Olympiad problems

2002 IMO Shortlist, 1

What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]

2008 International Zhautykov Olympiad, 2

Tags: polynomial , algebra , factorization , sum of cubes

A polynomial $ P(x)$ with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable $ x$) with integer coefficients.For example,the polynomials $ x^3 \minus{} 1$ and $ 9x^3 \minus{} 3x^2 \plus{} 3x \plus{} 7 \equal{} (x \minus{} 1)^3 \plus{} (2x)^3 \plus{} 2^3$ are good. a)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7$ good? b)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7 \plus{} 3x^{2008}$ good? Justify your answers.

2007 Moldova Team Selection Test, 1

Tags: algebra , geometry , 3d geometry , factorization , sum of cubes , number theory

Find the least positive integers $m,k$ such that a) There exist $2m+1$ consecutive natural numbers whose sum of cubes is also a cube. b) There exist $2k+1$ consecutive natural numbers whose sum of squares is also a square. The author is Vasile Suceveanu