| 1. T is an isosceles triangle. Another isosceles triangle T' has one vertex on each side of T. What is the smallest possible value of area T'/area T? |
| 2. A grasshopper hops about in the first quadrant (x, y >= 0). From (x, y) it can hop to (x+1, y-1) or to (x-5, y+7), but it can never leave the first quadrant. Find the set of points (x, y) from which it can never get further than a distance 1000 from the origin. |
| 3. In a group of people every person has less than 4 enemies. Assume that A is B's enemy iff B is A's enemy. Show that we can divide the group into two parts, so that each person has at most one enemy in his part. |
| 4. Let S be the set {0, 1}. Given any subset of S we may add its arithmetic mean to S (provided it is not already included - S never includes duplicates). Show that by repeating this process we can include the number 1/5 in S. Show that we can eventually include any rational number between 0 and 1. |
| 5. The real sequence x1 ≥ x2 ≥ x3 ≥ ... satisfies x1 + x4/2 + x9/3 + x16/4 + ... + xN/n ≤ 1 for every square N = n2. Show that it also satisfies x1 + x2/2 + x3 /3 + ... + xn/n ≤ 3. |
| 6. Given a finite set X of points in the plane. S is a set of vectors AB where (A, B) are some pairs of points in X. For every point A the number of vectors AB (starting at A) in S equals the number of vectors CA (ending at A) in S. Show that the sum of the vectors in S is zero. |
| 7. What is the smallest number of pieces that can be placed on an 8 x 8 chessboard so that every row, column and diagonal has at least one piece? [A diagonal is any line of squares parallel to one of the two main diagonals, so there are 30 diagonals in all.] What is the smallest number for an n x n board? |
| 8. a and b are real numbers. Find real x and y satisfying: (x - y (x2 - y2)1/2 = a(1 - x2 + y2)1/2 and (y - x (x2 - y2)1/2 = b(1 - x2 + y2)1/2. |
| 9. A set of square carpets have total area 4. Show that they can cover a unit square. |
| 10. xi are real numbers between 0 and 1. Show that (x1 + x2 + ... + xn + 1)2 ≥ 4(x12 + x22 + ... + xn2). |
| 11. m and n are relatively prime positive integers. The interval [0, 1] is divided into m + n equal subintervals. Show that each part except those at each end contains just one of the numbers 1/m, 2/m, 3/m, ... , (m-1)/m, 1/n, 2/n, ... , (n-1)/n. |
| 12. Given a point P in space and 1979 lines L1, L2, ... , L1979 containing it. No two lines are perpendicular. P1 is a point on L1. Show that we can find a point An on Ln (for n = 2, 3, ... , 1979) such that the following 1979 pairs of lines are all perpendicular: An-1An+1 and Ln for n = 1, ... , 1979. [We regard A-1 as A1979 and A1980 as A1.] |
13. Find a sequence a1, a2, ... , a25 of 0s and 1s such that the following sums are all odd:
a1a1 + a2a2 + ... + a25a25 a1a2 + a2a3 + ... + a24a25 a1a3 + a2a4 + ... + a23a25 ... a1a24 + a2a25 a1a25Show that we can find a similar sequence of n terms for some n > 1000. |
| 14. A convex quadrilateral is divided by its diagonals into four triangles. The incircles of each of the four are equal. Show that the quadrilateral has all its sides equal. |
| 15. [unclear] |
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© John Scholes
jscholes@kalva.demon.co.uk
1 June 2002