(a, b, c) ∈ S iff (b, c, a) ∈ S;
(a, b, c) ∈ S iff (c, b, a) ∉ S;
(a, b, c) and (c, d, a) ∈ S iff (b, c, d) and (d, a, b) ∈ S.
Prove that there exists an injection g from A to the reals, such that g(a) < g(b) < g(c) implies (a, b, c) ∈ S.
| A1. What is the smallest α such that two squares with total area 1 can always be placed inside a rectangle area α with sides parallel to those of the rectangle and with no overlap (of their interiors)? |
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| A2. Two circles have radii 1 and 3 and centers a distance 10 apart. Find the locus of all points which are the midpoint of a segment with one end on each circle. |
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| A3. There are six courses on offer. Each of 20 students chooses some all or none of the courses. Is it true that we can find two courses C and C' and five students S1, S2, S3, S4, S5 such that each Si has chosen C and C', or such that each Si has chosen neither C nor C'? |
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A4. A is a finite set. S is a set of ordered triples (a, b, c) of distinct elements of A, such that:
(a, b, c) ∈ S iff (b, c, a) ∈ S; (a, b, c) ∈ S iff (c, b, a) ∉ S; (a, b, c) and (c, d, a) ∈ S iff (b, c, d) and (d, a, b) ∈ S. Prove that there exists an injection g from A to the reals, such that g(a) < g(b) < g(c) implies (a, b, c) ∈ S. |
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| A5. Let p be a prime ≥ 5. Prove that p2 divides ∑0[2p/3] pCr. |
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| A6. Let R be the reals and k a non-negative real. Find all continuous functions f : R → R such that f(x) = f(x2 + k) for all x. |
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| B1. Let N be the set {1, 2, 3, ... , n}. X is selfish if |X| ∈ X. How many subsets of N are selfish and have no proper selfish subsets. |
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| B2. Let f(n) = ( (2n+1)/e )(2n+1)/2. Show that for n > 0: f(n-1) < 1·3·5 ... (2n-1) < f(n). |
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| B3. (x1, x2, ... , xn) is a permutation of (1, 2, ... , n). What is the maximum of x1x2 + x2x3 + ... + xn-1xn + xnx1? |
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| B4. Let B be the 2 x 2 matrix (bij) with b11 = b22 = 1, b12 = 1996, b21 = 0. Can we find a 2 x 2 matrix A such that sin A = B? [We define sin A by the usual power series: A - A3/3! + A5/5! - ... .] |
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| B5. We call a finite string of the symbols X and O balanced iff every substring of consecutive symbols has a difference of at most 2 between the number of Xs and the number of Os. For example, XOOXOOX is not balanced, because the substring OOXOO has a difference of 3. Find the number of balanced strings of length n. |
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| B6. The origin lies inside a convex polygon whose vertices have coordinates (ai, bi) for i = 1, 2, ... , n. Show that we can find x, y > 0 such that a1xa1yb1 + a2xa2yb2 + ... + anxanybn = 0 and b1xa1yb1 + b2xa2yb2 + ... + bnxanybn = 0. |
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To avoid possible copyright problems, I have changed the wording, but not the substance of all the problems. The original text of the problems and the official solutions are in American Mathematical Monthly 104 (1997) 746.
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© John Scholes
jscholes@kalva.demon.co.uk
16 Dec 1998