VTMO 2020 — Grade 8 — vtmo

How many integers $n$ make $\frac{2^{2020}}{3 n - 1}$ an integer?

Natural numbers $a , b$ satisfy $7 a + 11 b = 2020$. When $11 a + 7 b$ is maximal, find $ab$.

Triangle side lengths are selected from $\{ 4 , 5 , 6 , 7 , 8 \}$, allowing repetitions. How many obtuse triangles are there?

Parallelogram $ABCD$ has area $24 \,\mathrm{cm}^{2} , O$ its diagonal intersection. Construct parallelogram $BOCE$, and let $M = AE \cap BC$. Find area $MCE$.

Natural numbers satisfy $a + b = c + d = 80 , S = ab + cd$. Which value cannot be $S$? $A = 2019 B = 2020 C = 2021 D = 2022 E = 2023$.

Trapezoid $ABCD$ has $AB$ parallel $CD$, perpendicular diagonals $AC = 15 \,\mathrm{cm} , BD = 20 \,\mathrm{cm}$. Find its height.

Among plates $0001 , \ldots , 9999$, how many have equal left-pair and right-pair digit sums?

Seven unit squares form rows of four and three inside $ABC$ as pictured. Find area $ABC$.

A reduced positive fraction $\frac{a}{b}$ becomes half its old value when $36$ is added to numerator and denominator. Find $a + b$.

$C$ lies on $BD$ with $BD = 6 CD$. Equilateral triangles $ABC , ECD$ lie on the same side. Area $ACE = 5 \,\mathrm{cm}^{2}$. Find area $ABDE$.

$ABC$ is right isosceles at $A , AB = 6 \,\mathrm{cm}$. $M$ on $AB$ has $AM = 4 , N$ on $AC$ has $AN = 2$. $P , Q$ on $BC$ have $PM , QN$ perpendicular $MN$. Find area $MNQP$.

Even $n$ has $15$ positive divisors. What maximum number of positive divisors can $3 \frac{n}{2}$ have?

Natural numbers $a , b , c$ satisfy exactly one true statement per pair: $X$ a $^{2} + b^{2} + c^{2} = 165$ or $bc = 28$; $Y c \gt b$ or $9$ divides $b + c$; $Z = 8$ divides $c$ or $c = 4$. Find $a + 2 b + 3 c$.

Given $100$ distinct positive integers, $a$ nice pair has ratio $2$ or $3$. What maximum number of nice pairs is possible?

The VI asks for the smallest ten-digit number with distinct digits divisible by $2 , \ldots , 9$; the EN asks for the greatest. For digits $a = 1$..$a = 10$, find $a = 4 \cdot a = 6 \cdot a = 8$.

Twenty-five giraffes have distinct heights. Each round ranks five by height. Find the three tallest in order using seven rounds.

$ABC$ is isosceles at $A , M$ midpoint $AB , F$ on $AC , E$ on the ray opposite $BA$. VI states $BE = CF$; EN states $BE = 2 CF$. Reflect A in $F$ to $D$, set $K = EF \cap BC , N = DK \cap AB$. Prove $MN = CF$.