VTMO 2021 — Grade 9 — vtmo

Question 1

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

Question 2

Real $x , y$ satisfy $y = \sqrt{\frac{2020 x + y}{2020 x - y}} + \sqrt{\frac{y + 2020 x}{y - 2020 x}} + 2020$. Find $xy$.

$2020$
$- 2020$
$1010$
$- 1010$
$4040$

Question 3

Two points $A , B$ on $y = 2 x + 3$ are equidistant from the two coordinate axes. Find area $OAB$.

Question 4

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

$224$
$225$
$226$
$227$
$228$

Question 5

$2020$ points are equally spaced on a circle. How many regular polygons have all vertices in this set?

$12$
$1251$
$1254$
$1253$
$1151$

Question 6

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

$120$
$60$
$720$
$360$
$36$

Question 7

How many primes of form $n^{n} + 1$ are less than $8^{20}$ for positive integer $n$?

Question 8

An, Nam, Minh start together and run in the same direction around a lake at $5 , 7 , 9 \frac{\,\mathrm{km}}{h}$. How many distinct positions do their joint meetings mark?

Question 9

Circles centered at $O$ and I of radii $2$ and $3$ meet at $A , B$ with $\angle AOI = 90 ^{\circ}$. A secant through A meets them again at $M , C$, with $MA = MC$. Find $AC$.

$3 \sqrt{5}$
$8 \frac{\sqrt{5}}{3}$
$2 \sqrt{5}$
$4 \frac{\sqrt{5}}{3}$
$\sqrt{5}$

Question 10

$ABC$ is right at A with $AB = 30 , AC = 40$. Its incircle centered at I touches $BC$ at $K$. $M \in KC$; $AM$ meets the incircle at $E , F$ in order and $AE = MF$. Find $EF$.

$4 \sqrt{5}$
$4 \sqrt{15}$
$16$
$8 \sqrt{15}$
$8 \sqrt{5}$

Question 11

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 $3$.

Question 12

$A = \{ 1 , \ldots , 2020 \}$. How many integer pairs $a \lt b$ have $\frac{b}{a}$ a power of $2$?

Question 13

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

Three differently sized rectangles of common area $x \,\mathrm{cm}^{2}$ are arranged as pictured, the left one square. All sides are integers and total width $AB = x$ cm. Find $x$.

Question illustration

Question 15

$ABC$ is right at $A , AB = 24 , AC = 70$. Choose $D$ on $AC$ so the incircles of $ABD$ and $BDC$ are tangent. Find the inradius of $BDC$.

Question 16

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 $a ) n = 30$; $b ) n = 28$?

Question 17

$ABC$ is acute and isosceles at A on circle $\left(O\right)$. $D$ lies on minor arc $BC$ with $DB \lt DC$. Lines through $B , C$ parallel to $AD$ meet $\left(O\right)$ at $E , F$. $K$ lies inside minor arc $AE$; $M = KE \cap AC , N = KF \cap AB$. Prove $MN , KD , BC$ concur.