Let triangle be circumscribed about circle , and let be the orthocenter of . The circle touches line at . The tangent to the circle at meets at . Let be the midpoint of , and let the line meet again at . The tangent to parallel to meets the line at . Prove that is tangent to .

Let be two nonzero real numbers such that Consider the functions and satisfying the condition (a) Prove that there exists a real number such that (b) Find all functions and that satisfy the given condition.

Let with is coprime to . It is known that for every , there always exist satisfying Prove that is a cube.

Let be a square and a point lies on the interior of such that triangle is equilateral. Evaluate

Find the probability that if four numbers from to (inclusive) are selected randomly without repetitions, then either all of them will be odd, or all will be divisible by , or all will be divisible by

Let be a scalene acute triangle with incenter and circumcircle . is the midpoint of small arc on and is the projection of onto the line passing through the midpoints of and . A circle with center is internally tangent to at , and touches segment . If the circle with diameter meets again at , prove that bisects .

Proposed by David Anghel

Passing memes between social platforms (and posting in October)...

Let be the set of rooted binary trees, where we include the empty tree with 0 nodes. There is an obvious bijection between nonempty trees in and pairs of trees in , by taking the left and right subtree of . So there is an obvious bijection between and {the empty tree, plus pairs of trees in }.

Let's write this as , so . In other words, is a sixth root of unity.

So . Unfortunately this is absurd because there is more than one 6-tuple of trees.

But, let's try again and write . And in fact:

• There is a "very explicit" bijection between the set of 7-tuples of trees, and the set of all trees. Here "very explicit" means that to map 7 trees to 1, you just need to inspect the 7 trees down to a fixed finite depth, independent of what the trees actually are, to construct some part of the 1 tree. Then you can paste the subtrees beneath that depth to subtrees of what you constructed.
• This is tight: there exists a very explicit bijection between the set of -tuples of trees and trees iff .

Now the question is, of course: how did treating , an infinite set of combinatorial objects, as a complex number produce a reasonable theorem?

It turns out there's some ring theory you can pull to see why this is the case. Some arXiv papers on the subject: https://arxiv.org/pdf/math/9405205v1.pdf https://arxiv.org/abs/math/0212377