By Dawson C.N., Martinez-Canales M.L.

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A6. Deﬁne a sequence (an )n≥0 by an = 2 if n = (2 + 3)m for some integer m ≥ 0, and an = 3 otherwise. Prove that (an ) satisﬁes the self-generation property. B1. To guess the answer, try the problem with 1993 replaced by smaller numbers, and look for a pattern. Which m forces n to be large? B2. Player B wins by making each move so that A cannot possibly win on the next move. B3. Sketch the set of (x, y) in the unit square for which the integer nearest x/y is even. Evaluate its area by comparing to Leibniz’s formula π 1 1 1 = 1 − + − + ··· .

B6. If A ∈ M, then A2 commutes with all elements of M, so A2 = I. Given any linear relation among the elements of M, other relations of the same length can be obtained by multiplying by elements of M, and then shorter relations can be obtained by subtraction. Eventually this leads to a contradiction, so the matrices in M are linearly independent. 43 Hints: The Fifty-Fourth Competition (1993) The Fifty-Fourth William Lowell Putnam Mathematical Competition December 4, 1993 A1. Let (b, c) be the rightmost intersection point.

0 (page 184) B5. Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers. (page 185) B6. Let S be a set of three, not necessarily distinct, positive integers. Show that one can transform S into a set containing 0 by a ﬁnite number of applications of the following rule: Select two of the three integers, say x and y, where x ≤ y and replace them with 2x and y − x. (page 188) 21 Problems: The Fifty-Fifth Competition (1994) The Fifty-Fifth William Lowell Putnam Mathematical Competition December 3, 1994 Questions Committee: Eugene Luks, Fan Chung, and Mark I.

### Acharacteristic-Galerkin Aproximation to a system of Shallow Water Equations by Dawson C.N., Martinez-Canales M.L.

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