RU-ALLRUSSIAN 1978

$\mathbf{a}_{\mathrm{n}}$ is the nearest integer to $\sqrt{\mathrm{n}}$ . Find $1 / \mathrm{a}_{1} + 1 / \mathrm{a}_{2} + \ldots +1 / \mathrm{a}_{1980}$ .

ABCD is a quadrilateral. M is a point inside it such that ABMD is a parallelogram. $\angle \mathrm{CBM} =$ $\angle \mathrm{CDM}$ . Show that $\angle \mathrm{ACD} = \angle \mathrm{BCM}$ .

Show that there is no positive integer n for which $1000^{\mathrm{n}} - 1$ divides $1978^{\mathrm{n}} - 1$ .

If P, Q are points in space the point [PQ] is the point on the line PQ on the opposite side of Q to P and the same distance from Q. $\mathrm{K}_{0}$ is a set of points in space. Given $\mathrm{K}_{\mathrm{n}}$ we derive $\mathrm{K}_{\mathrm{n} + 1}$ by adjoining all the points [PQ] with P and Q in $\mathrm{K}_{\mathrm{n}}$ .

(1) $\mathrm{K}_{0}$ contains just two points A and B, a distance 1 apart, what is the smallest n for which $\mathrm{K}_{\mathrm{n}}$ contains a point whose distance from A is at least 1000?

(2) $\mathrm{K}_{0}$ consists of three points, each pair a distance 1 apart, find the area of the smallest convex polygon containing $\mathrm{K}_{\mathrm{n}}$ .

(3) $\mathrm{K}_{0}$ consists of four points, forming a regular tetrahedron with volume 1. Let $\mathrm{H}_{\mathrm{n}}$ be the smallest convex polyhedron containing $\mathrm{K}_{\mathrm{n}}$ . How many faces does $\mathrm{H}_{1}$ have? What is the volume of $\mathrm{H}_{\mathrm{n}}$ ?

Two players play a game. There is a heap of m tokens and a heap of $\mathrm{n}< \mathrm{m}$ tokens. Each player in turn takes one or more tokens from the heap which is larger. The number he takes must be a multiple of the number in the smaller heap. For example, if the heaps are 15 and 4, the first player may take 4, 8 or 12 from the larger heap. The first player to clear a heap wins. Show that if $\mathrm{m} > 2\mathrm{n}$ , then the first player can always win. Find all k such that if $\mathrm{m} > \mathrm{k}\mathrm{n}$ , then the first player can always win.

Show that there is an infinite sequence of reals $\mathrm{x}_{1}$ , $\mathrm{x}_{2}$ , $\mathrm{x}_{3}$ , ... such that $\mathrm{l}\mathrm{x}_{\mathrm{n}}\mathrm{l}$ is bounded and for any $\mathrm{m} > \mathrm{n}$ , we have $\mathrm{l}\mathrm{x}_{\mathrm{m}} - \mathrm{x}_{\mathrm{n}}\mathrm{l} > 1 / (\mathrm{m} - \mathrm{n})$ .

Let $\mathrm{p}(\mathrm{x}) = \mathrm{x}^{2} + \mathrm{x} + 1$ . Show that for every positive integer n, the numbers n, $\mathrm{p}(\mathrm{n})$ , $\mathrm{p}(\mathrm{p}(\mathrm{n}))$ , $\mathrm{p}(\mathrm{p}(\mathrm{p}(\mathrm{n})))$ , ... are relatively prime.

Show that for some k, you can find 1978 different sizes of square with all its vertices on the graph of the function $\mathrm{y} = \mathrm{k}\sin \mathrm{x}$ .

The set $\mathrm{S_0}$ has the single member (5, 19). We derive the set $\mathrm{S_{n + 1}}$ from $\mathrm{S_n}$ by adjoining a pair to $\mathrm{S_n}$ . If $\mathrm{S_n}$ contains the pair (2a, 2b), then we may adjoin the pair (a, b). If $\mathrm{S}$ contains the pair (a, b) we may adjoin (a+1, b+1). If $\mathrm{S}$ contains (a, b) and (b, c), then we may adjoin (a, c). Can we obtain (1, 50)? (1, 100)? If We start with (a, b), with a < b, instead of (5, 19), for which n can we obtain (1, n)?

