| 1. How many pairs of non-negative integers (m, n) each sum to 1492 without any carries? |
| 2. What is the greatest distance between the sphere center (-2, -10, 5) radius 19, and the sphere center (12, 8, -16) radius 87? |
| 3. A nice number equals the product of its proper divisors (positive divisors excluding 1 and the number itself). Find the sum of the first 10 nice numbers. |
| 4. Find the area enclosed by the graph of |x - 60| + |y| = |x/4|. |
| 5. m, n are integers such that m2 + 3m2n2 = 30n2 + 517. Find 3m2n2. |
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6. ABCD is a rectangle. The points P, Q lie inside it with PQ parallel to AB. Points X, Y lie on AB (in the order A, X, Y, B) and W, Z on CD (in the order D, W, Z, C). The four parts AXPWD, XPQY, BYQZC, WPQZ have equal area. BC = 19, PQ = 87, XY = YB + BC + CZ = WZ = WD + DA + AX. Find AB.
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| 7. How many ordered triples (a, b, c) are there, such that lcm(a, b) = 1000, lcm(b, c) = 2000, lcm(c, a) = 2000? |
| 8. Find the largest positive integer n for which there is a unique integer k such that 8/15 < n/(n+k) < 7/13. |
| 9. P lies inside the triangle ABC. Angle B = 90o and each side subtends an angle 120o at P. If PA = 10, PB = 6, find PC. |
| 10. A walks down an up-escalator and counts 150 steps. B walks up the same escalator and counts 75 steps. A takes three times as many steps in a given time as B. How many steps are visible on the escalator? |
| 11. Find the largest k such that 311 is the sum of k consecutive positive integers. |
| 12. Let m be the smallest positive integer whose cube root is n + k, where n is an integer and 0 < k < 1/1000. Find n. |
| 13. Given distinct reals x1, x2, x3, ... , x40 we compare the first two terms x1 and x2 and swap them iff x2 < x1. Then we compare the second and third terms of the resulting sequence and swap them iff the later term is smaller, and so on, until finally we compare the 39th and 40th terms of the resulting sequence and swap them iff the last is smaller. If the sequence is initially in random order, find the probability that x20 ends up in the 30th place. [The original question asked for m+n if the prob is m/n in lowest terms.] |
| 14. Let m = (104 + 324)(224 + 324)(344 + 324)(464 + 324)(584 + 324) and n = (44 + 324)(164 + 324)(284 + 324)(404 + 324)(524 + 324). Find m/n. |
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15. Two squares are inscribed in a right-angled triangle as shown. The first has area 441 and the second area 440. Find the sum of the two shorter sides of the triangle.
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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
12 July 2003
Last updated/corrected 12 Jan 04