Found problems: 966
1952 Putnam, A7
Directed lines are drawn from the center of a circle, making angles of $0, \pm 1, \pm 2, \pm 3, \ldots$ (measured in radians from a prime direction). If these lines meet the circle in points $P_0, P_1, P_{-1}, P_2, P_{-2}, \ldots,$ show that there is no interval on the circumference of the circle which does not contain some $P_{\pm i}.$ (You may assume that $\pi$ is irrational.)
2005 Putnam, A5
Evaluate $\int_0^1\frac{\ln(x+1)}{x^2+1}\,dx.$
2006 Putnam, B5
For each continuous function $f: [0,1]\to\mathbb{R},$ let $I(f)=\int_{0}^{1}x^{2}f(x)\,dx$ and $J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$
2012 Putnam, 1
Let $d_1,d_2,\dots,d_{12}$ be real numbers in the open interval $(1,12).$ Show that there exist distinct indices $i,j,k$ such that $d_i,d_j,d_k$ are the side lengths of an acute triangle.
1999 Putnam, 3
Consider the power series expansion \[\dfrac{1}{1-2x-x^2}=\sum_{n=0}^\infty a_nx^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2+a_{n+1}^2=a_m.\]
1974 Putnam, A6
Given $n$, let $k = k(n)$ be the minimal degree of any monic integral polynomial
$$f(x)=x^k + a_{k-1}x^{k-1}+\ldots+a_0$$
such that the value of $f(x)$ is exactly divisible by $n$ for every integer $x.$ Find the relationship between $n$ and $k(n)$. In particular, find the value of $k(n)$ corresponding to $n = 10^6.$
2019 Putnam, A1
Determine all possible values of $A^3+B^3+C^3-3ABC$ where $A$, $B$, and $C$ are nonnegative integers.
1955 Putnam, B1
A sphere rolls along two intersecting straight lines. Find the locus of its center.
1950 Putnam, A4
Answer either (i) or (ii).
(i) In a right prism with triangular base, given the sum of the areas of three mutually adjacent faces (that is, of two lateral faces and one base), show that these faces are of equal area and perpendicular to each other when the volume attains its maximum.
(ii) Show that \[ \frac{\frac x1 + \frac {x^3} {1 \cdot 3} + \frac {x^5} {1 \cdot 3 \cdot 5} + \frac {x^7} {1 \cdot 3 \cdot 5 \cdot 7} + \cdots }{1 + \frac {x^2} 2 + \frac {x^4}{2 \cdot 4} + \frac{x^6}{2 \cdot 4 \cdot 6} + \cdots} = \int_0^x e^{-t^2} \mathrm dt.\]
1974 Putnam, B2
Let $y(x)$ be a continuously differentiable real-valued function of a real variable $x$. Show that if $y'(x)^2 +y(x)^3 \to 0$ as $x\to \infty,$ then $y(x)$ and $y'(x) \to 0$ as $x \to \infty.$
2010 Putnam, B2
Given that $A,B,$ and $C$ are noncollinear points in the plane with integer coordinates such that the distances $AB,AC,$ and $BC$ are integers, what is the smallest possible value of $AB?$
1961 Putnam, A5
Let $\Omega$ be a set of $n$ points, where $n>2$. Let $\Sigma$ be a nonempty subcollection of the $2^n$ subsets of $\Omega$ that is closed with respect to the unions, intersections and complements. If $k$ is the number of elements of $\Sigma,$ what are the possible values of $k?$
Putnam 1939, A7
Do either $(1)$ or $(2)$:
$(1)$ Let $C_a$ be the curve $(y - a^2)^2 = x^2(a^2 - x^2).$ Find the curve which touches all $C_a$ for $a > 0.$ Sketch the solution and at least two of the $C_a.$
$(2)$ Given that $(1 - hx)^{-1}(1 - kx)^{-1} = \sum_{i\geq0}a_i x^i,$ prove that $(1 + hkx)(1 - hkx)^{-1}(1 - h^2x)^{-1}(1 - k^2x)^{-1} = \sum_{i\geq0} a_i^2 x^i.$
2014 Putnam, 5
In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible $n\times n$ matrices with entries in the field $\mathbb{Z}/p\mathbb{Z}$ of integers modulo $p,$ where $n$ is a fixed positive integer and $p$ is a fixed prime number. The rules of the game are:
(1) A player cannot choose an element that has been chosen by either player on any previous turn.
