OFM 2023

Let $u_0, u_1, u_2, \ldots$ be integers such that $u_0 = 100$ ; $u_{k+2} \geqslant 2 + u_k$ for all $k \geqslant 0$ ; and $u_{\ell+5} \leqslant 5 + u_\ell$ for all $\ell \geqslant 0$ . Find all possible values for the integer $u_{2023}$ .

On her blackboard, Alice has written $n$ integers strictly greater than $1$ . Then, she can, as often as she likes, erase two numbers $a$ and $b$ such that $a \neq b$ , and replace them with $q$ and $q^2$ , where $q$ is the product of the prime factors of $ab$ (each prime factor is counted only once). For instance, if Alice erases the numbers $4$ and $6$ , the prime factors of $ab = 2^3 \times 3$ and $2$ and $3$ , and Alice writes $q = 6$ and $q^2 =36$ . Prove that, after some time, and whatever Alice's strategy is, the list of numbers written on the blackboard will never change anymore.

Note: The order of the numbers of the list is not important.

Let $\Gamma$ and $\Gamma'$ be two circles with centres $O$ and $O'$ , such that $O$ belongs to $\Gamma'$ . Let $M$ be a point on $\Gamma'$ , outside of $\Gamma$ . The tangents to $\Gamma$ that go through $M$ touch $\Gamma$ in two points $A$ and $B$ , and cross $\Gamma'$ again in two points $C$ and $D$ . Finally, let $E$ be the crossing point of the lines $AB$ and $CD$ . Prove that the circumcircles of the triangles $CEO'$ and $DEO'$ are tangent to $\Gamma'$ .

Find all integers $n \geqslant 0$ such that $20n+2$ divides $2023n+210$ .

Let $P(X) = a_n X^n + a_{n-1} X^{n-1} + \cdots + a_1 X + a_0$ be a polynomial with real coefficients such that $0 \leqslant a_i \leqslant a_0$ for $i = 1, 2, \ldots, n$ . Prove that, if $P(X)^2 = b_{2n} X^{2n} + b_{2n-1} X^{2n-1} + \cdots + b_{n+1} X^{n+1} + \cdots + b_1 X + b_0$ , then $4 b_{n+1} \leqslant P(1)^2$ .

Let $k$ be a positive integer. Scrooge McDuck owns $k$ gold coins. He also owns infinitely many boxes $B_1, B_2, B_3, \ldots$ Initially, bow $B_1$ contains one coin, and the $k-1$ other coins are on McDuck's table, outside of every box. Then, Scrooge McDuck allows himself to do the following kind of operations, as many times as he likes:

• if two consecutive boxes $B_i$ and $B_{i+1}$ both contain a coin, McDuck can remove the coin contained in box $B_{i+1}$ and put it on his table;
• if a box $B_i$ contains a coin, the box $B_{i+1}$ is empty, and McDuck still has at least one coin on his table, he can take such a coin and put it in box $B_{i+1}$ . As a function of $k$ , which are the integers $n$ for which Scrooge McDuck can put a coin in box $B_n$ ?

Let $ABCD$ be a convex quadrilateral, with $\measuredangle ABC > 90^\circ$ , $\measuredangle CDA > 90^\circ$ and $\measuredangle DAB = \measuredangle BCD$ . Let $E$ , $F$ and $G$ be the reflections of $A$ with respect to the lines $BC$ , $CD$ and $DB$ . Finally, let the line $BD$ meet the line segment $AE$ at a point $K$ , and the line segment $AF$ at a point $L$ . Prove that the circumcircles of the triangles $BEK$ and $DFL$ are tangent to each other at $G$ .

Do there exist integers $a$ and $b$ such that none of the numbers $a,a+1,\ldots,a+2023,b,b+1,\ldots,b+2023$ divides any of the $4047$ other numbers, but $a(a+1)(a+2)\cdots(a+2023)$ divides $b(b+1)\cdots(b+2023)$ ?