Question 1
Triangle $ABC$ is isosceles with $AB=AC$. Equilateral triangle $DEF$ lies inside it, with $D,E,F$ on $AB,BC,AC$, respectively, as shown. Given $\angle ADF=50^\circ$ and $\angle A=\tfrac12\angle DEF$, find $\angle BED$.
VTMO 2026 - Grade 8 - Semifinal - SET 3 - Self-study
Question 2
Parallelogram $ABCD$ has area $144\text{ cm}^2$. Point $M$ lies on $AB$. As shown, $O=AC\cap BD$, $E=AC\cap DM$ and $F=BD\cap CM$. The two hatched triangles $ADE$ and $BCF$ have total area $48\text{ cm}^2$. Find the area of quadrilateral $MFOE$.
Question 3
Triangle $ABC$ is right-angled at $A$ and satisfies $BC=AC+\tfrac12 AB$. Find $AB:AC$.
Question 4
Which of the following triangles, specified by their three side lengths, has the greatest area?
Question 5
If $x-y>x$ and $x+y>y$, which conclusion is correct?
Question 6
In 2023, Huy notices that his age and his teacher Hùng's age each equal the sum of the digits of that person's birth year. Find their age difference.
Question 7
Positive integers $a,b$ satisfy $a\lt b\lt100$ and $9a>7b$. Find the greatest possible value of $b-a$.
Question 8
Natural numbers $a,b$ satisfy $7a+11b=2020$. When $11a+7b$ is as large as possible, find $ab$.
Question 9
A string consists of digits 0 and 1. Its first and last digits are 0, it contains at least ten 1s, and every four consecutive digits contain a 0. What is its minimum length?
Question 10
When $A(x)=x^{10}-x^2-1$ is divided by $B(x)=x^3-2$, the remainder is $r(x)=ax^2+bx+c$. Find $a+b+c$.
Question 11
In square $PQRS$, $M,N$ are the midpoints of $PS,RS$, respectively. Triangle $SMN$ has area $18\text{ cm}^2$. Find the area of triangle $QMN$.
Question 12
How many non-integer rational numbers $q$ satisfy that $20q$ is a positive integer less than 170?
Question 13
A rectangle and a square have equal areas, and all their side lengths are positive integers. The rectangle's length exceeds its width by 38. Find the positive difference between their perimeters.
Question 14
Natural numbers $a,b,c$ are described by three students. X says (1) $a^2+b^2+c^2=165$, (2) $bc=28$. Y says (1) $c>b$, (2) $b+c$ is divisible by 9. Z says (1) $c$ is divisible by 8, (2) $c=4$. Each student makes exactly one true and one false statement. Find $a+2b+3c$.
Question 15
Find the units digit of $1+4+4^2+4^3+\cdots+4^{100}$.
Question 16
Find the greatest positive integer $n$ for which $\underbrace{11\ldots11}_{2n\text{ digits}}-\underbrace{77\ldots77}_{n\text{ digits}}$ is a perfect square.
Question 17
Consider the sequence $\frac34,\frac5{36},\frac7{144},\frac9{400},\ldots$. What is the numerator of the sum of its first 20 terms when written in lowest terms?
Question 18
The balanced mobiles shown have total hanging-object mass 112 grams. Ignore the rods and strings. How many grams does the star weigh?
Question 19
The lengths of the four sides and one diagonal of a quadrilateral are, in some order, $1,2,2.8,5,7.5$. What is the diagonal's length?
Question 20
In trapezoid $ABCD$, $AB\parallel CD$. Point $I$ is the midpoint of $DC$, $IH\perp AB$ with $H\in AB$, and $M,N$ are the midpoints of $AD,BC$. Given $IH=IM=IN=ID$, find $\angle DAB$.