VTMO 2026 - Grade 9 - Final - SET 6 - Self-study

Triangle $ABC$ has $AB=5$, $AC=7$, and $BC=8$. Let $I$ be its incentre.

a) Find the inradius $r$.

b) Points $M,N$ lie on $BC,AC$ with $BM=5$ and $AN=5$. Lines $AM,BN$ meet at $K$. Prove that $K$ lies on the incircle of $ABC$.

a) Solve $x^2+3x\sqrt{x+1}=\dfrac{27}4(x+1)$.

b) Find all pairs of primes $(p,q)$ such that $p^2+5pq+4q^2$ is a perfect square.

a) Solve
$$\begin{cases}(2x+3)\sqrt{4x-1}+(2y+3)\sqrt{4y-1}=2\sqrt{(2x+3)(2y+3)},\\x+y=4xy.\end{cases}$$
b) Let $x,y,z\gt 1$ satisfy $x+y+z=\dfrac43xyz$. Find the minimum of
$$P=\frac{y-2}{x^2}+\frac{z-2}{y^2}+\frac{x-2}{z^2}.$$

Solve $(\sqrt{x+24}+\sqrt x)(\sqrt{x+15}-\sqrt{x+3})=12$.

There are three piles containing 51, 49, and 5 stones. In one operation, two piles may be merged, or a pile containing an even number of stones may be divided into two equal piles. Find the smallest number of stones in a pile that can be obtained by a sequence of these operations.