Given triangle ABC inscribed in (O) with orthocenter H. Let K be the midpoint of AH. Take E,F on AC, AB so that BKE=CKF=90. Prove that E,O,F are collinear.

Determine all positive integers and such that

Let be a real number. Determine all natural numbers for which there exist distinct natural numbers , , , such that

(where denotes the greatest common divisor of natural numbers and ).

In triangle with , point lies on the circumcircle of such that . The line through parallel to intersects in and in . Prove that the center of the circumcircle of triangle lies on the circumcircle of triangle .

Proposed by Prithwijit De

Are there any contest questions that seem simple but are actually difficult? :-D

Let three real numbers , no two of which are . Prove that:

Inspiration

points and some chords between them is given.
In every step we can take two chords with a common point other than and erase exactly one of and draw let be the minimum of chords after some steps. Find the maximum of over all initial positions.
In every step we can take two chords with a common point other than and erase both of and draw let be the minimum of chords after some steps. Find the maximum of over all initial positions.

Proposed by Afrouz Jabalameli, Abolfazl Asadi

Let be a positive integer and let and be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer such that at least one of the numbers and is divisible by

Cătălin Liviu Gherghe

Help me with this:

Let is a positive integer. Prove that