Explaining, convincing and proving

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

The nth term of a sequence is given by the formula n^3 + 11n. Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

Explore a number pattern which has the same symmetries in different bases.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.