An n- gon area A is inscribed in a circle radius R. We take a point on each side of the polygon to form another n- gon. Show that it has perimeter at least 2A/R.

Two players play a game by moving a piece on an n x n chessboard. The piece is initially in a corner square. Each player may move the piece to any adjacent square (which shares a side with its current square), except that the piece may never occupy the same square twice. The first player who is unable to move loses. Show that for even n the first player can always win, and for odd n the second player can always win. Who wins if the piece is initially on a square adjacent to the corner?

Given a set of n non- intersecting segments in the plane. No two segments lie on the same line. Can we successively add n- 1 additional segments so that we end up with a single non- intersecting path? Each segment we add must have as its endpoints two existing segment endpoints.

a and b are positive real numbers. $\mathbf{x}_i$ are real numbers lying between a and b. Show that $(\mathbf{x}_1 + \mathbf{x}_2 + \ldots + \mathbf{x}_n)(1 / \mathbf{x}_1 + 1 / \mathbf{x}_2 + \ldots + 1 / \mathbf{x}_n)\leq \mathrm{n}^2 (\mathrm{a} + \mathrm{b})^2 /4\mathrm{ab}$ .

$\mathrm{n} > 3$ is an integer. Let S be the set of lattice points (a, b) with $0\leq \mathrm{a},\mathrm{b}< \mathrm{n}$ . Show that we can choose n points of S so that no three chosen points are collinear and no four chosen points from a parallelogram.

Given any tetrahedron, show that we can find two planes such that the areas of the projections of the tetrahedron onto the two planes have ratio at least $\sqrt{2}$ .

$\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots , \mathbf{a}_{\mathrm{n}}$ are real numbers. Let $\mathbf{b}_{\mathrm{k}} = (\mathbf{a}_{1} + \mathbf{a}_{2} + \ldots +\mathbf{a}_{\mathrm{k}}) / \mathbf{k}$ for $\mathbf{k} = 1, 2, \ldots , \mathbf{n}$ . Let $\mathbf{C} = (\mathbf{a}_{1} - \mathbf{b}_{1})^{2} + (\mathbf{a}_{2} - \mathbf{b}_{2})^{2} + \ldots +(\mathbf{a}_{\mathrm{n}} - \mathbf{b}_{\mathrm{n}})^{2}$ , and $\mathbf{D} = (\mathbf{a}_{1} - \mathbf{b}_{\mathrm{n}})^{2} + (\mathbf{a}_{2} - \mathbf{b}_{\mathrm{n}})^{2} + \ldots +(\mathbf{a}_{\mathrm{n}} - \mathbf{b}_{\mathrm{n}})^{2}$ . Show that $\mathbf{C} \leq \mathbf{D} \leq 2\mathbf{C}$ .

Let $\mathbf{x}_{\mathrm{n}} = (1 + \sqrt{2} + \sqrt{3})^{\mathrm{n}}$ . We may write $\mathbf{x}_{\mathrm{n}} = \mathbf{a}_{\mathrm{n}} + \mathbf{b}_{\mathrm{n}}\sqrt{2} + \mathbf{c}_{\mathrm{n}}\sqrt{3} + \mathbf{d}_{\mathrm{n}}\sqrt{6}$ , where $\mathbf{a}_{\mathrm{n}}, \mathbf{b}_{\mathrm{n}}, \mathbf{c}_{\mathrm{n}}, \mathbf{d}_{\mathrm{n}}$ are integers. Find the limit as $\mathbf{n}$ tends to infinity of $\mathbf{b}_{\mathrm{n}} / \mathbf{a}_{\mathrm{n}}, \mathbf{c}_{\mathrm{n}} / \mathbf{a}_{\mathrm{n}}, \mathbf{d}_{\mathrm{n}} / \mathbf{a}_{\mathrm{n}}$ .