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.

Each task is worth 7 points

Problem 1

An integer n> 100 is given. Vanya wrote the numbers n, n + 1, …, 2n on n + 1 cards, each one once. Then he shuffled a deck of these cards and divided it into two piles. Prove that at least one of the two piles contains two cards, the sum of the numbers on which is an exact square.

Task 2

Prove that for any real numbers x


, …, x


the inequality holds

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


and O


– centers

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

EF and O




intersect at one point.

Day 2

Time to work: 4 hours 30 minutes.

Each task is worth 7 points

Problem 4

Given a circle Γ with center I. A convex quadrilateral ABCD is such that each of the segments AB, BC, CD, and DA touches Γ. Let Ω be the circumcircle of triangle AIC. The extension of the segment BA beyond the point A intersects Ω at the point X, the extension of the segment BC beyond the point C intersects Ω at the point Z. The extensions of the segments AD and CD beyond the point D intersect Ω at the points Y and T, respectively.

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

Problem 5

Chip and Dale have collected a nut for the winter of 2021. Chip numbered the nuts from 1 to 2021 and dug 2021 a small hole around their favorite tree. The next morning, he found that Dale had put a nut in each hole without worrying about order. Frustrated, Chip decided to reorder the nuts using the following sequence from 2021 actions: during the kth action, he swaps the places of the nuts adjacent to the nut numbered k.

Prove that there is a number k such that during the kth action, the nuts with numbers a and b are swapped such that a

Problem 6

Given an integer m> 2. In a finite set A consisting of (not necessarily positive) integers, there are subsets B


, B


, B


, …, B


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


is equal to m


… Prove that A contains at least m / 2 elements.

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