Find a positive integer m such that is a perfect square number.

Hi! I found this interesting geometry problem and I would really appreciate help with the proof.

Let ABC be an acute triangle, and let T be the Fermat (Torricelli) point of triangle ABC. Let A1, B1, and C1 be the feet of the perpendiculars from T to the sides BC, AC, and AB, respectively. Let ω be the circle passing through points A1, B1, and C1. Let A2, B2, and C2 be the second points where ω intersects the sides BC, AC, and AB, respectively (different from A1, B1, C1).

Prove that triangle A2B2C2 is equilateral.

A large sphere with radius 7 contains three smaller balls each with radius 3 . The three balls are each externally tangent to the other two balls and internally tangent to the large sphere. There are four right circular cones that can be inscribed in the large sphere in such a way that the bases of the cones are tangent to all three balls. Of these four cones, the one with the greatest volume has volume . Find .

Let be an acute triangle inscribed in a circle of center . If the altitudes intersect at and the circumcenter of is , prove that is a parallelogram.

Prove that there is not a positive integer such that numbers are both perfect squares.

We say a positive integer is good if there exists a permutation of such that is perfect square for all . Determine all the good numbers in the set .

We denote by the set of all positive real numbers.

Find all functions which have the property:

for all positive real numbers and .

Proposed by Nikolai Nikolov, Bulgaria