INMO 1988

Let $m_1,m_2,m_3,\dots,m_n$ be a rearrangement of the numbers $1,2,\dots,n$ . Suppose that $n$ is odd. Prove that the product $$\left(m_1-1\right)\left(m_2-2\right)\cdots \left(m_n-n\right)$$ is an even integer.

Prove that the product of 4 consecutive natural numbers cannot be a perfect cube.

Five men, $A$ , $B$ , $C$ , $D$ , $E$ are wearing caps of black or white colour without each knowing the colour of his cap. It is known that a man wearing black cap always speaks the truth while the ones wearing white always tell lies. If they make the following statements, find the colour worn by each of them: $A$ : I see three black caps and one white cap. $B$ : I see four white caps $C$ : I see one black cap and three white caps $D$ : I see your four black caps.

If $a$ and $b$ are positive and $a + b = 1$ , prove that $$\left(a+\frac{1}{a}\right)^2+\left(b+\frac{1}{b}\right)^2 \geq \frac{25}{2}$$

Show that there do not exist any distinct natural numbers $a$ , $b$ , $c$ , $d$ such that $a^3+b^3=c^3+d^3$ and $a+b=c+d$ .

If $a_0,a_1,\dots,a_{50}$ are the coefficients of the polynomial $$\left(1+x+x^2\right)^{25}$$ show that $a_0+a_2+a_4+\cdots+a_{50}$ is even.