In triangle , draw exterior equilateral triangles on sides and to obtain and , respectively. Let be the intersection of the altitude through and the median through . Let be the intersection of the altitude through and line . Let be the intersection of the median through and the line . Prove that , , and all lie on a line.

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Let be a triangle inscribed in a circle, where the tangents to the circle at points and intersect at the point . Let be a point on the arc (not containing ) such that and . Let the lines and intersect at point . Let be the reflection of with respect to the line . The lines and intersect at point , and intersects the circumcircle of again at point .

Prove that the point lies on the circumcircle of .

A quadruple of positive integers with is called good if we can colour each integer red, blue, green or purple, in such a way that
of each consecutive integers at least one is coloured red;
of each consecutive integers at least one is coloured blue;
of each consecutive integers at least one is coloured green;
of each consecutive integers at least one is coloured purple.
Determine all good quadruples with

Given are positive real numbers satisfied . Prove that:

The number is written on the blackboard. Every time, Carmela can erase the number on the black board and replace it with a new number , if and only if is a perfect square. Prove or disprove that all positive integers can be written exactly once on the blackboard.

Let be a square with sides length . Let be a path within which does not meet itself and which is composed of line segments with . Suppose that for every point on the boundary of there is a point of at a distance from no greater than . Prove that there are two points and of such that the distance between and is not greater than and the length of the part of which lies between and is not smaller than .

Consider the numbers and the sets If has exactly three elements, prove that cannot be an integer.

Let be a positive integer not divisible by 3. Prove that there are infinitely many positive integers such that and are divisible by , where is the sum of digits of .

Dorel Miheț

Find the range of the function

At a maths contest books are given as prizes to students (each students gets one book). In how many ways can the organisers give these prizes if they have copies of one book and other books each in one copy?

Consider . Show that there exists such that

Laurențiu Panaitopol

Determine the lengths of the edges of a right tetrahedron of volume so that the sum of its edges' lengths is minumum.