| 1. A lock has 10 buttons. A combination is any subset of 5 buttons. It can be opened by pressing the buttons in the combination in any order. How many combinations are there? Suppose it is redesigned to allow a combination to be any subset of 1 to 9 buttons. How many combinations are there? [The original question asked for the difference.] |
| 2. Let f(n) denote the square of the sum of the digits of n. Let f 2(n) denote f(f(n)), f 3(n) denote f(f(f(n))) and so on. Find f 1998(11). |
| 3. Given log2(log8x) = log8(log2x), find (log2x)2. |
| 4. xi are reals such that -1 < xi < 1 and |x1| + |x2| + ... + |xn| = 19 + |x1 + ... + xn|. What is the smallest possible value of n? |
| 5. Find the probability that a randomly chosen positive divisor of 1099 is divisible by 1088. [The original question asked for m+n, where the prob is m/n in lowest terms.] |
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6. The vacant squares in the grid below are filled with positive integers so that there is an arithmetic progression in each row and each column. What number is placed in the square marked * ?
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| 7. In the triangle ABC, the foot of the perpendicular from A divides the opposite side into parts length 3 and 17, and tan A = 22/7. Find area ABC. |
| 8. f(m, n) is defined for positive integers m, n and satisfies f(m, m) = m, f(m, n) = f(n, m), f(m, m+n) = (1 + m/n) f(m, n). Find f(14, 52). |
| 9. Find the smallest positive cube ending in 888. |
| 10. The truncated cuboctahedron is a convex polyhedron with 26 faces: 12 squares, 8 regular hexagons and 6 regular octagons. There are three faces at each vertex: one square, one hexagon and one octagon. How many pairs of vertices have the segment joining them inside the polyhedron rather than on a face or edge? |
| 11. A line L in the complex plane is a mean line for the points w1, w2, ... , wn if there are points z1, z2, ... , zn on L such that (w1 - z1) + ... + (wn - zn) = 0. There is a unique mean line for the points 32 + 170i, -7 + 64i, -9 + 200i, 1 + 27i, -14 + 43i which passes through the point 3i. Find its slope. |
| 12. P is a point inside the triangle ABC. The line PA meets BC at D. Similarly, PB meets CA at E, and PC meets AB at F. If PD = PE = PF = 3 and PA + PB + PC = 43, find PA·PB·PC. |
| 13. x2 - x - 1 is a factor of a x17 + b x16 + 1 for some integers a, b. Find a. |
| 14. The graph xy = 1 is reflected in y = 2x to give the graph 12x2 + rxy + sy2 + t = 0. Find rs. |
| 15. The boss places letter numbers 1, 2, ... , 9 into the typing tray one at a time during the day in that order. Each letter is placed on top of the pile. Every now and then the secretary takes the top letter from the pile and types it. She leaves for lunch remarking that letter 8 has already been typed. How many possible orders there are for the typing of the remaining letters. [For example, letters 1, 7 and 8 might already have been typed, and the remaining letters might be typed in the order 6, 5, 9, 4, 3, 2. So the sequence 6, 5, 9, 4, 3, 2 is one possibility. The empty sequence is another.] |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
13 June 2003
Last updated/corrected 13 Jun 03