Problems

A numeric set $M$ containing 2003 distinct numbers is such that for every two distinct elements $a,b$ in $M$, the number $a^2+b\sqrt{2}$ is rational. Prove that for any $a$ in $M$ the number $q\sqrt{2}$ is rational.

Find $x^3+y^3$ if $x+y=5$ and $x+y+x^{2}y+x{y}^2=24$.

Compute the following: $$\frac{(2001\times 2021+100)(1991\times 2031+400)}{{2011}^4}.$$

Prove that, if $b=a-1$, then $$(a+b)(a^2+b^2)(a^4+b^4)\cdots (a^{32}+b^{32})=a^{64}-b^{64}.$$

Prove the following formulae are true: $$\begin{array}{rl}{a}^{n+1}-b^{n+1}& =(a-b)(a^n+a^{n-1}b+\cdots +b^n);\\ a^{2n+1}+b^{2n+1}& =(a+b)(a^{2n}-a^{2n-1}b+a^{2n-2}{b}^2-\cdots +b^{2n}).\end{array}$$

Solve the inequality: $\lfloor x\rfloor \times \{x\}<x-1$.