(2) A player can only choose an element that commutes with all previously chosen elements.
(3) A player who cannot choose an element on his/her turn loses the game.
Patniss takes the first turn. Which player has a winning strategy?
2009 Putnam, A4
Let $ S$ be a set of rational numbers such that
(a) $ 0\in S;$
(b) If $ x\in S$ then $ x\plus{}1\in S$ and $ x\minus{}1\in S;$ and
(c) If $ x\in S$ and $ x\notin\{0,1\},$ then $ \frac{1}{x(x\minus{}1)}\in S.$
Must $ S$ contain all rational numbers?
2008 Putnam, A3
Start with a finite sequence $ a_1,a_2,\dots,a_n$ of positive integers. If possible, choose two indices $ j < k$ such that $ a_j$ does not divide $ a_k$ and replace $ a_j$ and $ a_k$ by $ \gcd(a_j,a_k)$ and $ \text{lcm}\,(a_j,a_k),$ respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: $ \gcd$ means greatest common divisor and lcm means least common multiple.)
1972 Putnam, A4
Show that a circle inscribed in a square has a larger perimeter than any other ellipse inscribed in the square.
1967 Putnam, B3
If $f$ and $g$ are continuous and periodic functions with period $1$ on the real line, then
$$\lim_{n\to \infty} \int_{0}^{1} f(x)g (nx)\; dx =\left( \int_{0}^{1} f(x)\; dx\right)\left( \int_{0}^{1} g(x)\; dx\right).$$
1968 Putnam, A2
Given integers $a,b,c,d,m,n$ such that $ad-bc\ne 0$ and any real $\varepsilon >0$, show that one can find rational numbers $x,y$ such that $0<|ax+by-m|<\varepsilon$ and $0<|cx+dy-n|<\varepsilon$.
2024 Putnam, B1
Let $n$ and $k$ be positive integers. The square in the $i$th row and $j$th column of an $n$-by-$n$ grid contains the number $i+j-k$. For which $n$ and $k$ is it possible to select $n$ squares from the grid, no two in the same row or column, such that the numbers contained in the selected squares are exactly $1,\,2,\,\ldots,\,n$?
2006 Putnam, A2
Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)
1941 Putnam, B7
Do either (1) or (2):
(1) Show that any solution $f(t)$ of the functional equation
$$f(x+y)f(x-y)=f(x)^{2} +f(y)^{2} -1$$
for $x,y\in \mathbb{R}$ satisfies
$$f''(t)= \pm c^{2} f(t)$$
for a constant $c$, assuming the existence and continuity of the second derivative.
Deduce that $f(t)$ is one of the functions
$$ \pm \cos ct, \;\;\; \pm \cosh ct.$$
(2) Let $(a_{i})_{i=1,...,n}$ and $(b_{i})_{i=1,...,n}$ be real numbers. Define an $(n+1)\times (n+1)$-matrix $A=(c_{ij})$ by
$$ c_{i1}=1, \; \; c_{1j}= x^{j-1} \; \text{for} \; j\leq n,\; \; c_{1n+1}=p(x), \;\; c_{ij}=a_{i-1}^{j-1} \; \text{for}\; i>1, j\leq n,\;\;
c_{in+1}=b_{i-1}\; \text{for}\; i>1.$$
The polynomial $p(x)$ is defined by the equation $\det A=0$. Let $f$ be a polynomial and replace $(b_{i})$ with $(f(b_{i}))$. Then $\det A=0$ defines another polynomial $q(x)$. Prove that $f(p(x))-q(x)$ is a multiple of
$$\prod_{i=1}^{n} (x-a_{i}).$$
1955 Putnam, A5
If a parabola is given in the plane, find a geometric construction (ruler and compass) for the focus.
1959 Putnam, A7
If $f$ is a real-valued function of one real variable which has a continuous derivative on the closed interval $[a,b]$ and for which there is no $x\in [a,b]$ such that $f(x)=f'(x)=0$, then prove that there is a function $g$ with continuous first derivative on $[a,b]$ such that $fg'-f'g$ is positive on $[a,b].$
2017 Putnam, B3
Suppose that $$f(x) = \sum_{i=0}^\infty c_ix^i$$
is a power series for which each coefficient $c_i$ is $0$ or $1$. Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.