Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. Let $\ell$ be the tangent to $\omega$ parallel to $BC$ and distinct from $BC$. Let $D$ be the intersection of $\ell$ and $AC$, and let $M$ be the midpoint of $\overline{ID}$. Prove that $\angle AMD = \angle DBC$.

Let $\mathcal{F}$ be a family of (distinct) subsets of the set $\{1,2,\dots,n\}$ such that for all $A$, $B\in \mathcal{F}$,we have that $A^C\cup B\in \mathcal{F}$, where $A^C$ is the set of all members of ${1,2,\dots,n}$ that are not in $A$. Prove that every $k\in {1,2,\dots,n}$ appears in at least half of the sets in $\mathcal{F}$. Stijn Cambie, Mohammad Javad Moghaddas Mehr

We call a pair of distinct numbers $(a, b)$ a binary pair if $ab+1$ is a power of two. Given a set $S$ of $n$ positive integers, what is the maximum possible numbers of binary pairs in S?

Let $n$ be a positive integer. The numbers $1, 2, \dots, 2n+1$ are arranged in a circle in that order, and some of them are marked. We define, for each $k$ such that $1\leq k \leq 2n+1$ , the interval $I_k$ to be the closed circular interval starting at $k$ and ending in $k+n$ (taking remainders mod(2n+1)). We call in interval magical if it contains strictly more than half of all the marked elements. Prove that the following two statements are equivalent: 1. At least $n+1$ of the intervals $I_1, I_2, \dots, I_{2n+1}$ are magical 2. The number of marked numbers is odd

Let $\omega$ be a semicircle with diamater $AB$. Let $M$ be the midpoint of $AB$. Let $X,Y$ be points on the same semiplane with $\omega$ with respect to the line $AB$ such that $AMXY$ is a parallelogram. Let $XM\cap \omega = C$ and $YM \cap \omega = D$. Let $I$ be the incenter of $\triangle XYM$. Let $AC \cap BD= E$ and $ME$ intersects $XY$ at $T$. Let the intersection point of $TI$ and $AB$ be $Q$ and let the perpendicular projection of $T$ onto $AB$ be $P$. Prove that $M$ is midpoint of $PQ$

Find all functions $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x+yf(x)) = xf(1+y)$ for all x, y positive reals.