CAPS 2005

Let $n$ be a given positive integer. Solve the system $$x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,$$ $$x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2}$$ in the set of nonnegative real numbers.

A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$ and circumscribed to a circle with center $I$ . Its diagonals meet at $P$ . Prove that points $O, I$ and $P$ lie on a line.

Find all integers $n \ge 3$ for which the polynomial $$W(x) = x^n - 3x^{n-1} + 2x^{n-2} + 6$$ can be written as a product of two non-constant polynomials with integer coefficients.

We distribute $n\ge1$ labelled balls among nine persons $A,B,C, \dots , I$ . How many ways are there to do this so that $A$ gets the same number of balls as $B,C,D$ and $E$ together?

Given a convex quadrilateral $ABCD$ , find the locus of the points $P$ inside the quadrilateral such that $$S_{PAB}\cdot S_{PCD} = S_{PBC}\cdot S_{PDA}$$ (where $S_X$ denotes the area of triangle $X$ ).

Determine all pairs of integers $(x, y)$ satisfying the equation $$y(x + y) = x^3- 7x^2 + 11x - 3.$$