VTMO 2021 — Grade 8 — vtmo

Evaluate $2^{2021} - 2^{2020} - \ldots - 2 - 1$.

Given a $^{2} + 4 ab = 28 , 2 ab - b^{2} = 3$, find $| a - 2 b |$.

Ten pens, twelve rulers, fourteen notebooks form $18$ gifts with two different kinds each. How many pen–ruler gifts?

Rectangle $O \left(0 , 0\right) , M \left(0 , 5\right) , N \left(10 , 5\right) , P \left(10 , 0\right)$: count interior lattice points with $x + y \lt 10$.

Ants leave $B , C$ in opposite directions around $ABC$, first meet at $A$, then $M$. $AB = 20 dm , AC = 30 dm , BC = 40 dm$. Find $BM$.

How many positive-integer pairs $\left(x , y\right) , x \lt y , x + y = 2020$, have $20$ dividing $xy$?

Find the smallest prime factor of $1$, twenty zeros, and $1$.

Six distinct cars: one green, two yellow, three red. Count orders without adjacent cars of the same color.

Students $1 , \ldots , 100$: in round $k$, labels divisible by $k$ step left for odd $k$, right for even $k$. How many return after $100$ rounds?

$ABC$ has area $100 , M$ on $BC , N$ on $CA , P$ on $AB , BM \lt CM$; $APN , BPM , CMN$ each have area $21$. Find area $ABM$.

Three nested cubes have integer sides, box $1$ contains $2$, box $2$ contains $3$. Volume $1$ exceeds $3$ by $218 \,\mathrm{cm}^{3}$. Find volume $3$.

Find the largest positive $n$ for which $2 n$ ones minus $n$ sevens is square.

Eight-digit abcdxyzt has distinct digits and is divisible by $9999$. Find its block sum $abcd$+xyzt.

Integers $a , b$ satisfy $3 a + b = 30$. Minimize $3$ a $^{2} + b^{2}$.

$ABCD$ splits into $1 , 2$; $MNPQ$ into $3 , 4$ as shown. VI: perimeter $2$ twice $1$, area $2$ four times $1$; perimeter $4$ twice $3$, area $4$ four times $3$; $AD = MN$. Source EN reverses area $\frac{1}{2}$. Find $4 \frac{AB}{MQ}$.

Each $a_{i}$ is $1$ or $- 1$ and $S_{n} = a = 1 a = 2 + a = 2 a = 3 + \ldots + a_{n} a = 1$. Can it be zero for $a ) n = 30$; $b ) n = 28$?

$ABC , M$ on $AB , BN$ parallel $CM$ meets extended $AC$ at $N , E$ midpoint $MN , K = AE \cap BC$. Find $\frac{KC}{KB}$ for $a ) M$ midpoint $AB$; $b ) AM = \frac{AB}{\sqrt{2}}$.