In a triangle with , denote its incenter and midpoint of by , , respectively. Let be the reflected point of wrt . Let the lines and meet at . Suppose that . Prove that .

Epsilon 1.3. Let ABC be a triangle and let D, E, F be the feet of the altitudes, with D on BC, E on CA, and F on AB. Let the parallel through D to EF meet AB at X and AC at Y. Let T be the intersection of EF with BC and let M be the midpoint of side BC. Prove that the points T, M, X, Y are concyclic.

Let , where is a positive integer.
Prove that there exists a subset of with exactly elements such that the sum of its elements is a prime number.

Help me plss!
Given an acute triangle . Points and lie on segments and , respectively. Lines and intersect at point . The circumcircles of triangles and intersect at a second point . The circumcircles of triangles and intersect at a second point . Point lies on segment such that . Prove that triangles and are similar.

An acute angled triangle has circumcenter . The lines and intersect at , while and intersect at and and intersect at . Show that if the triangles and are similar(with vertices in that order), than is equilateral.

Determine all four-digit numbers which are perfect squares and for which the equality holds:
.

Define on a coordinate plane, the parabola and the domain
Suppose that two lines passing through the origin touch .

(1) Let be a mobile point on the parabola . Let denote the distances between the point and the lines respectively. Find the coordinate of the point giving the minimum value of

(2) Draw the domain of the set of the points on a coordinate plane such that for all points over the domain , the inequality holds.

is a triangle in the plane. Find the locus of point for which form a triangle whose area is equal to one third of the area of triangle .