VTMO 2026 - Grade 9 - Semifinal - SET 3 - Self-study

In the figure, triangle ABC is right-angled at A, H lies on BC with $AH\perp BC$, K lies on AC, and I is the foot of the perpendicular from A to BK. Given $AB=2BH$, find $\angle BIH+\angle BCA$.

How many nine-digit numbers $\overline{a_1a_2\cdots a_9}$ have distinct nonzero digits and satisfy all the following conditions: $a_1\in\{1,2\}$; $a_9\in\{8,9\}$; and, for $1\lt k\lt9$, $a_k\in\{k-1,k,k+1\}$?

How many prime numbers can be written as $n^n+1$ and are less than $8^{20}$?

A natural number is called super-even if every decimal digit is even. How many four-digit super-even numbers have a super-even product when multiplied by 3?

The positive integers are arranged in rows as shown: row 1 contains 1; row 2 contains 2,3,4; row 3 contains 5,6,7,8,9; row 4 contains 10,11,12,13,14,15,16; and the pattern continues. In which row does 2012 occur?

Solve the equation: $(1+x+x^2+x^3)(1+x+x^2+\cdots+x^9)=(1+x+x^2+\cdots+x^6)^2$.

Triangle ABC is right-angled at A, with AB=24 and AC=70. A point D lies on AC so that the incircles of triangles ABD and BDC are tangent to each other. Find the inradius of triangle BDC.

A bus has two types of seats: single seats and two-person seats. On one occasion, with 13 passengers, 9 seats are empty; on another, with 10 passengers, 6 seats are empty. How many seats does the bus have, excluding the driver’s seat?

A quadratic polynomial P(x) satisfies P(3)=0, P(5)=2 and P(7)=12. Find $P(9)+P(11)$.

Let x,y,z be pairwise distinct positive integers satisfying x+y+z=100. Find the greatest possible value of $P=xy+2xz+3yz$.

Real numbers x,y satisfy $y=\sqrt{\dfrac{2020x+y}{2020x-y}}+\sqrt{\dfrac{y+2020x}{y-2020x}}+2020$. Find xy.

Square ABCD has side AB=8 cm. M,N,P,Q are the midpoints of AB,BC,CD,DA respectively. X,Y,Z,T are the midpoints of AM,BN,CP,DQ respectively. Find the area, in square centimetres, of the intersection of squares MNPQ and XYZT.

Integers a,b satisfy 3a+b=30. Find the minimum of $P=3a^2+b^2$.

A triangle has side lengths x+1, x+2 and x+3. Its perimeter is greater than 21 and no greater than 30. Which statement correctly describes x?

Among triangles with side-length triples (a,b,c) listed below, which has the largest area?

Square ABCD has side AB=2. Three circles, with centres $O_1,O_2,O_3$ and the same radius r, lie inside it. Circle $O_1$ touches AB,AD and circle $O_3$; circle $O_2$ touches AB,BC and circle $O_3$; circle $O_3$ touches CD and the other two circles. Find r.

What is the smallest number of odd integers that must be selected from $\{123,124,125,126,\ldots,2014,2015,2016\}$ to guarantee that two selected numbers have sum 2016?

Three players on the girls’ football team at Euclid High School have this conversation.

Ashley: “I have just noticed that all our jersey numbers are two-digit primes.”
Bethany: “The sum of your two jersey numbers is the day of the month of my birthday; you have just attended it!”
Caitlin: “Yes, how funny: the sum of your two jersey numbers is also the day of my birthday, which is coming soon.”
Ashley: “I now realize that both of you were born in this month. And the sum of your two jersey numbers equals today’s day of the month!”

What is Caitlin’s jersey number?

Find all prime numbers p,q satisfying $2p+6q=(p-q)^3$.

For a natural number N, let A be N rounded to the nearest hundred and B be N rounded to the nearest ten. Given A−B=20 and N $\lt$ 1000, how many N satisfy the conditions?