IMO 2017 Problem 1

For each integer aₒ > 1, define the sequence aₒ, a₁, a₂, … for n ≥ 0 as

$a_{n+1}=\begin{cases}\sqrt{a_{n}}, if \sqrt{a_{n}}\in Z\\ a_{n}+3, otherwise\end{cases}$

Determine all values of aₒ such that there exist a number A such that aₙ equal to A for infinitely many values of n.

IMO 2017 Problem 2

Let R be a set of real numbers. Determine all function f: R → R such that, for any real numbers x and y, f(f(x)f(y)) + f(x + y) = f(xy).

IMO 2017 Problem 3

A hunter and an invisible rabbit play a game in the Euclidean Plane. The rabbit’s starting point is Aₒ and the hunter’s starting point is Bₒ, are the same. After and n-1 rounds of the game, the rabbits is at the point Aₙ₋₁ and the hunter is at the point Bₙ₋₁. In the nth round of the game, three things occur in order.

(1) The rabbit moves invisibly to appoint Aₙ such that the distance between Aₙ₋₁ and Aₙ is exactly one.
(2) A tracking device reports a point Pₙ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between Pₙ and Aₙ is at most one.
(3) The hunter moves visibly to a point Bₙ such that the distance between Bₙ₋₁ and Bₙ is exactly one.

Is it always possible, no matter how the rabbit moves and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after 10⁹ rounds, she can ensure that the distance between her and the rabbit is at most 100?

IMO 2017 Problem 4

Let R and S be different points on a circle Ω, such that RS is not a diameter. Let L be that tangent line to Ω at R. Point T is such that S is the midpoint of the line segment RT. Point J is chosen on the shorter arc RS of Ω, so that the circumcircle Γ of triangle JST intersects L at two distinct points. Let A be the common point of Γ and L that is closer to R. Line AJ meets again Ω at K. Prove that the line KT is tangent to Γ.

IMO 2017 Problem 5

An integer N ≥ 2 is given. A collection of N(N + 1) soccer players, no two of whom are of the same height is stand in a row. Sir Alex wants to remove N(N – 1) players from this row leaving a new row of 2N players, in which the following N conditions hold.

(1) No one stands between the two tallest players.
(2) No one stands between the third and fourth tallest players.
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.
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(N) No one stand between the two shortest players.

Show that this is always possible.

IMO 2017 Problem 6

An ordered pair (x, y) of integers is a primitive point, if the greatest common divisor of x and y is 1. Given a finite set S a primitive points. Prove that there exist a positive integer n and the integers aₒ, a₁, a₂, …, aₙ, for each (x, y) in S, we have

aₒxⁿ + a₁xⁿ⁻¹y + a₂xⁿ⁻² y² + … + aₙ₋₁xyⁿ⁻¹ + aₙyⁿ = 1.