Teaching Olympiad-Level Math since2020

One toy car costs as much as one ball and two cubes together, and two balls cost as much as one car and one cube together. How many cubes is one car worth?

Points A, B, C, and D are marked on a straight line in some order. It is known that AB = 100, AC = 12, BD = 35, and DC = 123. Find the length of segment BC.

How many natural numbers less than 20,000 are there whose digits consist only of odd digits?

The classroom is decorated with red, blue, and white balloons. The total number of balloons is more than the number of white ones by 31, more than the number of blue ones by 22, and more than the number of red ones by 27. How many balloons are there in total?

A store received 223 liters of oil in 10-liter and 17-liter containers. How many containers of oil did the store receive?

Armen has two 500-dram coins, three 200-dram coins, and four 100-dram coins. He wants to put the coins into his right and left pockets so that neither pocket is empty. In how many different ways can he do this?

In an isosceles triangle ABC with base BC, the altitude AH is drawn. Points K and L are chosen on the sides AB and AC respectively, such that ∠ KHB =∠ LHC. Prove that BL = KC.

A 3 × 3 white square is divided into 9 equal squares (cells). In how many ways is it possible to color some or all of the cells black so that every white cell has exactly two neighboring white cells (two cells are neighbors if they share a common side)?

The numbers from 1 to 50 are written in a row (1, 2, 3, …, 49, 50). At each step you are allowed to swap two numbers if there is exactly one number between them (for example, in the first step you may swap 2 and 4). Is it possible, after some number of steps, to obtain the sequence (50, 49, …, 3, 2, 1)?

Find the natural number if it is known that the sum of its three smallest divisors is 13, and the sum of its three largest divisors is 329.

On the sides   and   of a square  , points   and   are marked so that the rays AM and   divide the angle   into three equal parts.   is the altitude of triangle . Find the angle  .

On a chessboard, 17 squares are marked. Prove that it is possible to choose 2 of these squares such that it is impossible to move from one to the other in one or two knight moves.