MK-MO 2014

In a plane, 2014 lines are distributed in 3 groups. in every group all the lines are parallel between themselves. What is the maximum number of triangles that can be formed, such that every side of such triangle lie on one of the lines?

Solve the following equation in $\mathbb{Z}$ :

$$3^{2a + 1}b^2 + 1 = 2^c$$

Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$ , circles $k_1$ and $k_3$ intersect and $C$ and $P$ , circles $k_2$ and $k_3$ intersect at $B$ and $P$ . Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$ , such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$ .

a) Prove that $\triangle XYZ$ is simular to $\triangle O_1O_2O_3$ b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$ . Is it possible to reach equation? $*Note:$ P$ denotes the area of a triangle*

Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$ . Prove that:

$$\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}$$

From an equilateral triangle with side 2014 we cut off another equilateral triangle with side 214, such that both triangles have one common vertex and two of the side of the smaller triangles lie on two of the side of the bigger triangle. Is it possible to cover this figure with figures in the picture without overlapping (rotation is allowed) if all figures are made of equilateral triangles with side 1? Explain the answer!