olymmath

Putnam 1955

William Lowell Putnam Mathematical Competition 1955.

Session A

A1

  • Session A
  1. Prove that there is no set of integers m,n,pm, n, p except 0,0,00,0,0 for which mm +n2+p3=0+n \sqrt{2}+p \sqrt{3}=0.

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A2

  • Session A
  1. A1A2…AnA_{1} A_{2} \ldots A_{n} is a regular polygon inscribed in a circle of radius rr and center O.PO . P is a point on line OA1O A_{1} extended beyond A1A_{1}. Show that ∏i=1nPA‾i=OP‾n−rn\prod_{i=1}^{n} \overline{P A}_{i}=\overline{O P}^{n}-r^{n}

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A3

  • Session A
  1. Suppose that ∑i=1∞xi\sum_{i=1}^{\infty} x_{i} is a convergent series of positive terms which monotonically decrease (that is, x1≥x2≥x3≥⋯x_{1} \geq x_{2} \geq x_{3} \geq \cdots ). Let PP denote the set of all numbers which are sums of some (finite or infinite) subseries of ∑i=1∞xi\sum_{i=1}^{\infty} x_{i}. Show that PP is an interval if and only if xn≤∑i=n+1∞xi for every integer n. (page 403) x_{n} \leq \sum_{i=n+1}^{\infty} x_{i} \text { for every integer } n . \text { (page 403) }

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A4

  • Session A
  1. On a circle, nn points are selected and the chords joining them in pairs are drawn. Assuming that no three of these chords are concurrent (except at the endpoints), how many points of intersection are there?

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A6

  • Session A
  1. Find a necessary and sufficient condition on the positive integer nn that the equation xn+(2+x)n+(2−x)n=0x^{n}+(2+x)^{n}+(2-x)^{n}=0 have a rational root.

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Session B

B2

  • Session B
  1. Suppose that ff is a function with two continuous derivatives and f(0)=f(0)= 0 . Prove that the function gg, defined by g(0)=f′(0),g(x)=f(x)′xg(0)=f^{\prime}(0), g(x)=f(x)^{\prime} x for x≠x \neq 0 , has a continuous derivative.

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B3

  • Session B
  1. Prove that there exists no distance-preserving map of a spherical cap into the plane. (Distances on the sphere are to be measured along great circles on the surface.)

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B4

  • Session B
  1. Do there exist 1,000,0001,000,000 consecutive integers each of which contains a repeated prime factor?

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B5

  • Session B
  1. Given an infinite sequence of 0's and 1's and a fixed integer kk, suppose that there are no more than kk distinct blocks of kk consecutive terms. Show that the sequence is eventually periodic. (For example, the sequence 11011010101 followed by alternating 0 's and 1 's indefinitely, which is periodic beginning with the fifth term.)

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B6

  • Session B
  1. Prove: If f(x)>0f(x)>0 for all xx and f(x)→0f(x) \rightarrow 0 as x→∞x \rightarrow \infty, then there exists at most a finite number of solutions of f(m)+f(n)+f(p)=1f(m)+f(n)+f(p)=1 in positive integers m,nm, n, and pp.

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