VTMO 2026 - Grade 9 - Semifinal - SET 07 - Self-study

Let $[a]$ denote the greatest integer not exceeding $a$. For how many integers $n$ is $A=\left|\left[\dfrac{5n^2+9n+1}{2}\right]\right|$ prime?

The first box has 4 blue and 6 red balls of one size. The second has 8 blue and 2 red balls, all larger than those in the first box. Choose $m$ balls from the first box and $n$ from the second to guarantee two of the same color but different sizes. Minimize $m+n$.

Triangle $ABC$ has $\angle ACB=45^\circ$. Construct an isosceles right triangle $ACI$, right-angled at $C$, outside $ABC$. Let $E=AC\cap BI$. If $BI=\sqrt2\,BC$, find $AE/CE$.

Triangle $ABC$ is right isosceles at $A$, with $AB=6$cm. Points $M\in AB,N\in AC$ satisfy $AM=4$cm,$AN=2$cm. Points $P,Q\in BC$ satisfy $PM\perp MN$ and $QN\perp MN$. Find the area of $MNQP$ in square centimeters.

Evaluate $\left(1+\frac1{100}\right)\left(1+\frac1{101}\right)\cdots\left(1+\frac1{2012}\right)$.

A luggage code has three digits from 0 to 9. Guess 923 has exactly one correct digit in the correct position. Guess 958 has exactly one correct digit, but in the wrong position. Guess 379 has exactly two correct digits, both in wrong positions. Find the code.

Let $a,b,c,d$ satisfy $a+b+c+d=10$. Prove that $ab+bc+cd+da\ne50$.

A rectangle has area 104 square meters, and its length exceeds its width by 5m. Find its perimeter.

At 3: 15, what is the smaller angle between the hour and minute hands?

A monkey is satisfied when it receives three different kinds of fruit. There are 20 apples,30 pears,40 bananas and 50 peaches. Fruits cannot be split. What is the largest number of monkeys that can be satisfied?

Test a16-digit sequence as follows: reverse the digits; leave odd positions unchanged and double even positions; replace any two-digit product by the sum of its digits; then require the total to be divisible by 10. For the exercise sequence $2853\ 00QP\ P003\ 6002$, how many digit pairs $(P,Q)$ with $P\le Q$ pass the test?

Example supplied in the source itself: enter 4634 8932 1298 2767 and reverse it to 7672 8921 2398 4364. The odd-position digits sum to A=7+7+8+2+2+9+4+6=45. Doubling the even-position digits gives 12,4,18,2,6,16,6,8; their digit sum is B=(1+2)+4+(1+8)+2+6+(1+6)+6+8=45. Thus C=A+B=90 ends in 0, so this example passes the check. The source introduces this as the Luhn algorithm for detecting input errors in a long digit sequence.

Triangle $ABC$ has $\angle B=60^\circ$, $BC=8$cm and $AB+AC=12$cm. Find $AB$.

Seven distinct integers have the property that the sum of any four exceeds the sum of the other three. Find the smallest possible total of the seven integers.

A bag contains one red, one yellow and one blue ball. A ball is drawn and replaced each time. Red scores 5, yellow 3 and blue 1. Minh makes 6 draws and scores 20. How many ordered result sequences are possible?

Trains A and B start simultaneously from stations 800 km apart and travel toward each other. A travels three times as fast as B, and they meet after 4 hours. Find their speeds in km/h.

All 46 lights are initially off. Each move toggles exactly three chosen lights. What is the minimum number of moves needed to turn all 46 on?

Let $A=\dfrac{20}{\sqrt2+\sqrt3+\sqrt7}$. Which identity is correct?

How many triangles have perimeter 27 cm and integer side lengths $x\lt y\lt z$ in centimeters?

How many three-digit numbers have their first digit greater than their last digit? Examples given in the source: 412,521,834.

Let $a=2\sqrt7$, $b=\sqrt6+\sqrt8$, and $c=\dfrac{53}{10}$. Compare the three numbers.