VTMO 2026 - Grade 7 - Semifinal - SET 11 - Self-study

Let a, b, c, d, e, f be six pairwise distinct positive integers. Find the smallest positive integer x such that x=a+b+c=d+e+f.

During the summer, An solved some mathematics problems, including some incorrectly. Correcting two wrong answers would make 76% of the answers correct. Instead deleting those same two wrong answers would make 75% correct. How many problems did An solve?

A painter needs 20 hours to paint a room’s four walls and ceiling. At the same constant productivity, another room with twice the length and width but the same height takes 50 hours. How many hours are needed for the original ceiling?

Dice are placed side by side in a straight row on a table, as shown. Counting visible faces from all directions, two dice have 8 visible faces and three have 11. How many faces are visible with 30 dice?

A right triangle ABC has integer side lengths in centimetres. Two sides have prime lengths, and two sides differ by 50 cm. Find the smallest possible length of a side.

How many fractions in 1/108,2/108,3/108,…,106/108,107/108 are in lowest terms?

Three wheels A, B, C are shown. A and C each contain 1,2,3,4,−1,−2,−3,−4; B contains +,−,×, ÷. Nam spins A, B, C in order and applies the operation from B to the numbers from A and C in that order. For example,4×2=8. How many different outcomes produce a positive integer?

Let T be the set of odd primes less than 73. Each element is placed once in a circle in the diagram so that every drawn straight line containing five numbers has the same sum. Some entries, including A and B, are indicated. Find A−B.

Let M be the smallest composite number not divisible by any integer from 2 to 50. Find the sum of its digits.

Let A={1,2,3,…,30,31,32}. How many ways can two different elements be chosen to form a fraction with an integer value? Examples include 2/1,15/3,15/5.

Points A, B, C, D lie in that order on line d, as shown (not to scale). Given AB/AD=3/8 and BC/AD=2/5, find AB/AC.

The product of three consecutive odd numbers is $9177$. Find their sum.

There are $50$ coins, all face down. First, $25$ coins are turned over; next, $29$ coins are turned over; finally, $14$ coins are turned over. All coins are then face up. How many coins were turned over in all three rounds?

The product of the digits of each of $229$ and $419$ is $36$. How many three-digit natural numbers have digit product $36$?

A water park reduces its admission price on weekdays to attract more visitors. At the end of the day, the number of visitors is $50\%$ higher, but total ticket revenue is only $20\%$ higher. By what percentage was the ticket price reduced? All visitors pay the same ticket price.

A cubic fish tank has volume 27 dm³. It contains a non-porous decorative stone of volume 9 dm³. Water is poured in until the stone is just submerged, at a water depth of 2.5 dm. What will the water depth be after the stone is removed, in centimetres?

A point inside a regular hexagon is joined to all six vertices, forming six triangles of different areas. The areas of three triangles are shown as $14$, $16$ and $26$. Find the area of the triangle marked $X$.

A square has side length 100 m. An and Binh start simultaneously at opposite vertices and walk anticlockwise along its sides. An walks at 50 m/min and Binh at 40 m/min. At each vertex An rests for 1 minute and Binh for 2 minutes. After how many minutes does An first catch Binh?

How many pairs of positive integers (x, y) satisfy x$\lt$y, x+y=2020, and 20 divides xy?

For positive integers $a,b$, define $a\blacksquare b=a\times1+a\times2+\cdots+a\times b$. For example, $3\blacksquare2=3\times1+3\times2=9$ and $2\blacksquare3=2\times1+2\times2+2\times3=12$. Positive integers $m,n$ satisfy $m\blacksquare2021=2022\blacksquare n$. Find the smallest value of $m+n$.