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VTMO 2019 — Grade 9 — semifinal

Question 1

$a + b + c = 10$ and $ab + bc + ca = 15$. Find a $^{2} + b^{2} + c^{2}$.
  1. $70$
  2. $65$
  3. $60$
  4. $75$
  5. $85$

Question 2

A quadratic $P$ has $P \left(3\right) 0 , P \left(5\right) 2 , P \left(7\right) 12$. Find $P \left(9\right) + P \left(11\right)$.
  1. $81$
  2. $86$
  3. $89$
  4. $88$
  5. $82$

Question 3

$P \left(x\right) = x^{2} - 2019 x + c$ has two prime roots. Find $c$.
  1. $4034$
  2. $2019$
  3. $2021$
  4. $2017$
  5. $2020$

Question 4

$AD$ bisects angle A in triangle $ABC , D \in BC$ and $\angle ADB : \angle ADC = 4 : 5$. The perpendicular bisector of $BC$ meets $AC$ at $M$. Find $\angle ABM$.
  1. $10 ^{\circ}$
  2. $20 ^{\circ}$
  3. $50 ^{\circ}$
  4. $90 ^{\circ}$
  5. $100 ^{\circ}$

Question 5

$AB = BD = BC$ and $\angle DBC = 76 ^{\circ}$ in quadrilateral $ABCD$ as pictured. Find $\angle DAC$.

Question illustration
  1. $36 ^{\circ}$
  2. $38 ^{\circ}$
  3. $40 ^{\circ}$
  4. $52 ^{\circ}$
  5. $32 ^{\circ}$

Question 6

$A , B , C , D$ have $76$ candies total. Simultaneously they keep $1 , 2 , 3 , 4$ candies respectively and split their remainders equally among the other three. Final totals are equal. How many did A originally have?
  1. $13$
  2. $19$
  3. $25$
  4. $30$
  5. $33$

Question 7

Hoa travels at constant speed. Distances at $1$ pm, $2$ pm, $3$ pm are two-digit $ab$, reversed ba, and $a = 0 b$ respectively. Find her speed in km $/ h$.
  1. $54$
  2. $45$
  3. $63$
  4. $36$
  5. $40$

Question 8

How many three-digit numbers have adjacent digits differing by at most $1$? Examples $123 , 434 , 987 , 221 , 555$.
  1. $75$
  2. $60$
  3. $90$
  4. $80$
  5. $35$

Question 9

A diamond number remains unchanged after $180 ^{\circ}$ rotation. In the stated font, only $0 , 8$ and $6$↔$9$ are allowed; $11$ is excluded. How many six-digit diamond numbers exist?
  1. $45$
  2. $52$
  3. $48$
  4. $50$
  5. $65$

Question 10

$A , B , C , D$ lie in order on a unit circle; $AB \parallel CD , AB = \frac{CD}{2}$ and $AC \perp BD$. Find the area.
  1. $9 \frac{\sqrt{2}}{5}$
  2. $\frac{9}{5}$
  3. $\frac{9}{2}$
  4. $9 \frac{\sqrt{2}}{2}$
  5. $9 \frac{\sqrt{5}}{2}$

Question 11

A grasshopper starts at the origin, jumping between lattice points by distances at most $\sqrt{5}$. What is the fewest jumps to $\left(79 , 97\right)$?
  1. $58$
  2. $59$
  3. $60$
  4. $61$
  5. $62$

Question 12

A prime date has prime day and month numbers. How many prime dates are there in $2019$?
  1. $53$
  2. $55$
  3. $54$
  4. $60$
  5. $52$

Question 13

A ten-step staircase allows a stride of $1$ through $10$ steps. How many ways reach the top?
  1. $480$
  2. $512$
  3. $360$
  4. $1024$
  5. $720$

Question 14

$ABC$ has perimeter $100 \,\mathrm{cm}$. A tangent to its incircle parallel to $BC$ meets $AB , AC$ at $M , N$. $MN = 12 \,\mathrm{cm}$ and $BC \gt 2 MN$. Find $BC$.
  1. $20$
  2. $30$
  3. $24$
  4. $32$
  5. $36$

Question 15

Draw six times with replacement: red $5$ points, yellow $3$, blue $1$. How many color sequences total $20$ points?
  1. $120$
  2. $126$
  3. $128$
  4. $3$
  5. $18$

Question 16

A contains positive integers at most $2019$, and every product of two members is a square. What is the largest size?

Question 17

The source defines $S \left(n + 1\right)$ as the number of trailing zeros of $n$!. Find the least $n$ with $S \left(n + 1\right) - S \left(n\right) = 5$.

Question 18

A line through $\left(2 , 0\right)$ meets $y = ax^{2} , a \ne 0$ at two distinct points with abscissas $x _{1} , x _{2}$. Find $\frac{1}{x} _{1} + \frac{1}{x} _{2}$.

Question 19

Representatives of five countries sit around a circular table. Every pair of countries must have adjacent representatives. What is the fewest people?

Question 20

Two concentric circles have radii $1 , 2$, and $OM = 3$. Tangents from $M$ to the smaller circle meet the larger at $A , B$ and $C , D$, with $MA \lt MB , MC \lt MD$. Find $BD$.