1. Show that we can find n integers whose sum is 0 and whose product is n iff n is divisible by 4.
2. Show that (a + b)2/2 + (a + b)/4 ≥ a√b + b √a for all positive a and b.
3. ABC and A'B'C' are equilateral triangles and ABC and A'B'C' have the same sense (both clockwise or both counter-clockwise). Take an arbitrary point O and points P, Q, R so that OP is equal and parallel to AA', OQ is equal and parallel to BB', and OR is equal and parallel to CC'. Show that PQR is equilateral.
4. Take a large number of unit squares, each with one edge red, one edge blue, one edge green, and one edge yellow. For which m, n can we combine mn squares by placing similarly colored edges together to get an m x n rectangle with one side entirely red, another entirely bue, another entirely green, and the fourth entirely yellow.
5. Let A = cos2a, B = sin2a. Show that for all real a and positive x, y we have xAyB < x + y.
6. Two players play a game. Each takes it in turn to paint three unpainted edges of a cube. The first player uses red paint and the second blue paint. So each player has two moves. The first player wins if he can paint all edges of some face red. Can the first player always win?
7. n > 3 positive integers are written in a circle. The sum of the two neighbours of each number divided by the number is an integer. Show that the sum of those integers is at least 2n and less than 3n. For example, if the numbers were 3, 7, 11, 15, 4, 1, 2 (with 2 also adjacent to 3), then the sum would be 14/7 + 22/11 + 15/15 + 16/4 + 6/1 + 4/2 + 9/3 = 20 and 14 ≤ 20 < 21.
8. The incircle of the triangle ABC has center I and touches BC, CA, AB at D, E, F respectively. The segments AI, BI, CI intersect the circle at D', E', F' respectively. Show that DD', EE', FF' are collinear.
9. Find all integers m, n such that (5 + 3√2)m = (3 + 5√2)n.
10. x1 < x2 < x3 < ... < xn. yi is a permutation of the xi. We have that x1 + y1 < x2 + y2 < ... < xn + yn. Prove that xi = yi.
11. ABC is a triangle and P is any point. The lines PA, PB, PC cut the circumcircle of ABC again at A'B'C' respectively. Show that there are at most eight points P such that A'B'C' is congruent to ABC.
12. The positive reals x, y, z satisfy x2 + xy + y2/3 = 25, y2/3 + z2 = 9, z2 + zx + x2 = 16. Find the value of xy + 2yz + 3zx.
13. Starting with the polynomial x2 + 10x + 20, a move is to change the coefficient of x by 1 or to change the coefficient of x0 by 1 (but not both). After a series of moves the polynomial is changed to x2 + 20x + 10. Is it true that at some intermediate point the polynomial had integer roots?
14. The center of a coin radius r traces out a polygon with perimeter p which has an incircle radius R > r. What is the area of the figure traced out by the coin?
15. Each weight in a set of n has integral weight and the total weight of the set is 2n. A balance is initially empty. We then place the weights onto a pan of the balance one at a time. Each time we place the heaviest weight not yet placed. If the pans balance, then we place the weight onto the left pan. Otherwise, we place the weight onto the lighter pan. Show that when all the weights have been placed, the scales will balance. [For example, if the weights are 2, 2, 1, 1. Then we must place 2 in the left pan, followed by 2 in the right pan, followed by 1 in the left pan, followed by 1 in the right pan.]
16. A number is prime however we order its digits. Show that it cannot contain more than three different digits. For example, 337 satisfies the conditions because 337, 373 and 733 are all prime.
17. Find all pairs of digits (b, c) such that the number b ... b6c ... c4, where there are n bs and n cs is a square for all positive integers n.
18. A, B, C and D lie on a line in that order. Show that if X does not lie on the line then |XA| + |XD| + | |AB| - |CD| | > |XB| + |XC|.
19. The real sequence xn is defined by x1 = 1, x2 = 1, xn+2 = xn+12 - xn/2. Show that the sequence converges and find the limit.
20. The squares of a 1983 x 1984 chess board are colored alternately black and white in the usual way. Each white square is given the number 1 or the number -1. For each black square the product of the numbers in the neighbouring white squares is 1. Show that all the numbers must be 1.
21. A 3 x 3 chess board is colored alternately black and white in the usual way with the center square white. Each white square is given the number 1 or the number -1. A move consists of simultaneously changing each number to the product of the adjacent numbers. So the four corner squares are each changed to the number previously in the center square and the center square is changed to the product of the four numbers in the corners. Show that after finitely many moves all numbers are 1.
22. Is ln 1.01 greater or less than 2/201?
23. C1, C2, C3 are circles with radii r1, r2, r3 respectively. The circles do not intersect and no circle lies inside any other circle. C1 is larger than the other two. The two outer common tangents to C1 and C2 meet at A ("outer" means that the points where the tangent touches the two circles lie on the same side of the line of centers). The two outer common tangents to C1 and C3 intersect at B. The two tangents from A to C3 and the two tangents from B to C2 form a quadrangle. Show that it has an inscribed circle and find its radius.
24. Show that any cross-section of a cube through its center has area not less than the area of a face.
Solutions
Problem 2
Show that (a + b)2/2 + (a + b)/4 ≥ a√b + b √a for all positive a and b.
Answer
By AM/GM √(ab) ≤ (a+b)/2, so ½(a+b) + √(ab) ≤ a + b. Hence √(2a+2b) ≥ √a + √b (*).
By AM/GM (a + b) ≥ 2√(ab) and 2(a+b) + 1 ≥ 2√(2a+2b). Multiplying, (a+b)(2a+2b+1) ≥ 4√(ab)√(2a+2b). Then using (*) ≥ 4√(ab)(√a + √b).
Problem 13
Starting with the polynomial x2 + 10x + 20, a move is to change the coefficient of x by 1 or to change the coefficient of x0 by 1 (but not both). After a series of moves the polynomial is changed to x2 + 20x + 10. Is it true that at some intermediate point the polynomial had integer roots?
Answer
Yes.
Solution
We have x2 + (n+1)x + n = (x+n)(x+1), so x2 + ax + b has integer roots if a = b+1 (and a and b are integers). But initially a-b is -10 and it ends up as +10. Each move changes a-b by ±1, so it must pass through all values between -10 and +10.
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All Soviet Union Mathematical Olympiad Problems
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