| 1. X is the set {1, 2, 3, ... , n}. How many ordered 6-tuples (A1, A2, ... , A6) of subsets of X are there such that every element of X belongs to 0, 3 or 6 subsets in the 6-tuple? |
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| 2. Find the best possible k, k' such that k < v/(v + w) + w/(w + x) + x/(x + y) + y/(y + z) + z/(z + v) < k' for all positive reals v, w, x, y, z. |
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3. Given n > 1, find all real-valued functions fi(x) on the reals such that for all x, y we have:
f1(x) + f1(y) = f2(x) f2(y) f2(x2) + f2(y2) = f3(x)f3(y) f3(x3) + f3(y3) = f4(x)f4(y) ... fn(xn) + fn(yn) = f1(x)f1(y). |
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| 4. P is a point inside the triangle ABC. Show that there are unique points A1 on the line AB and A2 on the line CA such that P, A1, A2 are collinear and PA1 = PA2. Similarly, take B1, B2, C1, C2, so that P, B1, B2 are collinear, with B1 on the line BC, B2 on the line AB and PB1 = PB2, and P, C1, C2 are collinear, with C1 on the line CA, C2 on the line BC and PC1 = PC2. Find the point P such that the triangles AA1A2, BB1B2, CC1C2 have equal area, and show it is unique. |
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| 5. The integer sequence an satisfies an+1 = an3 + 1999. Show that it contains at most one square. |
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6. Find all non-negative real solutions to
x22 + x1x2 + x14 = 1 x32 + x2x3 + x24 = 1 x42 + x3x4 + x34 = 1 ... x19992 + x1998x1999 + x19984 = 1 x12 + x1999x1 + x14 = 1. |
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| 7. Find all positive integers m, n such that mn+m = nn-m. |
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| 8. P and Q are on the same side of the line L. The feet of the perpendiculars from P, Q to L are M, N respectively. The point S is such that PS = PM and QS = QN. The perpendicular bisectors of SM and SN meet at R. The ray RS meets the circumcircle of PQR again at T. Show that S is the midpoint of RT. | |
| 9. A valid set is a finite set of plane lattice points and segments such that: (1) the endpoints of each segment are lattice points and it is parallel to x = 0, y = 0, y = x or y = -x; (2) two segments have at most one common point; (3) each segment has just five points in the set. Does there exist an infinite sequence of valid sets, S1, S2, S3, ... such that Sn+1 is formed by adding one segment and one lattice point to Sn? |
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
29 July 2002
Last corrected/updated 30 Nov 03