x1 = xn + xn-1
x2 = x1 + xn
x3 = x2 + x1
...
xn = xn-1 + xn-2.
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1. Find all real solutions to:
x1 = xn + xn-1 x2 = x1 + xn x3 = x2 + x1 ... xn = xn-1 + xn-2. |
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| 2. Show that for any 4 points in the plane, we can choose a non-empty subset X such that no disk contains the points in X and not the points not in X. | |
| 3. Factor 1 + 5y2 + 25y4 + 125y6 + 625y8. |
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| 4. Find all real polynomials p(x) such that p(x)2 + 2p(x)p(1/x) + p(1/x)2 = p(x2)p(1/x2) for all non-zero x. | |
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5. ABC is an equilateral triangle. A1, B1, C1 are the midpoints of BC, CA, AB respectively. p is an arbitrary line through A1. q and r are lines parallel to p through B1 and C1 respectively. p meets the line B1C1 at A2. Similarly, q meets C1A1 at B2, and r meets A1B1 at C2. Show that the lines AA2, BB2, CC2 meet at some point X, and that X lies on the circumcircle of ABC.
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| 6. Find the number of 4-tuples (A, B, C, D), where A, B, C, D are subsets of {1, 2, 3, ... , n} such that A and B have at least one common element, B and C have at least one common element, and C and D have at least one common element. | |
| 7. Let s(n) be the number of integer solutions (x, y) to the equation 3y4 + 4ny3 + 2xy + 48 = 0 such that |x| is a square and y is square-free (so there is no prime p such that p2 divides y). Let s be the maximal value of s(n) (as n varies). Find all n such that s(n) = s. | |
| 8. C is the cube { (x, y, z) such that |x| ≤ 1, |y| ≤ 1, |z| ≤ 1}. Pn, where n = 1, 2, ... , 95, are any points of the cube. vn is the vector from (0, 0, 0) to Pn. S is the set of 295 v vectors of the form ±v1 ± v2 ± ... ± v95. Each vector v in S can be regarded as a vector from (0, 0, 0) to some point (a, b, c). Show that some vector in S has a2 + b2 + c2 ≤ d, where d = 48. Find the smallest d for which we can always find a vector in S with a2 + b2 + c2 ≤ d. | |
| 9. Show that (n-1)(m-1)(xn+m + yn+m) + (n+m-1)(xnym + xmyn) ≥ mn(xn+m-1y + xyn+m-1) for all positive integers m, n and all positive real numbers x, y. |
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