VTMO 2021 — Grade 7 — vtmo

Calculate $2020 - 2019 + 2018 - 2017 + \ldots + 1012 - 1011 + 1010$.

How many positive-integer pairs $\left(x , y\right)$ with $x \lt y$ satisfy $x + y = 2020$ and $20$ divides $xy$?

$ABC$ has $AB \lt BC \lt CA$. Its angle bisectors meet at I. $AIB = x ^{\circ} , BIC = y ^{\circ} , AIC = z ^{\circ}$, where $x , y , z$ are consecutive integers. Find $BAC$.

How many bouquets of seven roses contain all three colors: red, white, yellow?

On the $6 m \times 6 m$ grid, ant A moves only right/down and ant $B$ only up/left. They start together at equal constant speeds and meet after $3$ minutes. Find each speed in $m /$ min.

$ABC$ has $AB = 4 , BC = 6 , AC = 5$. For any point $P$, find the minimum of $PA + 2 PB + 3 PC$.

$ABC$ has altitudes $AH = 16.8$ cm, $BK = 8 \,\mathrm{cm}$, and $BC = 10 \,\mathrm{cm}$, in the shown configuration. Find $AB$.

Cars $A$ and $B$ start together from $HCMC$ toward Vung Tau and stop $50 \,\mathrm{km}$ from $HCMC$. Their total driving times are $3 h$ and $3 h = 20$ min. $B$ reaches the stop $8$ min after $A$. Find the full distance.

$A$ is the set of primes less than $10$. How many isosceles triangles have all side lengths in $A$?

How many rectangles have positive-integer sides, perimeter $50$, and square area?

Hieu packs $7$ pens, $8$ rulers, and $9$ notebooks into twelve two-item gifts, each of different kinds. How many contain a pen and ruler?

Equilateral $ABC$ has area $100 \,\mathrm{cm}^{2}$. I lies on $BC$ with $IC = 4 IB$. The line through I perpendicular to $AB$ meets line $AC$ at $F$. Find area $AIF$.

Integers $x , y , z$ satisfy $\frac{2020 x - 2019 y}{2021} = \frac{2019 z - 2021 x}{2020} = \frac{2021 y - 2020 z}{2019}$. Find the least positive $y$.

How many months in $2020$ have exactly four Saturdays, given January $1$ is Wednesday?

$ABCD$ is divided into rectangles $\left(1\right) , \left(2\right)$ as shown. Rectangle $2$ has twice the perimeter and four times the area of rectangle $1$. $AB = 10 \,\mathrm{cm}$. Find $AD$.

Write $1 , 3 , 5 , \ldots , 99$. Each move replaces two numbers with the remainder of their sum upon division by $7$. What remains after $49$ moves?

$M$ is inside equilateral $ABC$, with perpendicular feet $D , E , F$ on $BC , CA , AB$. $MD = 1 , ME = 2 , MF = 3$. $a$) Find area $ABC$. $b ) BH \perp AC$ at $H$ and $G$ lies on $BH$ with $BG = 2 GH$. Prove $GM \parallel AC$.