VTMO 2020 — Grade 7 — vtmo

Question 1

Replace different letters by different digits to minimize $VIETNAM + TITAN + MATH$, with $V , T , M$ nonzero. Find $T + I + T + A + N$.

$16$
$29$
$13$
$14$
$15$

Question 2

A two-digit number is divided by its digit sum. What is the largest possible remainder?

$13$
$14$
$15$
$16$
$17$

Question 3

An solves some problems. Correcting two wrong answers makes $76 \%$ correct; removing those two wrong problems makes $75 \%$ correct. How many problems were solved?

$40$
$45$
$48$
$50$
$56$

Question 4

$A = 4000$-liter tank is filled at $20$ liters/minute. After $20$ minutes it leaks $2$ liters/minute. How many minutes from the start until full?

$220$
$200$
$230$
$240$
$250$

Question 5

How many triangles have side lengths chosen from $S = \{ 3 , 4 , 6 , 8 \}$?

$3$
$4$
$16$
$17$
$20$

Question 6

Distribute $70$ candies among six students in distinct positive amounts. What is the least possible largest amount?

$13$
$14$
$15$
$16$
$17$

Question 7

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

$3975$
$2208$
$278$
$276$
$287$

Question 8

How many ordered fractions with distinct numerator and denominator from $\{ 1 , \ldots , 32 \}$ are integers?

$80$
$84$
$85$
$86$
$87$

Question 9

Collinear $A , B , C$ occur in that order, $AB = 2 , BC = 1$. $D , E$ lie on the same side of $BC$ and $ABD , BCE$ are equilateral. Find area $ADEC$.

$\frac{\sqrt{3}}{4}$
$7$
$3 \frac{\sqrt{3}}{4}$
$7 \frac{\sqrt{3}}{4}$
$2 \sqrt{3}$

Angle xOy $90 ^{\circ}$. Interior rays Oz, Ot satisfy xOz=tOy $30 ^{\circ}$. $K$ lies on Ox with $OK \sqrt{3}$. $KB$ is perpendicular to Oz at $B$; the line through $B$ perpendicular to Ot meets Oy at $D$. Find $OD$.

$\frac{1}{2}$
$\frac{2}{\sqrt{3}}$
$2 \sqrt{3}$
$\frac{3}{2}$
$\frac{\sqrt{3}}{2}$

Question 11

$A = 360 m^{2}$ rectangle is partitioned as shown into $S = 1 , S = 2 , S = 3 , S = 4$. $S = 1$ is square and the area ratio is $1 : 2 : 3 : 4$. Find the perimeter.

Question illustration

Question 12

A runner and cyclist start together at A around a circular lake in the same direction. The cyclist is $3.5$ times as fast. Mark every meeting place. How many distinct places occur?

Question 13

Five teams play a single round-robin. A win earns $3 , a$ draw $1 , a$ loss $0$. At some stage all five have distinct positive scores. What is the minimum number of matches?

Question 14

Plates range from $0001$ to $9999$. How many four-digit plates $abcd$ satisfy $a + b = c + d$, as $3681$ does?

Question 15

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

Question illustration

Question 16

A consists of five distinct integers. The sumset $S = \{ x + y : x , y \in A \}$, allowing $x = y$, has nine elements. Prove that the sum of $A$ is divisible by $5$.

Question 17

$MAB$ is isosceles at $M$ with $AMB \lt 90 ^{\circ}$. The perpendicular to $MB$ at $B$ meets ray $MA$ at $N$. $C$ lies on $MN$ with $NC = NB$. $a$) Prove $ABC = 45 ^{\circ}$. $b$) The perpendicular to $AM$ at $M$ meets ray $BC$ at $D$. Prove $AD = \sqrt{2} MD$.