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IMO Shortlist 1992 problem 9

Let $f(x)$ be a polynomial with rational coefficients and $\alpha$ be a real number such that $\alpha^3 - \alpha = [f(\alpha)]^3 - f(\alpha) = 33^{1992}.$ Prove that for each $n \geq 1,$ $\left [ f^{n}(\alpha) \right]^3 - f^{n}(\alpha) = 33^{1992},$ where $f^{n}(x) = f(f(\cdots f(x))),$ and $n$ is a positive integer.

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IMO Shortlist 1992 problem 16

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

IMO Shortlist 1992 problem 15

Does there exist a set $M$ with the following properties?

(i) The set $M$ consists of 1992 natural numbers.
(ii) Every element in $M$ and the sum of any number of elements have the form $m^k$ $(m, k \in \mathbb{N}, k \geq 2).$

IMO Shortlist 1992 problem 12

Let $f, g$ and $a$ be polynomials with real coefficients, $f$ and $g$ in one variable and $a$ in two variables. Suppose

$f(x) - f(y) = a(x, y)(g(x) - g(y)) \forall x,y \in \mathbb{R}$

Prove that there exists a polynomial $h$ with $f(x) = h(g(x)) \text{ } \forall x \in \mathbb{R}.$

IMO Shortlist 1967 problem 3

The $n$ points $P_1,P_2, \ldots, P_n$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $D_n$ between any two of these points has its largest possible value $D_n.$ Calculate $D_n$ for $n = 2$ to 7. and justify your answer.

IMO Shortlist 1966 problem 57

Is it possible to choose a set of $100$ (or $200$ ) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

IMO Shortlist 1966 problem 54

We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$