7 1 2 1 7 1 1 1 1 1 2 2 3 4 1 2 3 1 2 1 1 1 1 1 2 2 3 4Show that the fourth row is always the same as the second row.
| 1. A chess board is placed on top of an identical board and rotated through 45 degrees about its center. What is the area which is black in both boards? | |
| 2. AB is a diameter of the circle C. M and N are any two points on the circle. The chord MA' is perpendicular to the line NA and the chord MB' is perpendicular to the line NB. Show that AA' and BB' are parallel. | |
| 3. Find an example of m and n such that m is the product of n consecutive positive integers and also the product of n+2 consecutive positive integers. Show that we cannot have n = 2. | |
4. Write down a row of arbitrary integers (repetitions allowed). Now construct a second row as follows. Suppose the integer n is in column k in the first row. In column k in the second row write down the number of occurrences of n in row 1 in columns 1 to k inclusive. Similarly, construct a third row under the second row (using the values in the second row), and a fourth row. An example follows:
7 1 2 1 7 1 1 1 1 1 2 2 3 4 1 2 3 1 2 1 1 1 1 1 2 2 3 4Show that the fourth row is always the same as the second row. |
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| 5. Let S be the set of points (x, y) given by y ≤ - x2 and y ≥ x2 - 2x + a. Find the area of the rectangle with sides parallel to the axes and the smallest possible area which encloses S. | |
| 6. ABC, CDE, EFG are equilateral triangles (not necessarily the same size). The vertices are counter-clockwise in each case. A, D, G are collinear and AD = DG. Show that BFD is equilateral. | |
| 7. 1000 people live in a village. Every evening each person tells his friends all the news he heard during the day. All news eventually becomes known (by this process) to everyone. Show that one can choose 90 people, so that if you give them some news on the same day, then everyone will know in 10 days. | |
| 8. The reals a and b are such that a cos x + b cos 3x > 1 has no real solutions. Show that |b| ≤ 1. | |
| 9. ABCD is a convex quadrilateral. K is the midpoint of AB and M is the midpoint of CD. L lies on the side BC and N lies on the side AD. KLMN is a rectangle. Show that its area is half that of ABCD. | |
| 10. The sequence an of positive integers is such that (1) an ≤ n3/2 for all n, and (2) m-n divides km - kn (for all m > n). Find an. | |
| 11. Is it possible to color half the cells in a rectangular array white and half black so that in each row and column more than 3/4 of the cells are the same color? | |
| 12. ACPH, AMBE, AHBT, BKXM and CKXP are parallelograms. Show that ABTE is also a parallelogram (vertices are labeled anticlockwise). | |
| 13. Find all solutions (x, y) in positive integers to x3 - y3 = xy + 61. |
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| 14. Eighteen teams are playing in a tournament. So far, each team has played exactly eight games, each with a different opponent. Show that there are three teams none of which has yet played the other. | |
| 15. ABC is a triangle. A' lies on the side BC with BA'/BC = 1/4. Similarly, B' lies on the side CA with CB'/CA = 1/4, and C' lies on the side AB with AC'/AB = 1/4. Show that the perimeter of A'B'C' is between 1/2 and 3/4 of the perimeter of ABC. | |
| 16. The positive reals x, y satisfy x3 + y3= x - y. Show that x2 + y2 < 1. |
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| 17. A convex polygon is drawn inside the unit circle. Someone makes a copy by starting with one vertex and then drawing each side successively. He copies the angle between each side and the previous side accurately, but makes an error in the length of each side of up to a factor 1±p. As a result the last side ends up a distance d from the starting point. Show that d < 4p. | |
| 18. An integer is initially written at each vertex of a cube. A move is to add 1 to the numbers at two vertices connected by an edge. Is it possible to equalise the numbers by a series of moves in the following cases? (1) The initial numbers are (1) 0, except for one vertex which is 1. (2) The initial numbers are 0, except for two vertices which are 1 and diagonally opposite on a face of the cube. (3) Initially, the numbers going round the base are 1, 2, 3, 4. The corresponding vertices on the top are 6, 7, 4, 5 (with 6 above the 1, 7 above the 2 and so on). | |
| 19. Find 21 consecutive integers, each with a prime factor less than 17. | |
| 20. Each of the numbers from 100 to 999 inclusive is written on a separate card. The cards are arranged in a pile in random order. We take cards off the pile one at a time and stack them into 10 piles according to the last digit. We then put the 1 pile on top of the 0 pile, the 2 pile on top of the 1 pile and so on to get a single pile. We now take them off one at a time and stack them into 10 piles according to the middle digit. We then consolidate the piles as before. We then take them off one at a time and stack them into 10 piles according to the first digit and finally consolidate the piles as before. What can we say about the order in the final pile? | |
| 21. Given 6 points inside a 3 x 4 rectangle, show that we can find two points whose distance does not exceed √5. | |
| 22. What is the smallest value of 4 + x2y4 + x4y2 - 3x2y2 for real x, y? Show that the polynomial cannot be written as a sum of squares. [Note the candidates did not know calculus.] | |
| 23. ABCDEF is a prism. Its base ABC and its top DEF are congruent equilateral triangles. The side edges are AD, BE and CF. Find all points on the base wich are equidistant from the three lines AE, BF and CD. |
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(C) John Scholes
jscholes@kalva.demon.co.uk
4 July 2002