IMO 2023

Problem 1. Determine all composite integers $n>1$ that satisfy the following property: if $d_1,d_2,\dots ,d_k$ are all the positive divisors of $n$ with $1=d_1<d_2<\cdots <d_k=n$, then $d_i$ divides $d_{i+1}+d_{i+2}$ for every $1\le i\le k-2$.

Problem 2. Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E\ne A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P\ne B$.

Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.

Problem 3. For each integer $k\ge 2$, determine all infinite sequences of positive integers $a_1,a_2,\dots$ for which there exists a polynomial $P$ of the form $P(x)=x^k+c_{k-1}{x}^{k-1}+\cdots +c_{1}x+c_0$, where $c_0,c_1,\dots ,c_{k-1}$ are non-negative integers, such that

$P(a_n)=a_{n+1}{a}_{n+2}\cdots a_{n+k}$

for every integer $n\ge 1$.

Problem 4. Let $x_1,x_2,\dots ,x_{2023}$ be pairwise different positive real numbers such that

$a_n=\sqrt{(x_1+x_2+\cdots +x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}\right)}$

is an integer for every $n=1,2,\dots ,2023$. Prove that $a_{2023}\ge 3034$.

Problem 5. Let $n$ be a positive integer. A Japanese triangle consists of $1+2+\cdots +n$ circles arranged in an equilateral triangular shape such that for each $i=1,2,\dots ,n$, the $i^{\text{th}}$ row contains exactly $i$ circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of $n$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with $n=6$, along with a ninja path in that triangle containing two red circles.

In terms of $n$, find the greatest $k$ such that in each Japanese triangle there is a ninja path containing at least $k$ red circles.

Problem 6. Let $ABC$ be an equilateral triangle. Let $A_1$, $B_1$, $C_1$ be interior points of $ABC$ such that $B{A}_1=A_{1}C$, $C{B}_1=B_{1}A$, $A{C}_1=C_{1}B$, and

$\angle B{A}_{1}C+\angle C{B}_{1}A+\angle A{C}_{1}B=480^{\circ }.$

Let $B{C}_1$ and $C{B}_1$ meet at $A_2$, let $C{A}_1$ and $A{C}_1$ meet at $B_2$, and let $A{B}_1$ and $B{A}_1$ meet at $C_2$.

Prove that if triangle $A_1{B}_1{C}_1$ is scalene, then the three circumcircles of triangles $A{A}_1{A}_2$, $B{B}_1{B}_2$ and $C{C}_1{C}_2$ all pass through two common points.

(Note: a scalene triangle is one where no two sides have equal length.)