olymmath

Putnam 1958

William Lowell Putnam Mathematical Competition 1958.

Session A

A1

  • Session A
  1. If a0,a1,…,ana_{0}, a_{1}, \ldots, a_{n} are real numbers satisfying a01+a12+⋯+ann+1=0\frac{a_{0}}{1}+\frac{a_{1}}{2}+\cdots+\frac{a_{n}}{n+1}=0 show that the equation a0+a1x+a2x2+⋯+anxn=0a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}=0 has at least one real root.

Answer, solutions and source

A2

  • Session A
  1. Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be "lost"? Assume the coefficient of friction is such that no slipping occurs.

Answer, solutions and source

A3

  • Session A
  1. Real numbers are chosen at random from the interval (0≤x≤1)(0 \leq x \leq 1). If after choosing the nnth number the sum of the numbers so chosen first exceeds 1 , show that the expected or average value for nn is ee.

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A4

  • Session A
  1. If a1,a2,…,ana_{1}, a_{2}, \ldots, a_{n} are complex numbers such that ∣a1∣=∣a2∣=⋯=∣an∣=r≠0\left|a_{1}\right|=\left|a_{2}\right|=\cdots=\left|a_{n}\right|=r \neq 0 and if nTs{ }_{n} T_{s} denotes the sum of all products of these nn numbers taken ss at a time, prove that ∣nTsnTn−s∣=r2s−n\left|\frac{{ }_{n} T_{s}}{{ }_{n} T_{n-s}}\right|=r^{2 s-n} whenever the denominator of the left-hand side is different from zero.

Answer, solutions and source

A5

  • Session A
  1. Show that the integral equation f(x,y)=1+∫0x∫0yf(u,v)dudvf(x, y)=1+\int_{0}^{x} \int_{0}^{y} f(u, v) d u d v has at most one solution continuous for 0≤x≤1,0≤y≤10 \leq x \leq 1,0 \leq y \leq 1.

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A6

  • Session A
  1. What is the smallest amount that may be invested at interest rate ii, compounded annually, in order that one may withdraw 1 dollar at the end of the first year, 4 dollars at the end of the second year, …,n2\ldots, n^{2} dollars at the end of the nnth year, in perpetuity?

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Session B

B1

  • Session B
  1. (i) Given line segments A,B,C,DA, B, C, D, with AA the longest, construct a quadrilateral with these sides and with AA and BB parallel, when possible. (ii) Given any acute-angled triangle ABCA B C and one altitude AHA H, select any point DD on AHA H, then draw BDB D and extend until it intersects ACA C in EE, and draw CDC D and extend until it intersects ABA B in FF. Prove angle AHE=A H E= angle AHFA H F.

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B3

  • Session B
  1. In a round-robin tournament with nn players (each pair of players plays one game) in which there are no draws, the numbers of wins scored by the players are s1,s2,…,sns_{1}, s_{2}, \ldots, s_{n}. Prove that a necessary and sufficient condition for the existence of 3 players, A,B,CA, B, C, such that AA beat B,BB, B beat CC, and CC beat AA is s12+s22+⋯+sn2<(n−1)(n)(2n−1)′6s_{1}^{2}+s_{2}^{2}+\cdots+s_{n}^{2}<(n-1)(n)(2 n-1)^{\prime} 6

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B5

  • Session B
  1. Given an infinite number of points in a plane, prove that if all the distances determined between them are integers then the points are all in a straight line.

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B6

  • Session B
  1. A projectile moves in a resisting medium. The resisting force is a function of the velocity and is directed along the velocity vector. The equation xx =f(t)=f(t) gives the horizontal distance in terms of the time tt. Show that the vertical distance yy is given by y=−gf(t)∫dtf′(t)+g∫f(t)f′(t)dt+Af(t)+By=-g f(t) \int \frac{d t}{f^{\prime}(t)}+g \int \frac{f(t)}{f^{\prime}(t)} d t+A f(t)+B where AA and BB are constants and gg is the acceleration due to gravity.

Answer, solutions and source