VTMO 2026 - Grade 8 - Semifinal - SET 07 - Self-study

How many three-digit palindromes are there? A palindrome reads the same forwards and backwards.

Positive integers are written consecutively in rows: row 1 contains 1; row 2 contains 2, 3; row 3 contains 4, 5, 6; and so on. Which row contains 2015?

How many triangles have positive integer side lengths $x\lt y\lt z$ and perimeter $27$?

Seven bags contain 10, 15, 20, 25, 30, 35 and 40 candies. Choose six bags and give them to two children so that the children receive equal numbers of candies, without opening any bag. How many ways are there?

At 10:00, after how many minutes will the hands of a clock first be perpendicular?

Quadrilateral $ABCD$ has right angles at $A,C$, with $AB=11$, $BC=7$, $CD=9$. Find its area.

A cake is cut into equal pieces so that it can be shared equally among 3, 5 or 6 children. Which listed number of pieces is suitable?

A circle and a quadrilateral are drawn in the plane. What is the greatest possible number of regions into which they divide the plane?

A car travels at $55$ km/h. Its odometer initially reads the three-digit number $\overline{abc}$ and after a positive integer number of hours reads $\overline{cba}$. Given $a+b+c\le7$, find $a^2+b^2+c^2$.

A hexagon is cut repeatedly, each straight cut dividing one existing piece into two and not passing through a vertex. The total number of sides of all resulting pieces is 90. How many cuts were made?

In the shown configuration, $AB=BD=BC$ and $\angle DBC=76^\circ$. Find $\angle DBC$.

Select odd numbers from the integers 123 through 2016. What is the minimum number that guarantees two selected numbers have sum 2016?

Find all real pairs $(a,b)$ such that $|ax+by|+|bx+ay|=|x|+|y|$ for every real $x,y$.

A trapezoid has perpendicular diagonals of lengths 15 and 20. Find its height.

For which listed value of $x$ is $P(x)=(x-2)(x-1)x(x+1)(x+2)$ positive?

How many positive integer pairs $(x,y)$ satisfy $x\lt y$, $x+y=2020$, and $xy$ is divisible by $20$?

Points $A,B,C$ are collinear in that order. Equilateral triangles $ABD$ and $BCE$ lie on the same side of $AC$. A point $N$ on $BE$ satisfies $NE:NB=4:5$. Line $AN$ meets $BD$ at $M$ and $CE$ at $P$. If $AB=16$ and triangles $AMB,ENP$ have equal areas, find $BC$.

Rectangles $ABCD$ and $MNPQ$ are each divided into two rectangles, numbered 1,2 and 3,4 as shown. The area of rectangle 2 is four times that of rectangle 1, and its perimeter is twice that of rectangle 1. The same relations hold for rectangle 4 compared with rectangle 3. Given $AD=MN$, find $4AB/MQ$.

Points $B,C,D$ are collinear in that order, with $BD=6CD$. Equilateral triangles $ABC$ and $ECD$ lie on the same side of $BD$. If the area of triangle $ACE$ is $5$, find the area of quadrilateral $ABDE$.

In trapezoid $ABCD$, $AB\parallel CD$ and diagonals $AC,BD$ intersect at $I$. The area of triangle $ICD$ equals half the area of the trapezoid.
(a) Find $AB/CD$.
(b) Let $M$ be the midpoint of $AB$, and let ray $MI$ meet $CD$ at $N$. Choose $J$ on $IN$. The lines through $J$ parallel to $AC$ and $BD$ meet $AD$ and $BC$ at $E$ and $F$, respectively. If $E,I,F$ are collinear, prove $IM=JN$.