VTMO 2022 — Grade 7 — vtmo

Let $a,b,c\in\mathbb{Z}$ satisfy $\frac{4^4\times5^5\times6^6}{8^8\times9^9\times10^{10}}=2^a\times3^b\times5^c$. Find $a-b+c$.

Given $(a+b):(b+c):(c+a)=2:3:5$ and $a+b+c=15$, calculate $abc$.

The chart shows notebook sales $2017 \text{--} 2021$. Sales increase $60 \%$ in each of $2018 , 2019$ and decrease $25 \%$ in each of $2020 , 2021$. Sales $2021$ exceed $2017$ by $30800$. Find $2021$ sales. Source options: $A = 44000 , B = 70000 , C = 74800 , D = 108000 , E = 119000$.

$ABC$ is right at $A , AB = 1 \,\mathrm{cm} , BC \sqrt{13} \,\mathrm{cm}$, and $M$ is midpoint $AC$. Find $BMC$.

$A = 11 \times 12 \times \ldots \times 20 , B = 21 \times 22 \times \ldots \times 30$. Find the remainder of $B - A - 2022$ divided by $1001$.

Acute $ABC$ has $M$ midpoint $AC$ and $AH \perp BC$ at $H$. $AH = BM$ and $BCA = 37 ^{\circ}$. Find $BMC$.

Hanh writes $\frac{1}{998} , \frac{2}{997} , \frac{3}{996} , \frac{4}{995} , \ldots , \frac{995}{4} , \frac{996}{3} , \frac{997}{2} , \frac{998}{1}$. How many fractions use exactly two different digits? Examples $\frac{9}{990} , \frac{333}{666} , \frac{445}{544}$.

In $ABC , E$ lies on $BC$ with $BE = 2 EC$, and $M$ on $AE$ with $ME = 3 MA$. $BM$ meets $AC$ at $D$. Area $MEC = 297 \,\mathrm{cm}^{2}$. Find area $MAD$.

On the shown $4 \times 4$ grid, An travels from A only up/right and Binh from $B$ only down/left. They start together at equal speeds. How many path combinations lead to a meeting?

A car travels segments of lengths $s , 2 s , \ldots$, ns at speeds $v , 2 v , \ldots$, nv. Find $n$ so that overall average speed is $7 v$.

An integer $a \gt 1$ is nice if a $^{2}$ is the product of all positive divisors of a. For example $8^{2} = 1 \times 2 \times 4 \times 8$. How many nice integers are less than $85$?

Wheels $A$ and $C$ each contain $\{ - 4 , - 3 , - 2 , - 1 , 1 , 2 , 3 , 4 \}$; wheel $B$ contains−,+, $\times$, $\div$, as shown. Spin $A , B , C$ and apply the selected operation; for example $4 \times 2 = 8$. How many spin outcomes produce a positive integer?

Define $a \blacksquare b = a \times 1 + a \times 2 + \ldots + a \times b$. Examples $3 \blacksquare 2 = 9 , 2 \blacksquare 3 = 12$. Positive integers $m , n$ satisfy $m \blacksquare 2021 = 2022 \blacksquare n$. Find the least $m + n$.

$T = 123^{m} + 789^{n}$, with $m , n$ in $\{ 1 , \ldots , 21 \}$. How many distinct $T$ values are divisible by $10$?

$ABC$ is right with integer side lengths, two of which are prime. Two side lengths differ by $50 \,\mathrm{cm}$. Find the least possible side length.

$a ) 55 ^{\circ}$ alcohol means $55$ ml pure alcohol per $100$ ml mixture. It can be reduced to the same $T ^{\circ}$ either by adding $v$ units of water or removing $v$ units of pure alcohol. Find $T$. $b$) How many primes $p$ make $p^{2} + 59$ have exactly six positive divisors?

$P$ is inside right angle xOy. $A$ lies on Ox, $B$ on Oy, and $APB = 90 ^{\circ}$. On the rays opposite $AP$ and $BP$, take $C , D$ with $CA = OA$ and $BD = OB$. $a$) Prove $C , O , D$ collinear. $b ) M$ is midpoint $CD$; prove $AMB = 90 ^{\circ}$.