The triangle ABC has \(\widehat{B}-\widehat{C}=a\) . On the opposite beam of AC rays take D so that AD = AC . Caculate the \(\widehat{CBD}\) by a.

Write monomials uniformly with monomial x^{2}y such that at x = -1 and y = 1, the values of the monomers are natural numbers less than 10.

On the outside of a given triangle ABC , construct equilateral triangles ABE and ACF . Let M and P be the midpoints of BC and EF , respectively . Let H be orthogonal projection of A onto È . Prove that MP = MH.

How many positive integers n less than 2015 have the property that \(\dfrac{1}{3}+\dfrac{1}{n}\) can be simplified to a fraction with denominator less than n ?

Given an integer N greater than 1 , the sum of N and the second largest factor of N can be found . For example , with N = 55 , the sum is 55 + 11 = 66 . For how many integers is this sum equal to 42 ?

A square room has its floor paved with square tiles of the same size ( where only whole tiles are used and no tile is cut ) . There are two colors of tiles : white and blue . The blue tiles are used on the two diagonals of the floor . The rest of the floor is paved with white tiles . Find the number of tiles used for the whole floor , given that the number of white tiles is 839 more than the number of blue tiles.