VTMO 2026 - Grade 9 - Final - SET 5 - Self-study

Let $a,b$ be integers such that $x^2+ax+1-b=0$ has two nonzero integer roots. Prove that $a^2+b^2$ is composite.

Let $p,q$ be consecutive primes with $p\gt q\gt 2$. Prove that $p+q$ has at least four positive divisors.

Solve $(1+x+x^2+x^3)(1+x+x^2+\cdots+x^9)=(1+x+\cdots+x^6)^2$.

Let $ABCD$ be a square with centre $O$. Let $M$ lie on $AB$, distinct from $A,B$. The line through $M$ parallel to $BD$ meets $AC$ at $N$. Lines $DN$ and $CM$ meet at $E$, and $DN$ meets $AB$ at $F$.

a) Prove that angle $DEC$ is constant as $M$ varies.

b) Prove that $E$ lies on the circle with centre $O$ and radius $OA$.

c) Find the position of $M$ that maximizes $MF$.

Let the isosceles trapezoid $ABCD$, with $AB\parallel CD$ and $AB\lt CD$, be inscribed in a circle with centre $O$. Let $K$ be the intersection of its diagonals. Let $M,N$ be the circumcentres of triangles $KAB,KCD$, respectively, and let $P,Q$ be the circumcentres of $KAD,KBC$, respectively.

a) Prove that $MQNP$ is a parallelogram.

b) Prove that $KM=ON$.