OIM 2020

Let $ABC$ be an acute scalene triangle such that $AB <AC$ . The midpoints of sides $AB$ and $AC$ are $M$ and $N$ , respectively. Let $P$ and $Q$ be points on the line $MN$ such that $\angle CBP = \angle ACB$ and $\angle QCB = \angle CBA$ . The circumscribed circle of triangle $ABP$ intersects line $AC$ at $D$ ( $D\ne A$ ) and the circumscribed circle of triangle $AQC$ intersects line $AB$ at $E$ ( $E \ne A$ ). Show that lines $BC, DP,$ and $EQ$ are concurrent.

Nicolás De la Hoz, Colombia

Let $T_n$ denotes the least natural such that $$n\mid 1+2+3+\cdots +T_n=\sum_{i=1}^{T_n} i$$ Find all naturals $m$ such that $m\ge T_m$ .

Proposed by Nicolás De la Hoz

Let $n\ge 2$ be an integer. A sequence $\alpha = (a_1, a_2,..., a_n)$ of $n$ integers is called Lima if $\gcd \{a_i - a_j \text{ such that } a_i> a_j \text{ and } 1\le i, j\le n\} = 1$ , that is, if the greatest common divisor of all the differences $a_i - a_j$ with $a_i> a_j$ is $1$ . One operation consists of choosing two elements $a_k$ and $a_{\ell}$ from a sequence, with $k\ne \ell$ , and replacing $a_{\ell}$ by $a'_{\ell} = 2a_k - a_{\ell}$ . Show that, given a collection of $2^n - 1$ Lima sequences, each one formed by $n$ integers, there are two of them, say $\beta$ and $\gamma$ , such that it is possible to transform $\beta$ into $\gamma$ through a finite number of operations.

Notes. The sequences $(1,2,2,7)$ and $(2,7,2,1)$ have the same elements but are different. If all the elements of a sequence are equal, then that sequence is not Lima.

Show that there exists a set $\mathcal{C}$ of $2020$ distinct, positive integers that satisfies simultaneously the following properties: $\bullet$ When one computes the greatest common divisor of each pair of elements of $\mathcal{C}$ , one gets a list of numbers that are all distinct. $\bullet$ When one computes the least common multiple of each pair of elements of $\mathcal{C}$ , one gets a list of numbers that are all distinct.

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(xf(x-y))+yf(x)=x+y+f(x^2),$$ for all real numbers $x$ and $y.$

Let $ABC$ be an acute, scalene triangle. Let $H$ be the orthocenter and $O$ be the circumcenter of triangle $ABC$ , and let $P$ be a point interior to the segment $HO.$ The circle with center $P$ and radius $PA$ intersects the lines $AB$ and $AC$ again at $R$ and $S$ , respectively. Denote by $Q$ the symmetric point of $P$ with respect to the perpendicular bisector of $BC$ . Prove that points $P$ , $Q$ , $R$ and $S$ lie on the same circle.