VTMO 2026 - Grade 9 - Semifinal - SET 10 - Self-study

Triangle ABC is isosceles at A. D lies on BC with BC=3BD. CE is perpendicular to AD at E∈AD. Given angle AEB=128°, find angle BAC.

Find the sum of all prime values p of $p=|n(n-12)|$ for integer n.

Each of50 students belongs to at least one of Art, Music, and Science. Every Music member belongs to Art. Eight belong to all three, nine belong only to Science,20 belong to Music, and18 to Science. How many belong only to Art?

Equilateral triangle ABC is inscribed in a circle of radius3 centered at O. M lies on minor arc BC and MB=1. Find $P=MA^2+MC^2$.

For natural numbers a,b define $a*b=(a+1)(b+1)$. If a$>$b and a*b=13, find a.

P is quadratic with P(3)=0, P(5)=2, and P(7)=12. Find A=P(9)+P(11).

Right triangle P1QR1 has its right angle at Q, with P1Q=2 cm and R1Q=5 cm. Along ray QP1, successive points P2,…,Pn+1 are spaced1 cm apart; along ray QR1, successive R2,…,Rn+1 are likewise spaced1 cm apart, in the order shown. Find n for which quadrilateral PnPn+1Rn+1Rn has area2016 cm².

Find the area of the triangle formed by y=3x+4, y=2x+4, and the x-axis.

Binh has25 cards labeled1 through25. What is the maximum number of cards in a row if the greatest common divisor of every adjacent pair must be prime?

Evaluate $\sqrt{4-2\sqrt3}+\sqrt{7-4\sqrt3}$.

For all real x,y, $f(x+y)=x+f(y)$, and f(0)=2. Find f(2014).

Given f(4x+3)=2x+1, find f(−9).

512 people stand in a queue. Move the first person to the back, then seat the next person. Repeat these two steps until only one person remains standing. What was that person's original position?

A5×5 board has one lamp in each cell. Touching a lamp toggles it and every edge-adjacent lamp. In the figure, black means off and white means on: only the central cell is black, and all other24 cells are white. Can all lamps be switched off? Prove your answer.

Twenty-seven unit cubes form a3×3×3 cube with64 lattice vertices. How many pairs of vertices are at distance √5?

Phuong's luggage code has three digits from0–9. For923, exactly one digit is correct and correctly placed. For958, exactly one digit is correct but misplaced. For379, exactly two digits are correct but misplaced. For142, no digit is correct. For127, exactly one digit is correct but misplaced. Find the code.

Hai earns5/8 of Tuan's salary. Hai saves40% of her salary and spends half as much as Tuan spends. What percentage of his salary does Tuan save?

Hieu has10 pens,12 rulers, and14 notebooks, packed into18 gifts of two different kinds of item each. How many gifts contain a pen and a ruler?

Evaluate $\dfrac{(2102+2012)^2-(2102-2012)^2}{2102\times2012}$.

A boat takes3 hours downstream from A to B and4 hours upstream back to A. How long does a floating clump of water plants take to drift from A to B?