Let $u, v$ be arbitrary positive real numbers. Prove that \[\min{(u, \frac{100}{v}, v+\frac{2023}{u})} \leq \sqrt{2123}.\]

In an acute triangle $ABC$ the points $D, E$ are the feet of the altitudes through $B, C$ respectively. Let $L$ be the point on segment $BD$ such that $AD=DL$. Similarly, let $K$ be the point on segment $CE$ such that $AE=EK$. Let $M$ be the midpoint of $KL$. The circumcircle of $ABC$ intersect the lines $AL$ and $AK$ for a second time at $T, S$ respectively. Prove that the lines $BS, CT, AM$ are concurrent.

Five girls and five boys took part in a competition. Suppose that we can number the boys and girls $1, 2, 3, 4, 5$ such that for each $1 \leq i,j \leq 5$, there are exactly $|i-j|$ contestants that the girl numbered $i$ and the boy numbered $j$ both know. Let $a_i$ and $b_i$ be the number of contestants that the girl numbered $i$ knows and the number of contestants that the boy numbered $i$ knows respectively. Find the minimum value of $\max(\sum\limits_{i=1}^5a_i, \sum\limits_{i=1}^5b_i)$. (Note that for a pair of contestants $A$ and $B$, $A$ knowing $B$ doesn't mean that $B$ knows $A$ and a contestant cannot know themself.)

Find all functions $f : \mathbb{R} \to \mathbb{R}$ and $h : \mathbb{R}^2 \to \mathbb{R}$ such that \[f(x+y-z)^2=f(xy)+h(x+y+z, xy+yz+zx)\]for all real numbers $x,y,z$.

There are $n$ students in a class, and some pairs of these students are friends. Among any six students, there are two of them that are not friends, and for any pair of students that are not friends there is a student among the remaining four that is friends with both of them. Find the maximum value of $n$.

Let $m$ be a positive integer. We say that a sequence of positive integers written on a circle is good , if the sum of any $m$ consecutive numbers on this circle is a power of $m$. 1. Let $n \geq 2$ be a positive integer. Prove that for any good sequence with $mn$ numbers, we can remove $m$ numbers such that the remaining $mn-m$ numbers form a good sequence. 2. Prove that in any good sequence with $m^2$ numbers, we can always find a number that was repeated at least $m$ times in the sequence.