Let be a positive integer. Consider a figure of a equilateral triangle of side and splitted in small equilateral triangles of side . One will mark some of the vertices of the small triangles, such that for every integer , there is not any trapezoid(trapezium), whose the sides are , with all the vertices marked. Furthermore, there are no small triangle(side ) have your three vertices marked. Determine the greatest quantity of marked vertices.

In the triangle is the center of the inscribed circle, point lies on the side of , with . Prove that the distance from point to line is equal to the diameter of the circle inscribed in triangle

Determine all functions from the set of non-negative integers to itself such that , whenever , are non-negative integers satisfying .

Let be the set of real numbers. Let be a function such that for every . Assume that the inequality is strict for some .

Prove that either for every or for every .

Given trapezium ABCD with basis AB and CD parallel. Choose a point E on side BC and a point F on side AD such that AE Is parallel to FC . Prove that DE Is parallel to FB.

Let be an integer. Ion draws a regular -gon and all its diagonals. On every diagonal and edge, Ion writes a positive integer, such that for any triangle formed with the vertices of the -gon, one of the numbers on its edges is the sum of the two other numbers on its edges. Determine the smallest possible number of distinct values that Ion can write.

Find such that coprime, and .

Let be an acute-angled triangle and , and be the feet of the altitudes from , and , respectively. On the rays , and , we have points , and respectively, lying outside of , such that
If the intersections of and , and , and and are , and respectively, prove that , and have a common point.

Find all positive integers such that

For , , and , let denote the greatest common divisor of .
Prove that is an integer.

Dan Schwarz, Romania

Question:
Solve for f:Z-->Z
My solution:
At a=0,
Take t=f(b) to get
Therefore, f(x)=2x+n where n=f(0)
Could someone please clarify if this is right or wrong?

Given a pyramid where is a parallelogram.
The intersection of the diagonals of the base is point .
Point is connected to the midpoint of , point to the midpoint of ,
point to the midpoint of and point to the midpoint of .
a) Prove: the four lines are concurrent in a point .
b) Calulate .

Let be a trapezium such that . Let be the intersection of diagonals and . Suppose that and . Prove that the internal angle bisector of is perpendicular to .