RU-ALLRUSSIAN 1979

T is an isosceles triangle. Another isosceles triangle T' has one vertex on each side of T. What is the smallest possible value of area T'area T?

A grasshopper hops about in the first quadrant $(\mathrm{x},\mathrm{y}\geqslant = 0)$ . From $(\mathrm{x},\mathrm{y})$ it can hop to $(\mathrm{x} + 1,\mathrm{y} - 1)$ or to $(\mathrm{x} - 5,\mathrm{y} + 7)$ , but it can never leave the first quadrant. Find the set of points $(\mathrm{x},\mathrm{y})$ from which it can never get further than a distance 1000 from the origin.

In a group of people every person has less than 4 enemies. Assume that A is B's enemy iff B is A's enemy. Show that we can divide the group into two parts, so that each person has at most one enemy in his part.

Let S be the set {0, 1}. Given any subset of S we may add its arithmetic mean to S (provided it is not already included - S never includes duplicates). Show that by repeating this process we can include the number 1/5 in S. Show that we can eventually include any rational number between 0 and 1.

The real sequence $\mathrm{x}_1\geq \mathrm{x}_2\geq \mathrm{x}_3\geq \ldots$ satisfies $\mathrm{x}_1 + \mathrm{x}_4 / 2 + \mathrm{x}_9 / 3 + \mathrm{x}_{16} / 4 + \ldots +\mathrm{x}_\mathrm{N} / \mathrm{n}\leq 1$ for every square $\mathrm{N} = \mathrm{n}^2$ . Show that it also satisfies $\mathrm{x}_1 + \mathrm{x}_2 / 2 + \mathrm{x}_3 / 3 + \ldots +\mathrm{x}_\mathrm{n} / \mathrm{n}\leq 3$ .

Given a finite set X of points in the plane. S is a set of vectors AB where (A, B) are some pairs of points in X. For every point A the number of vectors AB (starting at A) in S equals the number of vectors CA (ending at A) in S. Show that the sum of the vectors in S is zero.

What is the smallest number of pieces that can be placed on an $8\mathrm{x}8$ chessboard so that every row, column and diagonal has at least one piece? [A diagonal is any line of squares parallel to one of the two main diagonals, so there are 30 diagonals in all.] What is the smallest number for an n x n board?

a and b are real numbers. Find real x and y satisfying: $(\mathrm{x} - \mathrm{y}(\mathrm{x}^2 - \mathrm{y}^2)^{1 / 2} = \mathrm{a}(1 - \mathrm{x}^2 + \mathrm{y}^2)^{1 / 2}$ and $(\mathrm{y} - \mathrm{x}(\mathrm{x}^2 - \mathrm{y}^2)^{1 / 2} = \mathrm{b}(1 - \mathrm{x}^2 + \mathrm{y}^2)^{1 / 2}$ .

A set of square carpets have total area 4. Show that they can cover a unit square.

$\mathbf{x}_{\mathrm{i}}$ are real numbers between 0 and 1. Show that $\left(\mathbf{x}_{1} + \mathbf{x}_{2} + \ldots +\mathbf{x}_{\mathrm{n}} + 1\right)^{2}\geq 4(\mathbf{x}_{1}^{2} + \mathbf{x}_{2}^{2} + \ldots +\mathbf{x}_{\mathrm{n}}^{2})$ .

m and n are relatively prime positive integers. The interval [0, 1] is divided into m + n equal subintervals. Show that each part except those at each end contains just one of the numbers 1/m, 2/m, 3/m, ..., (m- 1)/m, 1/n, 2/n, ..., (n- 1)/n.

Given a point P in space and 1979 lines $\mathbf{L}_{1}$ , $\mathbf{L}_{2}$ , ..., $\mathbf{L}_{1979}$ containing it. No two lines are perpendicular. $\mathbf{P}_{1}$ is a point on $\mathbf{L}_{1}$ . Show that we can find a point $\mathbf{A}_{\mathrm{n}}$ on $\mathbf{L}_{\mathrm{n}}$ (for $\mathrm{n} = 2$ , 3, ..., 1979) such that the following 1979 pairs of lines are all perpendicular: $\mathbf{A}_{\mathrm{n - 1}}\mathbf{A}_{\mathrm{n + 1}}$ and $\mathbf{L}_{\mathrm{n}}$ for $\mathrm{n} = 1$ , ..., 1979. [We regard $\mathbf{A}_{\mathrm{- 1}}$ as $\mathbf{A}_{1979}$ and $\mathbf{A}_{1980}$ as $\mathbf{A}_{1}$ .]

Find a sequence $\mathbf{a}_{1}$ , $\mathbf{a}_{2}$ , ..., $\mathbf{a}_{25}$ of 0s and 1s such that the following sums are all odd:

$\begin{array}{r l} & {\mathrm{a}_{1}\mathrm{a}_{1} + \mathrm{a}_{2}\mathrm{a}_{2} + \ldots +\mathrm{a}_{25}\mathrm{a}_{25}}\\ & {\mathrm{a}_{1}\mathrm{a}_{2} + \mathrm{a}_{2}\mathrm{a}_{3} + \ldots +\mathrm{a}_{24}\mathrm{a}_{25}}\\ & {\mathrm{a}_{1}\mathrm{a}_{3} + \mathrm{a}_{2}\mathrm{a}_{4} + \ldots +\mathrm{a}_{23}\mathrm{a}_{25}}\\ & {\ldots}\\ & {\mathrm{a}_{1}\mathrm{a}_{24} + \mathrm{a}_{2}\mathrm{a}_{25}} \end{array}$

Show that we can find a similar sequence of n terms for some $\mathrm{n} > 1000$ .

A convex quadrilateral is divided by its diagonals into four triangles. The incircles of each of the four are equal. Show that the quadrilateral has all its sides equal.