VTMO 2026 - Grade 9 - Semifinal - SET 2 - Self-study

Question 1

An isosceles trapezoid $ABCD$, with $AB\parallel CD$, is circumscribed about a circle with centre $O$. Given $AB=6$ and $CD=10$, find the diagonal $AC$.

1. 10
2. $8\sqrt2$
3. $3\sqrt{10}$
4. $2\sqrt{31}$
5. $4\sqrt6$

Question 2

In the coordinate plane Oxy, the line $y=-2x+3$ meets Ox and Oy at A and B respectively. A point I satisfies $[IAB]=[OAB]$. On which pair of lines must I lie?

1. $y=-2x$ and $y=x+3$
2. $y=-2x$ and $y=-2x+6$
3. $y=-2x+6$ and $y=x+3$
4. $y=x+3$ and $y=-x$
5. $y=-x$ and $y=-2x+6$

Question 3

The quadratic $3x^2-5x+1=0$ has distinct roots $x_1,x_2$. Let $g(x)=x^3-x+1$. Evaluate $M=g(x_1)g(x_2)$.

1. $\frac{53}{27}$
2. $\frac{23}{9}$
3. $\frac{53}{9}$
4. $\frac83$
5. $\frac13$

Question 4

Let a and b be distinct positive numbers. Which of the following is largest?

1. $\frac{2ab}{a+b}$
2. $\sqrt{ab}$
3. $\frac{a^2+b^2}{a+b}$
4. $\frac{a+b}{2}$
5. $\sqrt{\frac{a^2+b^2}{2}}$

Question 5

The number $20142014\ldots2014$ has 1500 digits. What is the greatest number of digits that can be deleted so that the sum of the remaining digits is 2014?

1. 860
2. 868
3. 996
4. 611
5. 375

Question 6

An equilateral triangle has side length 1. Find the radius of its incircle.

1. 1
2. $\frac12$
3. $\frac{\sqrt3}{2}$
4. $\frac1{2\sqrt3}$
5. $\frac{\sqrt3}{4}$

Question 7

Which listed value of x satisfies $\sqrt{(x-10)^2}=10-x$?

1. 15
2. 13
3. 11
4. 9
5. 17

Question 8

Two circles with centres O,I and radii 4,10 respectively intersect at A,B. The internal bisector of $\angle OAI$ meets the circles again at C,D respectively. Given $OI=2\sqrt{39}$, and J is the circumcenter of triangle BCD, find $OJ^2$.

Question 9

A 500-page book has its pages numbered from 1 to 500. How many occurrences of the digit 5 are used in these page numbers?

1. 100
2. 101
3. 200
4. 201
5. 50

Question 10

Bình travels from home to work. At 40 km/h he arrives 1 minute late; at 45 km/h he arrives 1 minute early. How far is his workplace from home?

1. 72 km
2. 48 km
3. 24 km
4. 18 km
5. 12 km

Question 11

A cyclic quadrilateral ABCD has the property that the incircles of triangles ABC, BCD, CDA and DAB all have radius $2-\sqrt2$. Find the smallest possible value of $AB\cdot AD$.

Question 12

The polynomial $P(x)=x^2-2019x+c$ has two real roots, both prime numbers. Find c.

1. 4034
2. 2019
3. 4038
4. 2017
5. 2020

Question 13

Real numbers x,y,z satisfy $x+y+z=0$ and $\dfrac{x(x+2y)}{y(y+2z)}=\dfrac1{2024}$. Find $P=\dfrac{z^2+2xz}{x^2+2xy}$.

1. 2024
2. −2023
3. 2023
4. −2025
5. 2025

Question 14

Points A,B,C,D lie in this order on a circle of radius 1. Given $AB\parallel CD$, $AB=\frac12CD$ and $AC\perp BD$, find the area of ABCD.

1. $\frac{9\sqrt2}{5}$
2. $\frac92$
3. $\frac95$
4. $\frac{9\sqrt2}{2}$
5. $\frac{9\sqrt5}{2}$

Question 15

Let $B=\{a,a+1,a+2,\ldots,a+1050\}$, where a is a nonnegative integer. Find the smallest a for which B contains exactly 10 perfect squares.

Question 16

Real numbers x,y,z satisfy $x+y+z=5$ and $x^2+y^2+z^2=9$. (a) Prove $1\le x,y,z\le7/3$. (b) Find the greatest and smallest values of $P=xyz$.

Question 17

Choose 100 distinct positive integers. A pair is called a “nice pair” when the ratio of its larger member to its smaller member is 2 or 3. What is the greatest possible number of nice pairs? One number may belong to several nice pairs.

Question 18

A is a set of positive integers not exceeding 2019 such that the product of any two of its elements is a perfect square. What is the greatest possible number of elements of A?

Question 19

Acute triangle ABC has $\angle BAC=60^\circ$. The altitudes BE and CF meet at H. Given $EF+BC=60$ cm, find BC.

1. 48 cm
2. 40 cm
3. 50 cm
4. 36 cm
5. 42 cm

Question 20

Find the units digit of $1\cdot3\cdot5\cdot7+3\cdot5\cdot7\cdot9+5\cdot7\cdot9\cdot11+\cdots+2011\cdot2013\cdot2015\cdot2017$.

1. 0
2. 2
3. 4
4. 6
5. 8