Let $n$ be a positive integer. $n$ letters are written around a circle, each $A$, $B$, or $C$. When the letters are read in clockwise order, the sequence $AB$ appears $100$ times, the sequence $BA$ appears $99$ times, and the sequence $BC$ appears $17$ times. How many times does the sequence $CB$ appear?

Let $ABCD$ be a rhombus. Eight additional points $X_1$, $X_2$, $Y_1$, $Y_2$, $Z_1$, $Z_2$, $W_1$, $W_2$ were chosen so that the quadrilaterals $AX_1BX_2$, $BY_1CY_2$, $CZ_1DZ_2$, $DW_1AW_2$ are squares. Prove that the eight new points lie on two straight lines.

Bart wrote the digit "$1$" $2024$ times in a row. Then, Lisa wrote an additional $2024$ digits to the right of the digits Bart wrote, such that the resulting number is a square of an integer. Find all possibilities for the digits Lisa wrote.

A $100\times \sqrt{3}$ rectangular table is given. What is the minimum number of disk-shaped napkins of radius $1$ required to cover the table completely? Remark: The napkins are allowed to overlap and protrude the table's edges.

$2024$ otters live in the river. Some are friends with each other. Is it possible that, for any collection of $1012$ otters, there is exactly one additional otter that is friends with all $1012$ otters?

Let $a$, $b$ and $c$ be non-negative numbers such that $ab+ac+bc+abc=4.$ . Prove that: $$\sqrt{\frac{ab+ac+1}{a+2}}+\sqrt{\frac{ab+bc+1}{b+2}}+\sqrt{\frac{ac+bc+1}{c+2}}\leq3.$$PSDedicated to dear KhuongTrang

For a positive integer $n$, let $A_n$ be the set of quadruplets $(a,b,c,d)$ of integers, satisfying the following properties simultaneously: $0\le a\le c\le n,$ $0\le b\le d\le n,$ $c+d>n,$ and $bc=ad+1.$ Moreover, define $$\alpha_n=\sum_{(a,b,c,d)\in A_n}\frac{1}{ab+cd}.$$Find all real numbers $t$ such that $\alpha_n>t$ for every positive integer $n$.