Given two charges of any magnitude , a third charge collinear with them , exists such that it is in equillibirum , Prove that if a fourth charge in the same line exists such that it is in equillibrium , then the 3rd charge and the fourth charge are harmonic conjugates with respect to the two fixed charges . , For example if two +q charges are fixed then if in their midpoint placed a charge -q , it is in equillibrium , also if the same charge -q is placed at infinity the system is again in equillibrium , and the midpoint and the point at infinity are harmonic conjugates .

Problem. Let Prove that When does equality hold?

Find all solutions in the nonnegative integers to .

(Shining Sun, USA)

Let be a triangle, and let be an arbitrary point on line , where is the circumcenter of . Define and as the orthocenters of triangles and . Prove that passes through a fixed point which is independent of the choice of .

(Kritesh Dhakal, Nepal)

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 a positive integer greater than . Find all functions satisfying:
for all integers

Let be a polynomial with integer coefficients. Define and for .

If there exists a natural number such that , then prove that either or .

Let be a parallelogram such that Let and be the midpoints of and respectively. Assuming that is a cyclic quadrilateral, find