VTMO 2026 - Grade 6 - Final - SET 4 - Self-study

In the addition $\overline{GET}+\overline{THE}=\overline{MATH}$, different letters stand for different decimal digits, and leading digits are nonzero. Given $T=3$, find the four-digit number $\overline{MATH}$.

Some children have collected mushrooms. If An gives Binh half of An's mushrooms, all the children will have the same number of mushrooms. If instead An gives Chau all of An's mushrooms, Chau will have as many mushrooms as all the other children together. How many children are there?

Model clarification: the children have collected at least one mushroom in total.

A large cube is built from unit cubes. Some of its outer faces are painted completely. After the cube is taken apart, exactly 45 unit cubes have no paint. How many unit-cube faces have been painted?

In the diagram, the sum of the three numbers on each side of the square is 43. Different letters represent different prime numbers. Find $B\times C$.

Phuong Vy walks from her home $V$ to her grandmother's home $B$ along a route with level, uphill and downhill sections. Her speeds are 5 km/h on level ground, 4 km/h uphill and 6 km/h downhill. The outward journey takes 1 hour 36 minutes; the return along the same route takes 1 hour 39 minutes. The level sections total 2.5 km. Find the length of the route from $V$ to $B$.

a) Positive integers $a,b$ satisfy $56a=65b$. Prove that $a+b$ is composite.

b) The diagram shows a square of side 14 cm and two quarter-circles. Find the positive difference between the areas of regions (I) and (II), using $\pi=22/7$.

Find positive integers $a,b$ such that: a) $a+1$ is divisible by $b$; b) $a=2b+5$; c) $a+7b$ is prime.

In triangle $ABC$, $M$ is the midpoint of $BC$ and $N$ is the midpoint of $AC$. Segments $AM$ and $BN$ intersect at $O$.

a) Compare the areas of triangles $AON$ and $BOM$.

b) Find $OA/OM$.

a) Find the six-digit natural number $N$ formed only from digits 8 and 9 that is divisible by $2^6$.

b) Use digits $1,2,3,\ldots,9$ exactly once altogether to form three three-digit numbers $a,b,c$ such that $a:b:c=1:3:5$. Find $a,b,c$.

Fill the small circles around a ring with all elements of $S=\{1,2,3,\ldots,2018,2019\}$, using each once. The difference between any two adjacent numbers (larger minus smaller) is at most 2. The positions of 2015, $x$ and $y$ are shown. Find $x+y$.

Exactly one of the four dwarfs $A,B,C,D$ stole the king's gold. They each make two statements:

$A$: “$B$ is a liar.” “$C$ or $D$ stole the gold.”

$B$: “$C$ is a liar.” “$D$ or $A$ stole the gold.”

$C$: “$D$ is a liar.” “$A$ or $B$ stole the gold.”

$D$: “$A$ is a liar.” “$B$ or $C$ stole the gold.”

A liar makes at least one false statement; a truthful dwarf makes two true statements. How many of the eight statements are true?

a) On day 1, one alien virus is placed in an experimental bottle. On day 2 the number doubles; on day 3 it triples; on day 4 it becomes four times the previous day's number. This continues, so on day 10 the number becomes ten times that on day 9. Can the viruses on day 10 be divided equally among 24 test tubes? Explain.

b) A basket contains 13 apples. A scale can measure the combined mass of any two apples. Show how to find the total mass of all 13 apples using eight weighings.

Viet buys an old book originally numbered from page 1 to page 50. At home, he finds that one leaf (two pages) is missing. Odd page numbers are on the right and even page numbers on the left. The sum of the remaining page numbers is 1240. Which page numbers are on the missing leaf?