Let be relatively prime positive integers. Calculate

Integers are different at . If are also different at , we call the ordered -tuple lucky. For which positive integers , one can find a lucky -tuple?

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 .

In an acute-angled triangle , be the orthocenter of it and be any point on the side . The points are on the segments , respectively, such that the points and are cyclic. The segments and intersect at is a point on such that is tangent to the circumcircle of triangle at and intersect at . Prove that the points and lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus

. Prove that


(I don't know whether the equality holds)

Let denote a triangular grid of side length consisting of pegs. Charles the Otter wishes to place some rubber bands along the pegs of such that every edge of the grid is covered by exactly one rubber band (and no rubber band traverses an edge twice). He considers two placements to be different if the sets of edges covered by the rubber bands are different or if any rubber band traverses its edges in a different order. The ordering of which bands are over and under does not matter.
For example, Charles finds there are exactly different ways to cover using exactly two rubber bands; the full list is shown below, with one rubber band in orange and the other in blue.
IMAGE
Let denote the total number of ways to cover with any number of rubber bands. Compute the remainder when is divided by .

Ethan Lee

Let be a triangle, and let be its altitudes. Let be its orthocenter, and let and be the circumcenters of triangles and . Let be the second intersection of the circumcircles of triangles and . Prove that the lines , , and -median of are concurrent.

Let be an integer and let be real numbers such that . Prove that
Proposed by Kapil Pause

Let be a positive integer which is not a perfect square, and consider the equation Let be the set of positive integers for which the equation admits a solution in with , and let be the set of positive integers for which the equation admits a solution in with . Show that .

A game of solitaire is played with red cards, white cards, and blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of and the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.