Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .

Matías has a rectangular sheet of paper , with .Initially, he folds the sheet along a straight line , where is a point on the side , so that vertex is located on side , as shown in the figure. Then folds the sheet again along a straight line , where is a point on side , so that vertex lies on the line ; and finally folds the sheet along the line . Matías observed that the vertices and were located on the same point of segment after making the folds. Calculate the measure of the angle .

Find all functions such that for any positive real numbers and ,

Proposed by ThE-dArK-lOrD, and Papon Tan Lapate, Thailand

We say that a finite set of points in the plane is balanced if, for any two different points and in , there is a point in such that . We say that is centre-free if for any three different points , and in , there is no points in such that .

(a) Show that for all integers , there exists a balanced set consisting of points.

(b) Determine all integers for which there exists a balanced centre-free set consisting of points.

Proposed by Netherlands

Let be a sequence of real numbers, and let be a fixed positive integer less than . We say an index with is good if there exists some with such that , where the indices are taken modulo . Let be the set of all good indices. Prove that .

Proposed by Mark Sellke