VTMO 2013 — Grade 9 — semifinal

Question 1

Which equation is represented by the straight-line graph?

Question illustration

$x + y = 1$
$x + y = 2$
$x^{2} + y = 1$
$x - y = - 1$
$x + y = - 1$

Question 2

A rectangle has area $104 m^{2}$ and length $5 m$ greater than width. Find its perimeter.

$21 m$
$25 m$
$30 m$
$42 m$
$50 m$

Question 3

How many five-digit numbers with distinct digits can be formed from $0 , 1 , 2 , 3 , 4 , 5$?

$120$
$600$
$720$
$360$
$480$

Question 4

A herd of cows and ducks has $200$ heads and $522$ legs. How many of each?

$61$ cows, $139$ ducks
$139$ cows, $61$ ducks
$60$ cows, $140$ ducks
$62$ cows, $138$ ducks
$139$ cows, $61$ ducks

Question 5

Find a such that $x^{2} + ax + a - 1 = 0$ has exactly one real root.

$0$
$- 1$
$1$
$2$
$- 2$

Question 6

$A$ is a nine-digit number congruent to $2$ modulo $9$. Let $B$ be its digit sum and $C$ the digit sum of $B$. Which value can $C$ have?

$1$
$3$
$11$
$9$
$5$

Question 7

A triangle has sides $x + 1 , x + 2 , x + 3$ and perimeter greater than $21$ and at most $30$. Choose the correct range.

Question illustration

$5 \lt x \lt 8$
$5 \le x \lt 8$
$5 \lt x \le 8$
$7 \lt x \le 10$
$x = 8$

Question 8

A boat takes $3 h$ downstream from A to $B$ and $4 h$ upstream. How many hours does drifting vegetation take from A to $B$?

$3$
$4$
$7$
$12$
$24$

Question 9

Seven consecutive integers have sum $33$ for their three smallest members. Find the sum of their three largest members.

$37$
$39$
$42$
$45$
$48$

Question 10

$BA = 6 \,\mathrm{cm}$ is tangent to circle $\left(O\right)$ at $A$. Secant $BCD$ has $CD = 9 \,\mathrm{cm}$. Find $BC$.

Question illustration

$2 \,\mathrm{cm}$
$3 \,\mathrm{cm}$
$5 \,\mathrm{cm}$
$6 \,\mathrm{cm}$
$4 \,\mathrm{cm}$

$A , B , C , D$ are concyclic; $P$ is the exterior intersection of $AB$ and $DC$. $\angle ADB = 54 ^{\circ} , \angle ACD = 58 ^{\circ} , \angle PBC = 80 ^{\circ}$. Find $\angle APD$.

Question illustration

$32 ^{\circ}$
$16 ^{\circ}$
$84 ^{\circ}$
$42 ^{\circ}$
$40 ^{\circ}$

Question 12

$ABCD$ is cyclic and $AC \perp BD$ at $I$, with $IA = 3 \,\mathrm{cm} , IB = 4 \,\mathrm{cm} , ID = 6 \,\mathrm{cm}$. Find $CD$.

Question illustration

$8 \,\mathrm{cm}$
$7 \,\mathrm{cm}$
$12 \,\mathrm{cm}$
$10 \,\mathrm{cm}$
$18 \,\mathrm{cm}$

Question 13

Ten distinct positive integers sum $100$. What maximum largest term is possible?

$45$
$55$
$65$
$75$
$91$

Question 14

For a three-digit number, deleting either the first or last digit leaves a perfect square. Find the sum of all such numbers.

$1013$
$1177$
$1465$
$1993$
$2016$

Question 15

There are $60$ seats around a circular table. Seat students so each has exactly one occupied neighboring seat. What is the maximum number?

$10$
$20$
$30$
$40$
$60$

Question 16

Which ratio can represent the successive angles of a cyclic quadrilateral?

$1 : 2 : 3 : 4$
$1 : 2 : 4 : 3$
$1 : 3 : 2 : 4$
$1 : 4 : 3 : 2$
$1 : 4 : 2 : 3$

Question 17

Two parallel lines contain six and four marked points. How many triangles have vertices among them?

Question illustration

$6$
$15$
$36$
$60$
$96$

Question 18

The $1500$-digit number $20142014 \ldots 2014$ is shortened by deleting digits. What maximum deletions leave digit sum $2014$?

$860$
$868$
$996$
$611$
$375$

Question 19

An equilateral triangle has side $1$. Find its inradius.

$1$
$\frac{1}{2}$
$\frac{\sqrt{3}}{2}$
$\frac{1}{2 \sqrt{3}}$
$\frac{\sqrt{3}}{4}$

Question 20

A unit circle circumscribes an equilateral triangle. Find the area of the indicated circular segment.

Question illustration

$\frac{\pi - 3 \sqrt{3}}{12}$
$\frac{4 \pi - 3 \sqrt{3}}{12}$
$\frac{\pi - \sqrt{3}}{3}$
$\frac{2 \pi - \sqrt{3}}{12}$
$\frac{4 \pi - 3 \sqrt{3}}{3}$