Olympiad problems

2011 USAMTS Problems, 2

Tags: algebra , 3d geometry , factorization , difference of cubes , geometric series , geometry

Let $x$ be a complex number such that $x^{2011}=1$ and $x\neq 1$. Compute the sum \[\dfrac{x^2}{x-1}+\dfrac{x^4}{x^2-1}+\dfrac{x^6}{x^3-1}+\cdots+\dfrac{x^{4020}}{x^{2010}-1}.\]

1990 Baltic Way, 15

Tags: geometry , modular arithmetic , algebra , 3d geometry , factorization , difference of cubes , special factorizations

Prove that none of the numbers $2^{2^n}+ 1$, $n = 0, 1, 2, \dots$ is a perfect cube.

1996 Hungary-Israel Binational, 2

Tags: special factorizations , geometry , algebra , 3d geometry , factorization , difference of cubes , modular arithmetic

$ n>2$ is an integer such that $ n^2$ can be represented as a difference of cubes of 2 consecutive positive integers. Prove that $ n$ is a sum of 2 squares of positive integers, and that such $ n$ does exist.

2022 AMC 10, 13

Tags: prime , difference of cubes

The positive difference between a pair of primes is equal to $2$, and the positive difference between the cubes of the two primes is $31106$. What is the sum of the digits of the least prime that is greater than those two primes? $\textbf{(A) } 8 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 16$