RU-ALLRUSSIAN 1962

ABCD is any convex quadrilateral. Construct a new quadrilateral as follows. Take A' so that A is the midpoint of DA'; similarly, B' so that B is the midpoint of AB'; C' so that C is the midpoint of BC'; and D' so that D is the midpoint of CD'. Show that the area of A'B'C'D' is five times the area of ABCD.

Given a fixed circle C and a line L throught the center O of C. Take a variable point P on L and let K be the circle center P through O. Let T be the point where a common tangent to C and K meets K. What is the locus of T?

Given integers $\mathrm{a_0}$ , $\mathrm{a_1}$ , ..., $\mathrm{a_{100}}$ , satisfying $\mathrm{a_1 > a_0}$ , $\mathrm{a_1 > 0}$ , and $\mathrm{a_{r + 2} = 3}$ $\mathrm{a_{r + 1}} - 2\mathrm{a_r}$ for $\mathrm{r} = 0$ , 1, ..., 98. Prove $\mathrm{a_{100} > 2^{99}}$ .

Prove that there are no integers a, b, c, d such that the polynomial $\mathrm{ax^3 + bx^2 + cx + d}$ equals 1 at $\mathrm{x = 19}$ and 2 at $\mathrm{x = 62}$ .

Given an n x n array of numbers. n is odd and each number in the array is 1 or - 1. Prove that the number of rows and columns containing an odd number of - 1s cannot total n.

Given the lengths AB and BC and the fact that the medians to those two sides are perpendicular, construct the triangle ABC.

Given four positive real numbers a, b, c, d such that $\mathrm{abcd} = 1$ , prove that $\mathrm{a}^{2} + \mathrm{b}^{2} + \mathrm{c}^{2} + \mathrm{d}^{2} + \mathrm{ab} + \mathrm{ac} + \mathrm{ad} + \mathrm{bc} + \mathrm{bd} + \mathrm{cd} \geq 10$ .

Given a fixed regular pentagon ABCDE with side 1. Let M be an arbitary point inside or on it. Let the distance from M to the closest vertex be $\mathbf{r}_1$ , to the next closest be $\mathbf{r}_2$ and so on, so that the distances from M to the five vertices satisfy $\mathbf{r}_1 \leq \mathbf{r}_2 \leq \mathbf{r}_3 \leq \mathbf{r}_4 \leq \mathbf{r}_5$ . Find (a) the locus of M which gives $\mathbf{r}_3$ the minimum possible value, and (b) the locus of M which gives $\mathbf{r}_3$ the maximum possible value.

Given a number with 1998 digits which is divisible by 9. Let x be the sum of its digits, let y be the sum of the digits of x, and z the sum of the digits of y. Find z.

$\mathrm{AB} = \mathrm{BC}$ and M is the midpoint of AC. H is chosen on BC so that MH is perpendicular to BC. P is the midpoint of MH. Prove that AH is perpendicular to BP.

The triangle ABC satisfies $0\leq \mathrm{AB}\leq 1\leq \mathrm{BC}\leq 2\leq \mathrm{CA}\leq 3$ . What is the maximum area it can have?

Given unequal integers x, y, z prove that $\left(\mathrm{x - y}\right)^{5} + \left(\mathrm{y - z}\right)^{5} + \left(\mathrm{z - x}\right)^{5}$ is divisible by $5(\mathrm{x - y})(\mathrm{y - z})(\mathrm{z - x})$ .

Given $\mathrm{a_0}$ , $\mathrm{a_1}$ , ..., $\mathrm{a_n}$ , satisfying $\mathrm{a_0} = \mathrm{a_n} = 0$ , and and $\mathrm{a_{k - 1}} - 2\mathrm{a_k} + \mathrm{a_{k + 1}}\geq 0$ for $\mathrm{k = 0}$ , 1, ..., n- 1. Prove that all the numbers are negative or zero.

Given two sets of positive numbers with the same sum. The first set has m numbers and the second n. Prove that you can find a set of less than m+n positive numbers which can be arranged to part fill an m x n array, so that the row and column sums are the two given sets.

Example: row sums 1, 5, 3; column sums 2, 7. Array is:

x5 x1 21