VTMO 2026 - Grade 6 - Final - SET 2 - Self-study

Let $A=1+2+2^2+2^3+\cdots+2^{2014}+2^{2015}$. Find the remainder when $A$ is divided by 31.

Positive integers $a,b,c,d$ satisfy
$$\gcd(a,b,c)=b,\qquad \gcd(b,c,d)=c,$$
$$\operatorname{lcm}(a,b,c)=60,\qquad \operatorname{lcm}(b,c,d)=600.$$
Find the smallest possible value of $a+b+c+d$.

Two taps can fill a tank. The following plans each fill it:
• Plan 1: run tap A for 7 hours, close it, then run tap B for 14 hours.
• Plan 2: run tap A for 10 hours, close it, then run tap B for 2 hours.

For Plan 3, run tap A for 5 hours, close it, then run tap B. How long must tap B run?

Model clarification: each plan starts with the same empty tank; both taps have constant flow rates and no water is lost.

Vân travels from A to B and Phong from B to A.
• If they start at the same time, they meet after 3 hours.
• If Phong starts 90 minutes before Vân, they meet 2 hours 24 minutes after Vân starts.

How many minutes before Phong must Vân start so that, when they meet, Vân has travelled four times as far as Phong?

Model clarification: each travels at a constant speed, unchanged between the situations.

Two football teams, each with 10 players, play a match. Everyone on one team always tells the truth, and everyone on the other always lies. Afterwards each player is asked, “How many goals did you score?” Some answer “one”, Châu answers “two”, some answer “three”, and the rest answer “five”. Is Châu a truth-teller or a liar, given that the truth-telling team wins 20–17?

Model clarification: each team's score is the sum of the goals scored by its own players.

Place the numbers in $P=\{1,2,3,4,5,6,7,8,9\}$ into the nine cells of the shown $3\times3$ grid, using each number once. The sum in each row, each column and each of the two diagonals must be divisible by 9. The centre entry is $T$.

(a) Find every possible value of $T$.

(b) For each value, give an example of a completed grid.

Two vehicles leave A for B at the same time and travel at constant speeds of 75 km/h and 50 km/h. After 6 hours, the first vehicle meets a third vehicle travelling in the opposite direction at a constant speed. After another 1.5 hours, that third vehicle meets the second vehicle. Find the speed of the third vehicle.

Mother divides a bag of sweets between Đức and Thảo in the ratio $4:5$. Father then gives Đức an additional $1/3$ of the sweets Đức has, while Thảo eats $1/3$ of the sweets Thảo has. Their new combined number of sweets is 2 less than their original total. Find the original total.

The height of a cube is increased by 3 cm, turning it into a rectangular prism. Its total surface area increases by $72\text{ cm}^2$. Find the volume of the original cube.

Let $a,b,c$ be nonzero integers satisfying
$$2\left(\frac a{a'}\right)+3\left(\frac{b'}b\right)=5,$$
$$5\left(\frac b{b'}\right)+2\left(\frac{c'}c\right)=3.$$
Find $abc+a'b'c'$.

Notation clarification: the primed symbols $a',b',c'$ denote three further nonzero integers, separate from $a,b,c$; all six symbols are nonzero.

In a game, A, B, C and D each wear a shirt numbered 6, 7 or 8. Anyone wearing 7 always lies; anyone wearing 6 or 8 always tells the truth. Asked for the sum of the four shirt numbers, they answer:
• A: “The total is 28.”
• B: “The total is 27.”
• C: “The total is 26.”
• D: “The total is 25.”

Find each person's shirt number.

In rectangle $ABCD$, point $E$ lies on $AB$. Points $M,N$ lie on $CE$, in the order $C,M,N,E$, with $CM=MN=NE$. Line $DN$ meets side $AB$ at $F$, as shown. The two shaded triangles $DMN$ and $ENF$ have total area $12\text{ cm}^2$. Find the area of rectangle $ABCD$.

Mr Thanh saws a 4 m log into equal pieces 20 cm long. Each cut takes 8 minutes, and he rests for 2 minutes between consecutive cuts. He starts at 7:00 a.m. At what time does he finish?

Model clarification: cuts are made one at a time without stacking pieces; ignore the width of the saw blade. There is no rest before the first cut or after the last cut.