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

Teams A, B and C can complete the same job in 12, 15 and 21 days, respectively. Every worker has the same productivity. Team A has 30 more workers than team C. How many more workers does team A have than team B?

An integer $a>1$ is called nice if $a^2$ equals the product of all its positive divisors. For example, 8 is nice because $8^2=1\cdot2\cdot4\cdot8$. How many nice numbers are less than 85?

Positive integers $a,b,c,d$ satisfy $A=\frac ab+\frac cd=\frac ab\cdot\frac cd$ and $B=\frac ba+\frac dc=\frac ab\div\frac cd$. Find $A^2+B^2$.

If $a$ cows produce $b$ litres of milk in $c$ days, how many litres will $d$ cows produce in $e$ days at this same productivity?

Let $A$ be a set of five integers and let $S=\{x+y\mid x,y\in A\}$; the two chosen elements may be equal. If $S$ has exactly nine elements, prove that the sum of the elements of $A$ is divisible by 5.

What is the greatest possible number of consecutive integers each of which is a product of two distinct prime numbers?

A number is increased by 25%. By what percentage must the new number be decreased to return to the original number?

Find the remainder when $2^{2024}+3^{2023}+2^{2022}+3^{2021}$ is divided by 10.

Let $A=\frac1{2\cdot4}+\frac1{6\cdot8}+\frac1{10\cdot12}+\cdots+\frac1{194\cdot196}+\frac1{198\cdot200}$ and $B=\frac{50}{51}+\frac{51}{52}+\frac{52}{53}+\cdots+\frac{98}{99}+\frac{99}{100}$. Find $4A+B$.

A rectangle of area $360\,\mathrm{m}^2$ is divided as shown into four rectangles of areas $S_1,S_2,S_3,S_4$, where $S_1:S_2:S_3:S_4=1:2:3:4$. The region of area $S_1$ is a square. Find the perimeter of the original rectangle, in metres.

Ten distinct positive integers have sum 100. What is the greatest possible value of the largest integer?

For a positive integer $n$, define $n!=n(n-1)\cdots1$. What is the last digit of $1!+2!+3!+\cdots+2014!$?

A palindrome reads the same forwards and backwards: for example, 1221 and 181 are palindromes, but 122 is not. Let $A,B$ be three-digit palindromes and let $C=AB$ be a five-digit palindrome. If $C$ is as large as possible, find the sum of its digits.

Evaluate $1\cdot2-2\cdot3+3\cdot4-\cdots-48\cdot49+49\cdot50$.

Triangle $ABC$ is right-angled at $A$, with $AB=1$ cm and $BC=\sqrt{13}$ cm. Point $M$ is the midpoint of $AC$. Find $\angle BMC$, in degrees.

Rectangular box A has dimensions $10\times6\times8$ and rectangular box B has dimensions $4\times2\times3$, measured in centimetres. What is the greatest number of boxes B that can fit inside box A?

A hot-water tap alone fills a tank in 23 minutes, and a cold-water tap alone fills it in 17 minutes. The hot tap is opened first; the cold tap is opened later and both then run until the tank is full. How many minutes after the hot tap should the cold tap be opened so that the amount of hot water is 1.5 times the amount of cold water?

How many three-digit natural numbers have all their digits even?

It takes 165 minutes to saw a 72 cm iron rod into pieces of length 6 cm. How many minutes are needed to saw a 144 cm iron rod into pieces of length 6 cm?

Among the following expressions, in which can 7 be replaced by another integer while keeping the result unchanged?