VTMO 2020 — Grade 9 — vtmo

Question 1

Let $a = 2 \sqrt{7} , b \sqrt{6} + \sqrt{8} , c = \frac{53}{10}$. Choose the correct order.

$c \gt a \gt b$
$c \gt b \gt a$
$a \gt c \gt b$
$a \gt b \gt c$
$b \gt c \gt a$

Question 2

Find the triangle area formed by $y = 3 x + 4 , y = 2 x + 4$ and the $x$-axis.

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

Question 3

How many four-digit numbers have nonzero digits, end in $9$, and are divisible by all their digits?

Circle $\left(O\right)$ has radius $1 , M$ lies on it, and circle $\left(K\right)$ has diameter $MO$. A line meets the circles at four distinct ordered points $A , B , C , D$, with $A , D$ on $\left(O\right)$ and $AB = BC = CD$. Find the distance from $O$ to the line.

$\frac{1}{4} \,\mathrm{cm}$
$\frac{1}{3} \,\mathrm{cm}$
$\frac{1}{8} \,\mathrm{cm}$
$\frac{1}{6} \,\mathrm{cm}$
$\frac{2}{3} \,\mathrm{cm}$

Question 5

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

$20$
$24$
$16$
$30$
$28$

Question 6

Square $ABCD$ has side $8$. $M , N , P , Q$ bisect its sides, and $X , Y , Z , T$ bisect $AM , BN , CP , DQ$. Find the intersection area of squares $MNPQ$ and $XYZT$.

$26 \,\mathrm{cm}^{2}$
$28 \,\mathrm{cm}^{2}$
$29 \,\mathrm{cm}^{2}$
$30 \,\mathrm{cm}^{2}$
$31 \,\mathrm{cm}^{2}$

Question 7

Successive circles are tangent to both sides of angle xOy and each other as shown. $r _{1} 1 , r _{2} 2$. Find $r _{4}$.

Question illustration

$16$
$8$
$3$
$9$
$4$

Question 8

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 9

Ten quadratics $x^{2} + a _{i} x + b _{i} 0$ collectively have distinct roots±$1 , \ldots$, ±$10$. Let $M$ be the minimum sum of their constant terms. Find the units digit of $| M |$.

Question 10

Positive $a , b , c$ satisfy a $^{2} + b^{2} + c^{2} = 3$. Find the smallest $k$ such that $\min \left(a + b , b + c , c + a\right) \gt k$ guarantees a triangle.

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

Question 11

The six pairwise sums of real $a , b , c , d$ are $2 , 3 , 4 , 4 , 5 , 6$. Find a $^{2} + b^{2} + c^{2} + d^{2}$.

Question 12

$ABC$ 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 $MNQP$.

Question 13

A circle has radius $5 \,\mathrm{cm}$ and perpendicular diameters $AB , CD$. $M$ is the midpoint of $OD$; $AM$ meets the circle again at $K$. Find area $KAC$.

Question 14

How many polynomials $x^{2} - ax - b$ with positive integers $a , b$ have a positive root below $4$?

Question 15

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

Question 16

Place $22$ pieces on $a = 7 \times 7$ board, at most one per cell. Pieces attack in a shared row or column. Prove at least four are mutually nonattacking.

Question 17

Circle $\left(O\right)$ has diameter $AB$ and $D$ is the midpoint of arc $AB$. $M$ lies on $AB$ extended outside the circle; $MD$ meets the circle again at $C$. Circle $\left(I\right)$ through $D , C$ is tangent to $AB$ at $K$; $DK$ meets $\left(O\right)$ again at $E$. Prove $ME$ is tangent to $\left(O\right)$.