Let be a point in the plane, and a line not passing through . Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is not marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane.

(i) Can Evan construct* the reflection of over ?

(ii) Can Evan construct the foot of the altitude from to ?

*To construct a point, Evan must have an algorithm which marks the point in finitely many steps.

Proposed by Zack Chroman

Suppose is a cyclic quadrilateral with . Let and be the feet of perpendiculars from and to respectively. Prove that .

An infinite increasing sequence of positive integers is called central if for every positive integer , the arithmetic mean of the first terms of the sequence is equal to .

Show that there exists an infinite sequence of positive integers such that for every central sequence there are infinitely many positive integers with .

Let a,b be nonnegative real numbers such that . Prove that :

Let and Prove that

Let be positive integers. If for each positive integer we have that Prove that .

(Silouanos Brazitikos, Greece)

Ivan bought cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number . How many cats of each breed does he have?

Let and Prove that

Problem. Given non-negative real numbers satisfying Prove that

The number 2021 is fantabulous. For any positive integer , if any element of the set is fantabulous, then all the elements are fantabulous. Does it follow that the number is fantabulous?