US-TST 2021

Determine all integers $s \geq 4$ for which there exist positive integers $a, b, c, d$ such that $s=a+b+c+d$ and $s$ divides $a b c+a b d+a c d+b c d$.

Points $A, V_{1}, V_{2}, B, U_{2}, U_{1}$ lie fixed on a circle $\Gamma$, in that order, and such that $B U_{2}>A U_{1}>B V_{2}>A V_{1}$. Let $X$ be a variable point on the $\operatorname{arc} V_{1} V_{2}$ of $\Gamma$ not containing $A$ or $B$. Line $X A$ meets line $U_{1} V_{1}$ at $C$, while line $X B$ meets line $U_{2} V_{2}$ at $D$. Prove there exists a fixed point $K$, independent of $X$, such that the power of $K$ to the circumcircle of $\triangle X C D$ is constant.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the inequality $$f(y)-\left(\frac{z-y}{z-x} f(x)+\frac{y-x}{z-x} f(z)\right) \leq f\left(\frac{x+z}{2}\right)-\frac{f(x)+f(z)}{2}$$ for all real numbers $x<y<z$.