Mathematics Olympiad Problems, 1996

Vologda, Russia - for Grades 5-11

by Victor Guba

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These problems come from Professor Victor Guba, formerly of the Vologda Pedagogical University (Russia), and currently of Vanderbilt University. To request solutions or other information about the olympiad, please visit his web site

  1. A zoo has several ostriches and several giraffes. They have 30 eyes and 44 legs. How many ostriches and how many giraffes are in the zoo?

  2. Is there a configuration of 6 points and several closed intervals in the plane such that any point is joined exactly with a) 3 of the points, 4 of the points? (The intervals do not have inner intersections.)

  3. Is it true that n^3+5n-1 is prime for any natural n?

  4. A string of 1996 digits begins with the number 6. Any number formed by two consecutive digits is divisible by 17 or 23. What is the last digit?

  5. There are 3 boxes with balls inside them. They are labeled 'two whites', "two blacks', and 'black & white' to show the contents of the boxes. Someone changes the labels to make them all false. Find the contents of each box by removing only one of the balls.

  6. Divide a cube into 8 equal parts using a) four, b) three planar cuts.

  7. Is it true that

    1. for any sequence of 5 different numbers, one may choose an increasing or decreasing subsequence of three numbers?
    2. for any sequence of 9 different numbers one may choose an increasing or decreasing subsequence of four numbers?

  8. Is it possible to fill in a 5x5 table using numbers such that the sum of all them is positive, but the sum of any 4 numbers forming a 2x2 sub-table is negative?

  9. Solve the following multiplication example

    where all the asterisks are prime digits (2,3,5,7).

  10. Is there a closed polygonal line that intersects any of its edges exactly once and consists of a) six, b) seven edges?

  11. Is it true that for any of 100 integer numbers, there may be chosen a) 15, b) 16 of them such that the difference between any two of them is divisible by 7?

  12. There is a straight-line scratch on a chess board. What is the greatest possible number of scratched squares?

  13. Is it possible to fill 25 squares of squared paper such that each of the filled squares has a) an even and nonzero, b) an odd number of filled neighbors?

  14. Find 3 numbers such that each of them is a square of the difference of the two others.

  15. Two players compose a 2000-digit number by taking turns at writing a digit from 1 to 5. To win the game, Player 2 must cause the result to be divisible by 9, so Player 1 attempts to prevent this from occurring. If a) k = 10, b) k = 15, who will win?

  16. A convex quadrilateral is divided diagonally into 4 triangles. The areas of three of these triangles, proceeding clockwise, are 1, 2, and 3. What is the area of the fourth triangle?

  17. The game 'Sea Battle' takes place on an 8x8 board. What is the least number of shots needed to hit a ship 4x1 squares in size?

  18. Is there an infinite increasing geometric progression such that the first 100 elements of the sequence but none of the rest of the elements are integers?

  19. Is it possible to find a rectangle ABCD and a point M on a plane such that the distances from M to A,B,C,D respectively are

    1. 1,4,8,7
    2. 1,2,3,4
    3. 1,2,4,3 ?

  20. Present 100 as a sum of positive integers such that their product is maximal.

  21. Find the largest possible number n within which, for any permutation of the numbers 1,2,...,100, there are 10 consecutive numbers (with respect to a permutation) such that their sum is greater than or equal to n.

  22. A chess tournament involves only experts and grand masters. The number of experts is three times the number of grand masters. Only one game is played between each pair of players, with 1 point being awarded for a win, 0 points for a loss, and 1/2 point for a draw. The sum of the points of all the experts is 1.2 times greater than the sum of points of all the grand masters. How many participants does the tournament have?

  23. How many solutions does the equation sin x = x/9 have?

  24. Find the number of 10-digit numbers that involve only the numbers 2 and 3 but do not have two 3's next to each other.

  25. Is the number [(44+\sqrt{1996})^{100}] odd or even?

  26. Quadrilateral ABCD is inscribed in a circle. Let Hi (i=1,2,3,4) be the point of intersection of the three altitudes of triangles BCD, ACD, ABD, and ABC, respectively. Prove that quadrilateral H1H2H3H4 is equal to quadrilateral ABCD.

  27. A circle is inscribed in a triangle ABC. The circle touches AB and BC at points E and F, respectively. Angle bisector A intersects line EF at point K. Show that angle CKA is a right angle.

  28. The sum of the lengths of a system of intervals of a line is less than 1. Given a set of 100 points on the line, show that they can be shifted by a distance less than 50 such that no images of these points under the shift belong to the system of intervals.

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10 January 1997