VTMO 2026 - Grade 6 - Semifinal - SET 5 - Self-study

In which expression can the number 6 be replaced by another integer without changing the result?

Clarification: replace every 6 by the same nonzero integer, and require the result to stay unchanged for every such choice.

The diagram shows the roads between cities A and B. An starts at A and can only travel upward or rightward. Bình starts at B and can only travel downward or leftward. They start together at the same speed. How many different ways can they meet?

Clarification: the small grid segments have equal length; count pairs of routes up to their meeting, not just meeting locations.

For every integer from 30 through 50, subtract its units digit from its tens digit. What is the sum of all the results?

Let A be the set of all prime numbers less than 11. How many subsets does A have?

Find the smallest prime number that can be expressed as a sum of two primes, as a sum of three distinct primes, as a sum of four distinct primes, and as a sum of five distinct primes.

The pyramid shown is built from 84 unit cubes, each of side 1 cm. Every outer face is painted except the bottom of the pyramid. Find the total painted area in square centimetres.

During the summer, An solved some maths problems. After checking, some answers were wrong. If An corrects two wrong answers, 76% of the answers are correct. If An instead removes those two problems entirely, 75% of the remaining answers are correct. How many problems did An originally solve?

Find the missing number in the last triangle, using a rule consistent with the first three triangles.

A cube has 6 faces, 8 vertices and 12 edges. Khang writes eight consecutive natural numbers, one at each vertex. What is the greatest possible number of edges whose two endpoint numbers have an odd sum?

(a) (Exact division and division with remainder.) Find a six-digit natural number $N$ made only from the digits 8 and 9 such that $N$ is divisible by $2^6$.

(b) (Ratios, fractions and continued fractions.) Use each of the digits $1,2,3,\ldots,9$ exactly once to form three three-digit numbers $a,b,c$ such that $a:b:c=1:3:5$. Find $a,b,c$.

Nam writes all the consecutive natural numbers from 1 to 120. How many digits does he write altogether?

Hạnh writes the fractions $\frac1{98},\frac2{97},\frac3{96},\frac4{95},\ldots,\frac{95}4,\frac{96}3,\frac{97}2,\frac{98}1$. How many of the written fractions use exactly two different digits? Examples are $\frac9{90},\frac{33}{66},\frac{45}{54}$. Count digits as written, without simplifying the fractions.

A father is now 36 years old and his child is 6. After how many years will the father be three times as old as the child?

Simplify $\displaystyle\frac{1}{2+\frac{3}{4+\frac56}}$.

Use each of the digits 2, 3, 4, 5, 6 and 7 exactly once to form two numbers. What is the smallest positive difference between them?

Let $A=1\times2\times3\times\cdots\times89$. How many zero digits are at the end of A?

An has congruent right-trapezoid pieces, with two sides measuring 4 cm as shown. She uses four pieces to form the rectangle in the diagram. What is the perimeter of that rectangle?

Thanh draws a square. As shown, each side is extended by 2 cm, and the new endpoints are joined to form a larger square. The new square has area 36 square centimetres greater than the original. Find the area of the original square.

A box contains a positive number of sweets which can be shared equally among 3, 4, 6 or 7 children with none left over. What is the smallest possible number of sweets?

Minh uses each digit from 1 to 8 exactly once to form two four-digit natural numbers. What is their smallest possible sum?