VTMO 2026 - Grade 7 - Semifinal - SET 6 - Self-study

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 $m+n$.

Positive integers are arranged in the pattern shown, with row 1 containing 1, row 2 containing 2, 3, and so on. In which row does 2015 occur?

In the figure, $AB\parallel CD$, $AC\perp BD$, $AC=20$ cm, $BD=15$ cm, and $BE\perp CD$ at $E$. Find $BE$ in centimetres.

Fill in the blank: $2014\times2015=2014\times2000-2014+2014\times\boxed{\phantom{16}}$.

Write all the integers from 1 to 999 consecutively to form a digit string. How many digits does the string contain?

Use each of the digits 2, 3, 4, 5, 6, 7, 8 exactly once as the distinct digits $U,V,W,X,Y,M,N$. What is the smallest possible value of $\overline{UVW}-(\overline{XY}+\overline{MN})$?

Triangle $ABC$ has $AB=AC=7$ cm. The length of $BC$ is an integer number of centimetres. What is the greatest possible perimeter of $ABC$?

Let $A$ be the largest four-digit odd number whose digits are pairwise distinct. What is the sum of its digits?

How many three-digit numbers have the property that any two adjacent digits differ by 3? Examples include 141 and 258.

Consider $-\frac42$, $\left(\frac14\right)^0$, $\frac13$, $-\frac{-1}{2}$, and $2.1$. How many belong to the rational numbers $\mathbb Q$ but not to the integers $\mathbb Z$?

There are 65 students in a hall, each wearing either a white or a blue shirt. There are 25 boys in white shirts, 23 students in blue shirts, and 28 girls altogether. How many girls wear blue shirts?

Each cell of the $3\times3$ square below contains a different power of 2 with a natural-number exponent. The products along every row, every column, and each of the two diagonals are equal. What number is in cell $X$?

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

Triangle $ABC$ has $AB=5$ cm, $BC=8$ cm and $\angle ABC=30^\circ$. Find its area.

Hạnh writes the fractions $\frac1{998},\frac2{997},\frac3{996},\frac4{995},\ldots,\frac{995}{1},\frac{996}{3},\frac{997}{2},\frac{998}{1}$ on the board. How many of these written fractions use exactly two distinct digits? Examples: $\frac9{990},\frac{333}{666},\frac{445}{544}$.

In triangle $ABC$, the external bisector at $B$ meets the internal bisector at $C$ at $D$, and the external bisector at $C$ meets the internal bisector at $B$ at $E$. Find the correct relation between the angle measures $p,q,r$ shown in the figure.

The distance from Anh’s home to school is greater than the distance from Bình’s home to school by $\frac18$ of Bình’s distance. Bình’s journey time is less than Anh’s by $\frac16$ of Anh’s time. What is the ratio of Anh’s speed to Bình’s speed?

Working alone, team A needs 20 days to complete a project, and team B needs 30 days. They work on the project together, but A takes 3 days off and B takes 2 days off. There is no day when both teams are off. How many days does the project take altogether, rounded to the nearest integer?

Find the sum of the five marked angles $a,b,c,d,e$ in the star.

(a) Alcohol strength measures the millilitres of pure alcohol in 100 ml of an alcohol–water mixture. For example, Bàu Đá wine of strength $55^\circ$ contains 55 ml pure alcohol and 45 ml water per 100 ml. A bartender can reduce a bottle of strength $55^\circ$ to the same strength $T^\circ$ either by adding $v$ units of water or by removing $v$ units of pure alcohol. Find $T$. (b) How many prime numbers $p$ make $p^2+59$ have exactly 6 positive divisors?