VTMO 2026 - Grade 9 - Semifinal - SET 05 - Self-study

The product of five distinct odd primes is a five-digit number $\overline{abcab}$ with $c=0$. How many such sets of five primes exist?

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 $MF$.

A polynomial $P(x)$ leaves remainder $-4$ on division by $2x+1$ and remainder 5 on division by $5x-2$. Its quotient on division by $10x^2+x-2$ is $x$, with a remainder. Find $P(2)$.

Points $A,B,C,D$ lie in that order on a circle of radius1. Suppose $AB\parallel CD$, $AB=CD/2$ and $AC\perp BD$. Find the area of $ABCD$.

Representatives of five countries sit around a round table. Each country may have several representatives. Every pair of countries has representatives seated next to each other. What is the minimum number of people?

Both real roots of $P(x)=x^2-2019x+c$ are primes. Find $c$.

A three-digit integer becomes a perfect square both when its first digit is removed and when its last digit is removed. Find the sum of all such integers.

Real numbers $a,b,c$ satisfy $a+b+c=10$ and $ab+bc+ca=15$. Find $A=a^2+b^2+c^2$.

Using factorial notation, find the largest integer $n$ such that $2016^n$ divides $59!$.

For positive real numbers $a,b,c$ with $a^2+b^2+c^2=3$, find the smallest real $k$ such that $\min\{a+b,b+c,c+a\}\gt k$ guarantees that $a,b,c$ are side lengths of a triangle.

Distribute100 apples into20 bags so that each bag contains exactly3 apples or $x$ apples. How many values of $x$ are possible?

In the figure, triangle $ABC$ is isosceles with $AB=AC$. Equilateral triangle $DEF$ lies inside it, with $D\in AB,E\in BC,F\in AC$. Given $\angle ADF=50^\circ$ and $\angle A=\frac12\angle DEF$, find $\angle BED$.

Let $AD,BE$ be angle bisectors of triangle $ABC$. The circumcircles of triangles $ADC,BEC$ meet again at $F$ on $AB$. Find $\angle ACB$.

Evaluate $2012^2-2011\cdot2013$.

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

Three cube-shaped boxes have integer side lengths in centimeters. The first can contain the second, and the second can contain the third. The first box has volume $218\,\mathrm{cm}^3$ greater than the third. Find the third box's volume.

The sum of the squares of three consecutive odd natural numbers is a four-digit number with all digits equal. Let $S$ be the sum of those three numbers. Find the sum of the digits of $S$.

In triangle $ABC$, $AD$ bisects angle $A$, with $D\in BC$, and $\angle ADB:\angle ADC=4:5$. The perpendicular bisector of $BC$ meets $AC$ at $M$. Find $\angle ABM$.

An has ten segments of lengths $1,2,\ldots,10$. How many choices of three segments form a triangle?

A team plays90 matches and wins40 of its first50. How many remaining matches must it win to finish with exactly70% wins?