Let $n$ be a positive integer, and let $[n]=\{1,2,\dots,n\}$. Find the maximum posible cardinality of a subset $S$ of $[n]$ with the property that there aren't any distinct $a,b,c\in S$ such that $a+b=c$.

Let $ABCD$ be a fixed convex quadrilateral. Say a point $K$ is pastanaga if there's a rectangle $PQRS$ centered at $K$ such that $A\in PQ, B\in QR, C\in RS, D\in SP$. Prove there exists a circle $\omega$ depending only on $ABCD$ that contains all pastanaga points.

Let $x_1,x_2,\dots, x_n$ be positive real numbers such that $x_1+x_2+\cdots + x_n=1$. Prove that $$\sum_{i=1}^n \frac{\min\{x_{i-1},x_i\}\cdot \max\{x_i,x_{i+1}\}}{x_i}\leq 1,$$where we denote $x_0=x_n$ and $x_{n+1}=x_1$.

Let $ABC$ be a triangle, and let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to sides $AC$ and $AB$, respectively. Let $P$ and $Q$ be the intersections of $EF$ with the tangents from $B$ and $C$ to $(ABC)$, respectively. If $M$ is the midpoint of $BC$, prove that $(PQM)$ is tangent to $BC$ at $M$.

Find all functions $f\colon \mathbb{R}^+ \to \mathbb{R}^+$ such that \[ f(xf(x)+y^2) = x^2+yf(y) \]for any positive reals $x,y$.

For each positive integer $k$, define $a_k$ as the number obtained from adding $k$ zeroes and a $1$ to the right of $2024$, all written in base $10$. Determine wether there's a $k$ such that $a_k$ has at least $2024^{2024}$ distinct prime divisors.