Let be a hyperboloid of one sheet and let be another quadric that intersects the hyperboloid at the curve . Let be a point on , and let be a line through in one specific ruling of the hyperboloid. Let this line intersect again at , now define to be the line through in the opposite ruling. Similarily define . Prove that if then this is true for all initial choices of .

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Prove that equation doesn't have any solution on integers.

Let be positive integers such that is a square. Prove that is not prime.

Evan o'Dorney

Let be a real polynomial of degree that has only simple roots. It is possible to determine a positive quantity so that for every pair of positive real numbers , with , every transformed polynomial of the form has exactly or sign variations.

Find all positive integers such that

Proposed by Dorlir Ahmeti, Albania

with even-degree
Prove that there is a non-negative integer k such that

has no real root

Let be a real numbers such that Prove that
Where

Ok so there's been no geo marathon here for more than 2 years,so lets start one,rules remain same.
1st problem.
Let be a cyclic quadrilateral with and let and be the feet of altitudes from to the lines and ,.Prove bisects .
P.s._eeezy ,try without ss line.

Find all functions is strictly increasing : such that:
i)
ii)

For a positive integer , let be the set of positive integers that do not exceed and are coprime to . Define as the smallest positive integer that allows to be partitioned into disjoint subsets, each forming an arithmetic progression.

Prove that there exist infinitely many pairs satisfying , , and .