Let be real numbers such that : . Prove that :

find all pairs of such that

Let be a triangle. Let be the feet of the altitudes from respectively. Let be the projections of onto line . Show that .

Which positive integers have the property that is a perfect square for infinitely many positive integers ?

A teacher asks each of eleven pupils to write a positive integer with at most four digits, each on a separate yellow sticky note. Show that if all the numbers are different, the teacher can always submit two or more of the eleven stickers so that the average of the numbers on the selected notes are not an integer.

In Duckville there is a perpetual trophy with the words “Best child of Duckville” engraved on it. Each inhabitant of Duckville has a non-empty list (which never changes) of other inhabitants of Duckville. Whoever receives the trophy
gets to keep it for one day, and then passes it on to someone on their list the next day. Gregers has previously received the trophy. It turns out that each time he does receive it, he is guaranteed to receive it again exactly days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers for which we can be absolutely certain that she cannot receive the trophy again in days, given the above information.

Find all functions such that
for all real numbers

6. Circles and with equal radii are given. Let , be the intersection of the circles.
points and are on and such that they are inside and respectively.
Points , are on and respectively, such that and .Denote by as the other intersection of and . Prove that are concurrent.