Explaining, convincing and proving

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

Find all the solutions to the this equation.

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

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

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