IMO 2023

Determine all composite integers $n>1$ that satisfy the following property: if $d_1$ , $d_2$ , $\ldots$ , $d_k$ 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 $1 \leq i \leq k - 2$ .

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

For each integer $k\geq 2$ , determine all infinite sequences of positive integers $a_1$ , $a_2$ , $\ldots$ for which there exists a polynomial $P$ of the form $$P(x)=x^k+c_{k-1}x^{k-1}+\dots + c_1 x+c_0,$$ 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 $n\geq 1$ .

Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that $$a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}$$ is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$

Let $n$ be a positive integer. A Japanese triangle consists of $1 + 2 + \dots + n$ circles arranged in an equilateral triangular shape such that for each $i = 1$ , $2$ , $\dots$ , $n$ , the $i^{th}$ 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. [asy] // credit to vEnhance for the diagram (which was better than my original asy): size(4cm); pair X = dir(240); pair Y = dir(0); path c = scale(0.5)unitcircle; int[] t = {0,0,2,2,3,0}; for (int i=0; i<=5; ++i) { for (int j=0; j<=i; ++j) { filldraw(shift(iX+jY)c, (t[i]==j) ? lightred : white); draw(shift(iX+jY)c); } } draw((0,0)--(X+Y)--(2X+Y)--(3X+2Y)--(4X+2Y)--(5X+2Y),linewidth(1.5)); path q = (3,-3sqrt(3))--(-3,-3sqrt(3)); draw(q,Arrows(TeXHead, 1)); label(" $n = 6$ ", q, S); label(" $n = 6$ ", q, S); [/asy] 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.

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$ , $CB_1=B_1A$ , $AC_1=C_1B$ , and $$\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ$$ Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$ Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$ and $CC_1C_2$ all pass through two common points.

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

Proposed by Ankan Bhattacharya, USA