x + y2 + z4 = 0
y + z2 + x4 = 0
z + x2 + y5 = 0.
x14 + 14x1x2 + 1 = y14
x24 + 14x2x3 + 1 = y24
...
xn4 + 14xnx1 + 1 = yn4.
| 1. The distinct points X1, X2, X3, X4, X5, X6 all lie on the same side of the line AB. The six triangles ABXi are all similar. Show that the Xi lie on a circle. | |
| 2. Find all solutions in positive integers to aA = bB = cC = 19901990abc, where A = bc, B = ca, C = ab. | |
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3. Show that there are two real solutions to:
x + y2 + z4 = 0 y + z2 + x4 = 0 z + x2 + y5 = 0. |
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4. Find all solutions in positive integers to:
x14 + 14x1x2 + 1 = y14 x24 + 14x2x3 + 1 = y24 ... xn4 + 14xnx1 + 1 = yn4. |
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| 5. If a1, ... , an is a permutation of 1, 2, ... , n, call ∑ |ai - i| its modsum. Find the average modsum of all n! permutations. | |
| 6. p(x) is a polynomial with integer coefficients. The sequence of integers a1, a2, ... , an (where n > 2) satisfies a2 = p(a1), a3 = p(a2), ... , an = p(an-1), a1 = p(an). Show that a1 = a3. |
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| 7. Dn is a set of domino pieces. For each pair of non-negative integers (a, b) with a ≤ b ≤ n, there is one domino, denoted [a, b] or [b, a] in Dn. A ring is a sequence of dominoes [a1, b1], [a2, b2], ... , [ak, bk] such that b1 = a2, b2 = a3, ... , bk-1 = ak and bk = a1. Show that if n is even there is a ring which uses all the pieces. Show that for n odd, at least (n+1)/2 pieces are not used in any ring. For n odd, how many different sets of (n+1)/2 are there, such that the pieces not in the set can form a ring? | |
| 8. We are given a supply of a x b tiles with a and b distinct positive integers. The tiles are to be used to tile a 28 x 48 rectangle. Find a, b such that the tile has the smallest possible area and there is only one possible tiling. (If there are two distinct tilings, one of which is a reflection of the other, then we treat that as more than one possible tiling. Similarly for other symmetries.) Find a, b such that the tile has the largest possible area and there is more than one possible tiling. | |
| 9. a1, a2, ... , an is a sequence of integers such that every non-empty subsequence has non-zero sum. Show that we can partition the positive integers into a finite number of sets such that if xi all belong to the same set, then a1x1 + a2x2 + ... + anxn is non-zero. |
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
18 Dec 2002
Last corrected/updated 18 Dec 2002