|The 2nd All Russian Mathematical Olympiad
- 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 a0, a1, ... , a100, satisfying a1>a0, a1>0, and ar+2=3 ar+1 - 2 ar for r=0, 1, ... , 98. Prove a100 > 299 .
- Prove that there are no integers a, b, c, d such that the polynomial ax3+bx2+cx+d equals 1 at x=19 and 2 at 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 abcd=1, prove that a2 + b2 + c2 + d2 + ab + ac + ad + bc + bd + cd ≥ 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 r1, to the next closest be r2 and so on, so that the distances from M to the five vertices satisfy r1 ≤ r2 ≤ r3 ≤ r4 ≤ r5. Find (a) the locus of M which gives r3 the minimum possible value, and (b) the locus of M which gives r3 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.
- AB=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 ≤ AB ≤ 1 ≤ BC ≤ 2 ≤ CA ≤ 3. What is the maximum area it can have ?
- Given unequal integers x, y, z prove that (x-y)5 + (y-z)5 + (z-x)5 is divisible by 5(x-y)(y-z)(z-x).
- Given a0, a1, ... , an, satisfying a0 = an = 0, and and ak-1 - 2ak + ak+1 ≥ 0 for 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.