Olympiad problems

2024 Irish Math Olympiad, P5

Let $A,B,C$ be three points on a circle $\gamma$, and let $L$ denote the midpoint of segment $BC$. The perpendicular bisector of $BC$ intersects the circle $\gamma$ at two points $M$ and $N$, such that $A$ and $M$ are on different sides of line $BC$. Let $S$ denote the point where the segments $BC$ and $AM$ intersect. Line $NS$ intersects the circumcircle of $\triangle ALM$ at two points $D$ and $E$, with $D$ lying in the interior of the circle $\gamma$. (a) Prove that $M$ is the circumcentre of $\triangle BCD$. (b) Prove that the circumcircles of $\triangle BCD$ and $\triangle ADN$ are tangent at the point $D$.

2024 Irish Math Olympiad, P3

Tags: function , positive integers , irmo , modular arithmetic , number theory

Let $\mathbb{Z}_+=\{1,2,3,4...\}$ be the set of all positive integers. Determine all functions $f : \mathbb{Z}_+ \mapsto \mathbb{Z}_+$ that satisfy: [list] [*]$f(mn)+1=f(m)+f(n)$ for all positive integers $m$ and $n$; [*]$f(2024)=1$; [*]$f(n)=1$ for all positive $n\equiv22\pmod{23}$. [/list]

2024 Irish Math Olympiad, P6

Tags: combinatorics , irmo

Find all positive integers $n$ and $m$ such that $$\dbinom{n}{1} + \dbinom{n}{3} = 2^m.$$

2024 Irish Math Olympiad, P2

Tags: Integers , irmo

A non-negative integer $p$ is a [i]3-choice[/i] if $\dfrac{k(k-1)(k-2)}{6}$ for some positive integer $k$. Let $p$ and $q$ be 3-choices with $p<q$. Show there is an integer $n$ such that $p \leq n^2 < q$.

2024 Irish Math Olympiad, P4

Tags: number theory , Digits , irmo

How many 4-digit numbers $ABCD$ are there with the property that $|A-B|= |B-C|= |C-D|$? Note that the first digit $A$ of a four-digit number cannot be zero.

2024 Irish Math Olympiad, P9

Tags: geometry , cyclic quadrilateral , irmo

Let $K, L, M$ denote three points on the sides $BC$, $AB$ and $BC$ of $\triangle{ABC}$, so that $ALKM$ is a parallelogram. Points $S$ and $T$ are chosen on lines $KL$ and $KM$ respectively, so that the quadrilaterals $AKBS$ and $AKCT$ are both cyclic. Prove that $MLST$ is cyclic if and only if $K$ is the midpoint of $BC$.

2024 Irish Math Olympiad, P7

Tags: Coin , irmo

A game of coins is played as follows: You start with $1$ head and $1$ tail on a table. At each turn, you can perform any one of the following moves: [list=a] [*]You can turn over all the coins on the table. [*]You can triple the number of heads and tails at the table. [*]If there are at least $4$ tails on the table, you can turn over $4$ tails. [*]If there are at least $5$ tails on the table, you can turn over $3$ of the tails and discard $2$ of the tails. [/list] Knowing that at the end of the game you have $2024$ heads, what are all possible numbers of tails at the end of that game?