Problem Sheet 2

UK IMO squad training camp, Bath 2001

1. (Israel) Find all solutions of the pair of simultaneous equations

\{ \begin{matrix}x_{1} + x_{2} + ⋯ + x_{2000} & = & 2000 \\ x_{1}^{4} + x_{2}^{4} + ⋯ + x_{2000}^{4} & = & x_{1}^{3} + x_{2}^{3} + ⋯ + x_{2000}^{3} \end{matrix}

2. (Israel) Given 2001 real numbers x_{1}, x_{2}, …, x_{2001} such that 0 ≤ x_{n} ≤ 1 for each n = 1, 2, …, 2001, find the maximum value of

(\frac{1}{2001} ∑_{n=1}^{2001} x_{n}^{2}) − (\frac{1}{2001} ∑_{n=1}^{2001} x_{n})^{2} .

Where is this maximum attained?

3. (Hungary) Let t and r be the area and the inradius of the triangle ABC. Let r_{a} denote the radius of the circle touching the incircle, AB and AC. Define r_{b} and r_{c} in a similar fashion. The common tangent of the circles with radii r and r_{a} cuts a little triangle from ABC with area t_{a}. Quantities t_{b} and t_{c} are defined in a similar fashion. Prove that

\frac{t_{a}}{r_{a}} + \frac{t_{b}}{r_{b}} + \frac{t_{c}}{r_{c}} = \frac{t}{r}.

4. (Hungary-Israel competition) In this question, G_{n} is a simple undirected graph with n vertices, K_{n} is the complete graph with n vertices, and e(G_{n}) is the number of edges of the graph G_{n}. Let C_{n} denote circle with n verticies.

(i) The edges of K_{n}, n ≥ 3 are coloured with n colours, and every colour appears at least once. Prove that one can find a triangle whose sides are coloured with 3 different colours.

(ii) Suppose that n ≥ 5 is given. If e(G_{n}) ≥ n^{2}/4+2, prove that there exist two triangles which have exactly one common vertex.

(iii) Suppose that e(G_{n}) ≥ \frac{n\sqrt{n}}{\sqrt{2}} + n. Prove that G_{n} contains C_{4}.

5. (Israel) 2001 lines are given in the plane. No two of them are parallel and no three meet in a point. These lines partition the plane into some regions (not necessarily finite) bounded by segments of these lines. is called a map. Two regions on the map are called neighbours if they share a side. The set of intersection points of lines is called the set of verticies. Two vertices are called neighbours if they are found on the same side.

A legal colouring of the regions (one colour per region) must have neighbouring regions coloured differently. A legal colouring of the vertices (one colour per vertex) must have neighbouring vertices coloured differently.

(i) What is the minimum number of colours required for the legal colouring of any map?

(ii) What is the minimum number of colours required for the legal colouring of the vertices of any map?

6. (Israel) Triangle ABC in the plane Π is said to be good if it has the following property: for any point D in space, out of the plane Π, it is possible to construct a triangle with sides of lengths |AD|, |BD| and |CD|. Find all the good triangles.

7. (Israel) Find a pair of integers (x, y) such that 15x^{2} + y^{2} = 2^{2000}. Does there exist such a pair (x, y) with x odd?