RU-ALLRUSSIAN 1964

In the triangle ABC, the length of the altitude from A is not less than BC, and the length of the altitude from B is not less than AC. Find the angles.

If m, k, n are natural numbers and $\mathrm{n} > 1$ , prove that we cannot have $\mathrm{m(m + 1)} = \mathrm{k^n}$ .

Reduce each of the first billion natural numbers (billion $= 10^9$ ) to a single digit by taking its digit sum repeatedly. Do we get more 1s than 2s?

Given n odd and a set of integers $\mathrm{a_1}$ , $\mathrm{a_2}$ , ..., $\mathrm{a_n}$ , derive a new set $(\mathrm{a_1} + \mathrm{a_2}) / 2$ , $(\mathrm{a_2} + \mathrm{a_3}) / 2$ , ..., $(\mathrm{a_{n - 1}} + \mathrm{a_n}) / 2$ , $(\mathrm{a_n} + \mathrm{a_1}) / 2$ . However many times we repeat this process for a particular starting set we always get integers. Prove that all the numbers in the starting set are equal.

For example, if we started with 5, 9, 1, we would get 7, 5, 3, and then 6, 4, 5, and then 5, 4, 5, 5.5. The last set does not consist entirely of integers.

The convex hexagon ABCDEF has all angles equal. Prove that $\mathrm{AB - DE = EF - BC = CD - }$ FA.

(b) Given six lengths $a_{1}$ , ..., $a_{6}$ satisfying $a_{1} - a_{4} = a_{5} - a_{2} = a_{3} - a_{6}$ , show that you can construct a hexagon with sides $a_{1}$ , ..., $a_{6}$ and equal angles.

Find all possible integer solutions for $\mathrm{sqrt(x + sqrt(x\ldots(x + sqrt(x)) \ldots)) = y}$ , where there are 1998 square roots.

ABCD is a convex quadrilateral. A' is the foot of the perpendicular from A to the diagonal BD, B' is the foot of the perpendicular from B to the diagonal AC, and so on. Prove that A'B'C'D' is similar to ABCD.

Find all natural numbers n such that $\mathrm{n}^2$ does not divide n!.

Given a lattice of regular hexagons. A bug crawls from vertex A to vertex B along edges of the hexagons, taking the shortest possible path (or one of them). Prove that it travels a distance at least AB/2 in one direction. If it travels exactly AB/2 in one direction, how many edges does it traverse?

A circle center O is inscribed in ABCD (touching every side). Prove that angle AOB + angle COD equals 180 degrees.

The natural numbers a, b, n are such that for every natural number k not equal to b, b - k divides a - $\mathrm{k^n}$ . Prove that $\mathrm{a} = \mathrm{b^n}$ .

How many (algebraically) different expressions can we obtain by placing parentheses in $\mathrm{a_1 / a_2 / ... / a_n?}$

What is the smallest number of tetrahedrons into which a cube can be partitioned?

a) Find the smallest square with last digit not 0 which becomes another square (not zero) by the deletion of its last two digits. #(b) Find all squares, not containing the digits 0 or 5, such that if the second digit is deleted the resulting number divides the original one.

A circle is inscribed in ABCD. AB is parallel to CD, and BC = AD. The diagonals AC, BD meet at E. The circles inscribed in ABE, BCE, CDE, DAE have radius $\mathrm{r}_1$ , $\mathrm{r}_2$ , $\mathrm{r}_3$ , $\mathrm{r}_4$ respectively. Prove that $1 / \mathrm{r}_1 + 1 / \mathrm{r}_3 = 1 / \mathrm{r}_2 + 1 / \mathrm{r}_4$ .