International Mathematical Olympiad: IMO 2021

Created on 11 Sep, 2021
Last Update on 29 Sep, 2022

English (eng), day 1

Monday, 19. July 2021

Problem 1. Let n ≥ 100 be an integer. Ivan writes the numbers n, n+1, ..., 2n each on different cards. He then shuffles these n+1 cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

Problem 2. Show that the inequality

holds for all real numbers x1, ..., xn.

Problem 3. Let D be an interior point of the acute triangle ABC with AB > AC so that ∠DAB = ∠CAD. The point E on the segment AC satisfies ∠ADE = ∠BCD, the point F on the segment AB satisfies ∠FDA = ∠DBC, and the point X on the line AC satisfies CX = BX. Let O1 and O2 be the circumcentres of the triangles ADC and EXD, respectively. Prove that the lines BC, EF, and O1O2 are concurrent.

Language: English

Time: 4 hours and 30 minutes.
Each problem is worth 7 points.

English (eng), day 2

Tuesday, 20. July 2021

Problem 4. Let Γ be a circle with centre I, and ABCD a convex quadrilateral such that each of the segments AB, BC, CD and DA is tangent to Γ. Let Ω be the circumcircle of the triangle AIC. The extension of BA beyond A meets Ω at X, and the extension of BC beyond C meets Ω at Z. The extensions of AD and CD beyond D meet Ω at Y and T, respectively. Prove that AD + DT + TX + XA = CD + DY + YZ + ZC.

Problem 5. Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the k-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut k. Prove that there exists a value of k such that, on the k-th move, Jumpy swaps some walnuts a and b such that a < k < b.

Problem 6. Let m ≥ 2 be an integer, A be a finite set of (not necessarily positive) integers, and B1, B2, B3, ... , Bm be subsets of A. Assume that for each k = 1,2, ... , m the sum of the elements of Bk is mk. Prove that A contains at least m/2 elements.

Language: English

Time: 4 hours and 30 minutes.
Each problem is worth 7 points.

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