Let $\{a_n\}$ be a sequence defined by $a_1=0$ and $$a_n=\frac{1}{n}+\frac{1}{\lceil \frac{n}{2} \rceil}\sum_{k=1}^{\lceil \frac{n}{2} \rceil}a_k$$for any positive integer $n$. Find the maximal term of this sequence.

Find the smallest real $\lambda$, such that for any positive integers $n, a, b$, such that $n \nmid a+b$, there exists a positive integer $1 \leq k \leq n-1$, satisfying $$\{\frac{ak} {n}\}+\{\frac{bk} {n}\} \leq \lambda.$$

Let $ABC$ be a triangle with $AB<BC<CA$ and let $D$ be a variable point on $BC$. The point $E$ on the circumcircle of $ABC$ is such that $\angle BAD=\angle BED$. The line through $D$ perpendicular to $AB$ meets $AC$ at $F$. Show that the measure of $\angle BEF$ is constant as $D$ varies.

If a right triangle can be covered by two unit circles, find the maximal area of the right triangle.

Let $n,m,r$ be positive integers such that $n>m$ and both $n^2+r, m^2+r$ are powers of $2$. Show that $n>\frac{2m^2}{r}$.

Let $n$ be a positive integer. If $x_1, x_2, \ldots, x_n \geq 0$, $x_1+x_2+\ldots+x_n=1$ and, assuming $x_{n+1}=x_1$, find the maximal value of $$\sum_{k=1}^n \frac{1+x_k^2+x_k^4}{1+x_{k+1}+x_{k+1}^2+x_{k+1}^3+x_{k+1}^4}.$$

It is known that there are $2024$ pairs of friends among $100$ people. Show that is possible to split them into $50$ pairs so that: (a) There are at most $20$ pairs that are friends with each other; (b) There are at least $23$ pairs that are friends with each other; (c) There are exactly $22$ pairs that are friends with each other.