IMO Problems 2021

IMO Problems 2021

Day 1 19/July/2021

IMO 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.

IMO 2021 Problem 2

Show that the inequality
\sum ^{n}_{i=1}\sum ^{n}_{j=1}\sqrt{\left| x_{i}-x_{j}\right| }\leq \sum ^{n}_{i=1}\sum ^{n}_{j=1}\sqrt{\left| x_{i}+x_{j}\right| }
holds for all real numbers x₁, x₂, …, xₙ.

IMO 2021 Problem 3

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

Day 2 20/July/2021

IMO 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.

IMO 2021 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 favorite 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.

IMO 2021 Problem 6

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