International Mathematical Olympiad · 2023

Problems

Problem 1Determine 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 $1\le i\le k-2$ .Solutions: 1
Problem 2Let $ABC$ 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 $\angle BAC$ .Solutions: 1
Problem 3For each integer $k\ge2$ , determine all infinite sequences of positive integers $a_1,a_2,\ldots$ for which there exists a polynomial $P$ of the form $P(x)=x^k+c_{k-1}x^{k-1}+\cdots+c_1x+c_0$ , where $c_0,c_1,\ldots,c_{k-1}$ are non-negative integers, such that $P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k}$ for every integer $n\ge1$ .Solutions: 1
Problem 4Let $x_1,x_2,\ldots,x_{2023}$ be pairwise different positive real numbers such that $a_n=(x_1+x_2+\cdots+x_n)\left(\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_n}\right)$ is an integer for every $n=1,2,\ldots,2023$ . Prove that $a_{2023}\ge3034$ .Solutions: 1
Problem 5Let $n$ be a positive integer. A Japanese triangle consists of $1+2+\cdots+n$ circles arranged in an equilateral triangular shape such that for each $i=1,2,\ldots,n$ , the $i$ -th row contains exactly $i$ circles, exactly one of which is colored 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. 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.Solutions: 1
Problem 6Let $ABC$ 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 $A_2=BC_1\cap CB_1$ , $B_2=CA_1\cap AC_1$ , $C_2=AB_1\cap BA_1$ . Prove that if triangle $A_1B_1C_1$ is scalene, then the circumcircles of triangles $AA_1A_2$ , $BB_1B_2$ , and $CC_1C_2$ all pass through two common points.Solutions: 1