VTMO 2026 - Grade 6 - Semifinal - SET 11 - Self-study

Two security guards at TITAN start working on May 2, 2017. The first rests for one day after every six consecutive working days; the second rests for two consecutive days after every eleven consecutive working days. Holidays do not interrupt these schedules: a guard still works if a holiday is a scheduled workday. How many days in 2017 do both guards rest together? Count May 2 as the first working day for both.

A number consists of 2014 digits, all of them 7. What is the minimum number of digits to delete so that the remaining number is divisible by 63?

Find the prime numbers $x,y$ such that $x^2=y^2-45$.

A rhombus has area $24$ cm² and one diagonal of length $8$ cm. Find the difference between its longer and shorter diagonals.

A box contains $10$ blue, $13$ red and $12$ yellow balls. Pi draws one ball at a time without looking and without replacement. What is the minimum number of draws that guarantees $10$ balls of one colour?

Which pair has greatest common divisor 20?

Rectangle ABCD is divided into rectangles AEFD and BCFE as shown. The perimeter of BCFE is twice that of AEFD, and its area is four times that of AEFD. Given AD = 10 cm, find AB in centimetres.

Replace the symbols ■ and ▲ by operations chosen from +, −, ×, ÷ so that $6\,\blacksquare\,6\,\blacktriangle\,6\,\blacksquare\,6=2$ is true. What are ■ and ▲, respectively?

Two numbers satisfy $3\le x\le6$ and $8\le y\le10$. Find the smallest possible value of $y-x$.

For natural numbers $a,b,c,d$, calculate $a+b+c+d$ given $2^{12}=4^a=8^b=16^c=64^d$.

A group contains men and women. The women’s average age is 18 and the men’s average age is 24. There are twice as many men as women. What is the average age of the whole group?

Each segment in the park diagram is a path of length $100$ m. An wants to go from $A$ to $B$ without using any segment more than once. What is the greatest distance she can travel?

Find the units digit of $7^{2014}$.

A palindrome is a natural number that reads the same forwards and backwards. For example, 1221 is a palindrome. How many three-digit palindromes are there?

The figure consists of four equal squares, each of side $6$ cm. Find the total shaded area in cm².

A rectangular garden is divided into four rectangles $A,B,C,D$ as shown. Their areas satisfy $S_A=24$ m², $S_B=48$ m² and $S_C=12$ m². What is the total area of the garden?

In every cell of a $4\times5$ rectangular board, write one of $0,1,2$ so that the sum in each row and each column is divisible by $3$. What is the greatest possible number of entries equal to $1$?

Evaluate $1^{10}-1^{20}+1^{30}-1^{40}+\cdots+1^{90}-1^{100}$.

In a game, each of $A,B,C,D$ wears a shirt numbered $6$, $7$ or $8$. Anyone wearing $7$ always lies, while anyone wearing $6$ or $8$ always tells the truth. Asked for the sum of their four shirt numbers, they reply: A: “28”; B: “27”; C: “26”; D: “25”. Find each person’s shirt number.

An and Bình each travel $600$ m, starting and finishing at the same times. An runs for the first half of the total time and walks for the second half. Bình runs the first half of the distance and walks the second half. Both walk at $5$ km/h, and An runs at $10$ km/h. How fast does Bình run?