VTMO 2019 — Grade 8 — semifinal

Question 1

Real $a , b , c$ satisfy $a + b + c = 10 , ab + bc + ca = 15$. Find a $^{2} + b^{2} + c^{2}$.

$70$
$65$
$60$
$75$
$85$

Question 2

Calendar dates $a , b$; $c , d$ occupy $a = 2 \times 2$ block as pictured, $a + b + c + d = 52$. Find $a + b$.

Question illustration

$19$
$25$
$26$
$27$
$32$

Five pupils describe one quadrilateral: An an isosceles trapezoid; Binh a parallelogram; Cuong a rectangle; Danh a square; Minh a right trapezoid. The source says exactly one is wrong. Who? $A$. An $B$. Binh $C$. Cuong $D$. Danh $E$. Minh.

Question 4

Dice form one row on a table: two show eight faces, three eleven. If $50$ faces are visible, how many dice are there?

Question illustration

$12$
$13$
$14$
$15$
$16$

Question 5

Fifty coins start on one side. Three rounds flip $25 , 29 , 14$ coins, and all finish on the other side. How many were flipped three times?

$9$
$11$
$15$
$22$
$33$

Question 6

I is the incenter of $ABC$, and angles $BIC , CIA , AIB$ have ratio $8 : 9 : 13$. Find angle $A$.

$10$
$36$
$32$
$12$
$18$

Question 7

Two cars leave $A , B$ at $6 AM$ and meet at $7 : 12$. Car $1$ reaches $B$ at $8 AM$. When does car $2$ reach $A$?

$8 : 00$
$9 : 00$
$10 : 00$
$11 : 00$
$8 : 30$

Question 8

A binary string starts and ends $0$, has at least ten $1 s$, and every four consecutive digits contain $0$. Minimum length?

$11$
$12$
$14$
$15$
$16$

Question 9

How many three-digit numbers have adjacent digits differing by at most $1$?

$75$
$60$
$90$
$80$
$35$

Question 10

Start with $D = \{ 1 , 2 , 3 , 4 , 5 , 6 , 7 \}$. Each move decreases one entry $1$ and increases another $5$. Finally $D = \{ 2012 , 2013 , 2014 , 2015 , 2016 , 2017 , x \}$. Which $x$ is possible?

$2018$
$2017$
$2019$
$2020$
$2021$

Question 11

$a , b$ are the distinct real roots of $x^{2} - 10 x + 2$. Find $\frac{1}{a^{3} + 2 a} + \frac{1}{b^{3} + 2 b}$.

$1$
$2$
$\frac{1}{2}$
$\frac{5}{12}$
$\frac{12}{5}$

Question 12

$ABC$ is isosceles at $A , O$ inside, $OA = 3 \,\mathrm{cm} , OB = OC = 2 \,\mathrm{cm}$, and distance from $O$ to $AB = 1 \,\mathrm{cm}$. Find the distance from $O$ to $BC$.

$\frac{2 \sqrt{6} + 1}{3}$
$\frac{\sqrt{6} - 1}{3}$
$\frac{2 \sqrt{6} - 1}{3}$
$\frac{\sqrt{6} - 1}{2}$
$\frac{\sqrt{6} + 1}{2}$

Question 13

Quadrilateral $ABCD$ has $AB = BD = BC$ and angle $DBC = 76 ^{\circ}$. The source asks for angle $DBC$. Choices: $A = 36 ^{\circ} , B = 38 ^{\circ} , C = 40 ^{\circ} , D = 52 ^{\circ} , E = 32 ^{\circ}$.

Question illustration

Question 14

Trapezoid $ABCD$ has $AB$ parallel $CD , I$ midpoint $DC , IH$ perpendicular $AB , M , N$ midpoints $AD , BC$. Given $IH = IM = IN = ID$, find angle $DAB$.

$95$
$60$
$75$
$105$
$85$

Question 15

$S$ is a set of positive integers with minimum $5$, maximum $1023$, and no pair one double the other. What maximum size can $S$ have?

$680$
$768$
$512$
$678$
$350$