# Problems from the International Mathematical Olympiad 2021 (solved in the comments)

I propose to stretch your brains and, like last year, solve problems from the math olympiad in the comments to this article. There were 6 problems, and they were given 2 days of 4.5 hours each. (Chur, in the answers no peeking!)

This summer, St. Petersburg hosted the 62nd International Mathematical Olympiad with the following results:

- The first place was taken by the Chinese team, which won six gold medals (208 points).
- Russian schoolchildren took second place with five gold and one silver medals (183 points)
- In third place is the South Korean team with five gold and one silver medals (172 points)

The first such Olympiad was held in 1959 in Romania, and then representatives of only seven countries took part in it. In 2021, more than 619 schoolchildren from 107 countries took part in the Olympiad.

**Russian national team**

The Russian national team was coached by the mathematics teacher of the Presidential Physics and Mathematics Lyceum No. 239 of St. Petersburg Kirill Sukhov, teachers from the Center for Pedagogical Excellence in Moscow Vladimir Bragin and Andrei Kushnir. Russia at the Olympiad was represented by:

- Ivan Bakharev (grade 10, St. Petersburg) – gold medal;
- Aydar Ibragimov (grade 11, Kazan / Moscow) – gold medal;
- Matvey Isupov (grade 11, Izhevsk) – gold medal;
- Andrey Shevtsov (grade 11, Moscow) – silver medal;
- Danil Sibgatullin (grade 11, Kazan / Moscow) – gold medal;
- Maxim Turevsky (grade 10, St. Petersburg) – gold medal, absolute second place in the overall ranking.

### Day 1

Time to work: 4 hours 30 minutes.

### Problem 1

### Task 2

Prove that for any real numbers x

### Problem 3

A point D inside an acute-angled triangle ABC, in which AB> AC, is such that

∠DAB = ∠CAD. Point E on segment AC is such that ∠ADE = ∠BCD; point F on segment AB

is such that ∠F DA = ∠DBC; a point X on the line AC is such that CX = BX. Points O

the circumscribed circles of triangles ADC and EXD, respectively. Prove that lines BC,

### Day 2

Time to work: 4 hours 30 minutes.

### Problem 4

Prove that AD + DT + TX + XA = CD + DY + YZ + ZC.

### Problem 5

### Problem 6

such that for each k = 1, 2, …, m the sum of elements of the set B