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The 61st William Lowell Putnam Mathematical Competition Saturday, December 2, 2000

The document contains 6 problems labeled A-1 through A-6 and 5 problems labeled B-1 through B-6 related to mathematics. The problems cover topics such as series convergence, properties of polynomials and integers, geometry problems involving circles and triangles, and set theory questions.

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Sarah Kalinsky
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138 views1 page

The 61st William Lowell Putnam Mathematical Competition Saturday, December 2, 2000

The document contains 6 problems labeled A-1 through A-6 and 5 problems labeled B-1 through B-6 related to mathematics. The problems cover topics such as series convergence, properties of polynomials and integers, geometry problems involving circles and triangles, and set theory questions.

Uploaded by

Sarah Kalinsky
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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The 61st William Lowell Putnam Mathematical Competition Saturday, December 2, 2000

A-1 Let A be a positive real number. What are the possible values of j =0 x2 j , given that x0 , x1 , . . . are positive numbers for which j =0 xj = A? A-2 Prove that there exist innitely many integers n such that n, n + 1, n + 2 are each the sum of the squares of two integers. [Example: 0 = 02 + 02 , 1 = 02 + 12 , 2 = 12 + 12 .] A-3 The octagon P1 P2 P3 P4 P5 P6 P7 P8 is inscribed in a circle, with the vertices around the circumference in the given order. Given that the polygon P1 P3 P5 P7 is a square of area 5, and the polygon P2 P4 P6 P8 is a rectangle of area 4, nd the maximum possible area of the octagon. A-4 Show that the improper integral
B B

exist integers r, s, t such that raj + sbj + tcj is odd for at least 4N/7 values of j , 1 j N . B-2 Prove that the expression gcd(m, n) n n m is an integer for all pairs of integers n m 1. B-3 Let f (t) = j =1 aj sin(2jt), where each aj is real and aN is not equal to 0. Let Nk denote the number of k f zeroes (including multiplicities) of d . Prove that dtk N0 N1 N2 and lim Nk = 2N.
k N

lim

sin(x) sin(x2 ) dx
0

[Editorial clarication: only zeroes in [0, 1) should be counted.] B-4 Let f (x) be a continuous function such that f (2x2 1) = 2xf (x) for all x. Show that f (x) = 0 for 1 x 1. B-5 Let S0 be a nite set of positive integers. We dene nite sets S1 , S2 , . . . of positive integers as follows: the integer a is in Sn+1 if and only if exactly one of a 1 or a is in Sn . Show that there exist innitely many integers N for which SN = S0 {N + a : a S0 }. B-6 Let B be a set of more than 2n+1 /n distinct points with coordinates of the form (1, 1, . . . , 1) in ndimensional space with n 3. Show that there are three distinct points in B which are the vertices of an equilateral triangle.

converges. A-5 Three distinct points with integer coordinates lie in the plane on a circle of radius r > 0. Show that two of these points are separated by a distance of at least r1/3 . A-6 Let f (x) be a polynomial with integer coefcients. Dene a sequence a0 , a1 , . . . of integers such that a0 = 0 and an+1 = f (an ) for all n 0. Prove that if there exists a positive integer m for which am = 0 then either a1 = 0 or a2 = 0. B-1 Let aj , bj , cj be integers for 1 j N . Assume for each j , at least one of aj , bj , cj is odd. Show that there

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