest form. Compute the number of ordered pairs $(r,s)$ of rational numbers such that the polynomial $x^3+10x^2+rx+s$ is $p$-good for infinitely many primes $p.$

2024 Harvard-MIT Mathematics Tournament, 11

Tags: guts

Let $ABCD$ be a rectangle such that $AB = 20$ and $AD = 24.$ Point $P$ lies inside $ABCD$ such that triangles $PAC$ and $PBD$ have areas $20$ and $24,$ respectively. Compute all possible areas of triangle $PAB.$

2024 Harvard-MIT Mathematics Tournament, 1

Tags: guts

Compute the sum of all integers $n$ such that $n^2-3000$ is a perfect square.

2024 HMNT, 32

Tags: guts

Let $ABC$ be an acute triangle and $D$ be the foot of altitude from $A$ to $BC.$ Let $X$ and $Y$ be points on the segment $BC$ such that $\angle{BAX} = \angle{YAC}, BX = 2, XY = 6,$ and $YC = 3.$ Given that $AD = 12,$ compute $BD.$

2025 Harvard-MIT Mathematics Tournament, 9

Tags: guts

Let $P$ and $Q$ be points selected uniformly and independently at random inside a regular hexagon $ABCDEF.$ Compute the probability that segment $\overline{PQ}$ is entirely contained in at least one of the quadrilaterals $ABCD,$ $BCDE,$ $CDEF,$ $DEFA,$ $EFAB,$ or $FABC.$

2024 Harvard-MIT Mathematics Tournament, 13

Tags: guts

Mark has a cursed six-sided die that never rolls the same number twice in a row, and all other outcomes are equally likely. Compute the expected number of rolls it takes for Mark to roll every number at least once.

2024 HMNT, 16

Tags: guts

Compute $$\frac{2+3+\cdots+100}{1}+\frac{3+4+\cdots+100}{1+2}+\cdots+\frac{100}{1+2+\cdots+99}.$$

2024 LMT Fall, 33

Tags: guts

Let $a$ and $b$ be positive real numbers that satisfy \begin{align*} \sqrt{a-ab}+\sqrt{b-ab}=\frac{\sqrt{6}+\sqrt{2}}{4} \,\,\, \text{and}\,\,\, \sqrt{a-a^2}+\sqrt{b-b^2}=\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)^2. \end{align*} Find the ordered pair $(a, b)$ such that $a>b$ and $a+b$ is maximal.

2025 Harvard-MIT Mathematics Tournament, 14

Tags: guts

A parallelogram $P$ can be folded over a straight line so that the resulting shape is a regular pentagon with side length $1.$ Compute the perimeter of $P.$

2024 LMT Fall, 16

Tags: guts

A new meme is circling around social media known as the [i]DaDerek Convertible[/i]. The license plate number of the [i]DaDerek Convertible[/i] is such that the product of its nonzero digits times $5$ is equal to itself. Given that its license plate number has less than or equal to $3$ digits and that it has at least one nonzero digit, find the [i]DaDerek Convertible[/i]'s license plate number.