VTMO 2026 - Grade 9 - Final - SET 3 - Self-study

Find the prime numbers $p$ and $q$ satisfying $2p+6q=(p-q)^3$.

For two nonnegative real numbers $a,b$, their arithmetic and geometric means are $\dfrac{a+b}{2}$ and $\sqrt{ab}$, respectively. For example, the numbers 2 and 8 have arithmetic mean 5 and geometric mean 4. Find two distinct positive integers whose arithmetic and geometric means are two-digit numbers made from the same two digits in opposite orders.

A circle $(O)$ has a fixed chord $BC$ that is not a diameter. A point $A$ moves on the major arc $BC$, with $AB\ne AC$. Draw the angle bisector $AD$ and median $AM$ of $\triangle ABC$, where $D,M\in BC$. The circumcircle of $ADM$ meets the lines $AB,AC$ again at $E,F$, respectively.

a) Prove that $BE=CF$.

b) Let $I$ be the midpoint of $EF$, and let line $MI$ meet $AB,AC$ at $H,G$, respectively. Prove that the circumcircle of $AHG$ always passes through a fixed point as $A$ moves on the major arc $BC$.

Let $K$ be a set of $n$ elements, where $n\ge5$. Consider any $n+1$ distinct three-element subsets of $K$.

a) Prove that one of these subsets, $A$, has the property that each element of $A$ also belongs to another subset in the family.

b) For $n=5$, prove that two of the six subsets have exactly one common element.

c) For $n=9$, prove that two of the ten subsets have exactly one common element.

A quadrilateral $ABCD$ is circumscribed about a circle $(O)$. Let $M,N,P,Q$ be its points of tangency with the sides $AB,BC,CD,DA$, respectively. Let $I=NQ\cap AC$.

a) Prove that $\dfrac{IA}{IC}=\dfrac{QA}{NC}$.

b) Prove that the four lines $AC,BD,MP,NQ$ are concurrent.