Olympiad problems

1949 Putnam, A4

Given that $P$ is a point inside a tetrahedron with vertices at $A, B, C$ and $D$, such that the sum of the distances $PA+PB+PC+PD$ is a minimum, show that the two angles $\angle APB$ and $\angle CPD$ are equal and are bisected by the same straight line. What other pair of angles must be equal?

1969 Putnam, B3

Tags: Putnam , Sequences , pi

The terms of a sequence $(T_n)$ satisfy $T_n T_{n+1} =n$ for all positive integers $n$ and $$\lim_{n\to \infty} \frac{ T_{n} }{ T_{n+1}}=1.$$ Show that $ \pi T_{1}^{2}=2.$

1960 Putnam, B5

Tags: Putnam , Sequence , limit

Define a sequence $(a_n)$ by $a_0 =0$ and $a_n = 1 +\sin(a_{n-1}-1)$ for $n\geq 1$. Evaluate $$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k.$$

2008 Putnam, B5

Tags: Putnam , function , number theory , greatest common divisor , calculus , derivative , integration

Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)

2020 Putnam, B3

Tags: Putnam , probability , expected value , function , Putnam 2020

Let $x_0=1$, and let $\delta$ be some constant satisfying $0<\delta<1$. Iteratively, for $n=0,1,2,\dots$, a point $x_{n+1}$ is chosen uniformly form the interval $[0,x_n]$. Let $Z$ be the smallest value of $n$ for which $x_n<\delta$. Find the expected value of $Z$, as a function of $\delta$.

1988 Putnam, A1

Tags: Putnam

Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y| \leq 1$ and $|y| \leq 1$. Sketch the region $R$ and find its area.

1960 Putnam, A1

Tags: Putnam , equations , Integers

Let $n$ be a given positive integer. How many solutions are there in ordered positive integer pairs $(x,y)$ to the equation $$\frac{xy}{x+y}=n?$$

1953 Putnam, A4

Tags: Putnam , logarithms , trigonometry

From the identity $$ \int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2} \log \sin x \, dx + \int_{0}^{\pi \slash 2} \log \cos x \, dx +\int_{0}^{\pi \slash 2} \log 2 \, dx, $$ deduce the value of $\int_{0}^{\pi \slash 2} \log \sin x \, dx.$

1975 Putnam, B3

Tags: Putnam , supremum , inequalities

Let $n$ be a positive integer. Let $S=\{a_1,\ldots, a_{k}\}$ be a finite collection of at least $n$ not necessarily distinct positive real numbers. Let $$f(S)=\left(\sum_{i=1}^{k} a_{i}\right)^{n}$$ and $$g(S)=\sum_{1\leq i_{1}<\ldots<i_{n} \leq k} a_{i_{1}}\cdot\ldots\cdot a_{i_{n}}.$$ Determine $\sup_{S} \frac{g(S)}{f(S)}$.

1990 Putnam, B1

Tags: calculus , integration , derivative , function , algebra , functional equation , Putnam

Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, \[ \left( f(x) \right)^2 = \displaystyle\int_0^x \left[ \left( f(t) \right)^2 + \left( f'(t) \right)^2 \right] \, \mathrm{d}t + 1990. \]

1961 Putnam, A3

Tags: Putnam , limit , series

Evaluate $$\lim_{n\to \infty} \sum_{j=1}^{n^{2}} \frac{n}{n^2 +j^2 }.$$

1992 Putnam, B4

Tags: Putnam , algebra , polynomial , derivative

Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with $x^3 -x$. Let $$ \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 -x } \right) =\frac{f(x)}{g(x)}$$ for polynomials $f(x)$ and $g(x).$ Find the smallest possible degree of $f(x)$.

1950 Putnam, B5

Tags: Putnam

Answer either (i) or (ii). (i) Given that the sequence whose $n$th term is $(s_n + 2s_{n + 1})$ converges, show that the sequence $\{ s_n \}$ converges also. (ii) A plane varies so that it includes a cone of constant value equal to $\pi a^3 / 3$ with the surface the equation of which in rectangular coordinates is $2xy = z^2.$ Find the equation of the envelope of the various positions of this plane. State the result so that it applies to a general cone (that is, conic surface) of the second order.

1959 Putnam, B1

Tags: Putnam , Intersection of set , lines

Let each of $m$ distinct points on the positive part of the $x$-axis be joined to $n$ distinct points on the positive part of the $y$-axis. Obtain a formula for the number of intersection points of these segments, assuming that no three of the segments are concurrent.

1965 Putnam, A4

Tags: Putnam

At a party, assume that no boy dances with every girl but each girl dances with at least one boy. Prove that there are two couples $gb$ and $g'b'$ which dance whereas $b$ does not dance with $g'$ nor does $g$ dance with $b'$.

1953 Putnam, A3

Tags: Putnam , inequalities

$a, b, c$ are real, and the sum of any two is greater than the third. Show that $\frac{2(a + b + c)(a^2 + b^2 + c^2)}{3} > a^3 + b^3 + c^3 + abc$.