VTMO 2026 - Grade 9 - Final - SET 2 - Self-study

Let $A=\{1,2,\ldots,100\}$, and let $B_k$ be a $k$-element subset of $A$ such that the sum of any two distinct elements of $B_k$ is not divisible by their difference. Find the largest possible value of $k$.

Consider all four-digit numbers from $1111$ to $9999$. Concatenating them in any order gives a number with $35556$ digits. Prove that this number is divisible by $11$.

A number of students stand in a circle. Their teacher asks neighboring students to shake hands. Let $b$ and $g$ be the numbers of boys and girls, respectively, and let $B$ and $G$ be the numbers of boy-boy and girl-girl handshakes. Prove that $b-g=B-G$.

Let $a,b,c$ be distinct positive integers such that the product of any two of them plus 1 is a perfect square. Write $ab+1=r^2$, $ac+1=s^2$, $bc+1=t^2$, where $r,s,t$ are positive integers. Suppose $0<a<b<c$. Prove that $\dfrac{rs}{t}$ is not a positive integer.

Find all integer solutions with $x>y$ of $$2x^3-9x^2y+7xy^2-3x+3y-3=0.$$