Found problems: 966
2024 Putnam, A5
Consider the circle $\Omega$ with radius $9$ and center at the origin $(0,\,0)$, and a disk $\Delta$ with radius $1$ and center at $(r,\,0)$, where $0\leq r\leq 8$. Two points $P$ and $Q$ are chosen independently and uniformly at random on $\Omega$. Which value(s) of $r$ minimize the probability that the chord $\overline{PQ}$ intersects $\Delta$?
2012 Putnam, 6
Let $p$ be an odd prime number such that $p\equiv 2\pmod{3}.$ Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x)\equiv x^3\pmod{p}.$ Show that $\pi$ is an even permutation if and only if $p\equiv 3\pmod{4}.$
2005 Putnam, B1
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a\rfloor,\lfloor 2a\rfloor)=0$ for all real numbers $a.$
(Note: $\lfloor v\rfloor$ is the greatest integer less than or equal to $v.$)
1958 November Putnam, A5
Show that the number of non-zero integers in the expansion of the $n$-th order determinant having zeroes in the main diagonal and ones elsewhere is
$$n ! \left(1- \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + \frac{(-1)^{n}}{n!} \right) .$$
1950 Putnam, B6
Consider the closed plane curves $C_i$ and $C_o,$ their respective lengths $|C_i|$ and $|C_o|,$ the closed surfaces $S_i$ and $S_o,$ and their respective areas $|S_i|$ and $|S_o|.$ Assume that $C_i$ lies inside $C_o$ and $S_i$ inside $S_o.$ (Subscript $i$ stands for "inner," $o$ for "outer.") Prove the correct assertions among the following four, and disprove the others.
(i) If $C_i$ is convex, $|C_i| \le |C_o|.$
(ii) If $S_i$ is convex, $|S_i| \le |S_o|.$
(iii) If $C_o$ is the smallest convex curve containing $C_i,$ then $|C_o| \le |C_i|.$
(iv) If $S_o$ is the smallest convex surface containing $S_i,$ then $|S_o| \le |S_i|.$
You may assume that $C_i$ and $C_o$ are polygons and $S_i$ and $S_o$ polyhedra.
2021 Putnam, B4
Let $F_0,F_1,\dots$ be the sequence of Fibonacci numbers, with $F_0=0,F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for $n \ge 2$. For $m>2$, let $R_m$ be the remainder when the product $\prod_{k=1}^{F_m-1} k^k$ is divided by $F_m$. Prove that $R_m$ is also a Fibonacci number.
1992 Putnam, A5
For each positive integer $n$, let $a_n = 0$ (or $1$) if the number of $1$’s in the binary representation of $n$ is even (or
odd), respectively. Show that there do not exist positive integers $k$ and $m$ such that
$$a_{k+j}=a_{k+m+j} =a_{k+2m+j}$$
for $0 \leq j \leq m-1.$
1958 November Putnam, B1
Given
$$b_n = \sum_{k=0}^{n} \binom{n}{k}^{-1}, \;\; n\geq 1,$$
prove that
$$b_n = \frac{n+1}{2n} b_{n-1} +1, \;\; n \geq 2.$$
Hence, as a corollary, show
$$ \lim_{n \to \infty} b_n =2.$$
2007 Putnam, 2
Suppose that $ f: [0,1]\to\mathbb{R}$ has a continuous derivative and that $ \int_0^1f(x)\,dx\equal{}0.$
Prove that for every $ \alpha\in(0,1),$
\[ \left|\int_0^{\alpha}f(x)\,dx\right|\le\frac18\max_{0\le x\le 1}|f'(x)|\]
1963 Putnam, A5
i) Prove that if a function $f$ is continuous on the closed interval $[0, \pi]$ and
$$ \int_{0}^{\pi} f(t) \cos t \; dt= \int_{0}^{\pi} f(t) \sin t \; dt=0,$$
then there exist points $0 < \alpha < \beta < \pi$ such that $f(\alpha) =f(\beta) =0.$
ii) Let $R$ be a bounded, convex, and open region in the Euclidean plane. Prove with the help of i) that the centroid of $R$ bisects at least three different chords of the boundary of $ R.$
1947 Putnam, A6
A $3\times 3$ matrix has determinant $0$ and the cofactor of any element is equal to the square of that element. Show that every element in the matrix is $0.$
1955 Putnam, B7
Four forces acting on a body are in equilibrium. Prove that, if their lines of action are mutually skew, they are rulings of a hyperboloid.
2000 Putnam, 3
Let $f(t) = \displaystyle\sum_{j=1}^{N} a_j \sin (2\pi jt)$, where each $a_j$ is areal and $a_N$ is not equal to $0$.
Let $N_k$ denote the number of zeroes (including multiplicites) of $\dfrac{d^k f}{dt^k}$. Prove that \[ N_0 \le N_1 \le N_2 \le \cdots \text { and } \lim_{k \rightarrow \infty} N_k = 2N. \] [color=green][Only zeroes in [0, 1) should be counted.][/color]
2021 Putnam, B2
Determine the maximum value of the sum
\[
S=\sum_{n=1}^{\infty}\frac{n}{2^n}(a_1 a_2 \dots a_n)^{\frac{1}{n}}
\]
over all sequences $a_1,a_2,a_3,\dots$ of nonnegative real numbers satisfying
\[
\sum_{k=1}^{\infty}a_k=1.
\]
1946 Putnam, B5
Show that $\lceil (\sqrt{3}+1)^{2n})\rceil$ is divisible by $2^{n+1}.$
2024 Putnam, A4
Find all primes $p>5$ for which there exists an integer $a$ and an integer $r$ satisfying $1\leq r\leq p-1$ with the following property: the sequence $1,\,a,\,a^2,\,\ldots,\,a^{p-5}$ can be rearranged to form a sequence $b_0,\,b_1,\,b_2,\,\ldots,\,b_{p-5}$ such that $b_n-b_{n-1}-r$ is divisible by $p$ for $1\leq n\leq p-5$.
2001 Putnam, 4
Let $S$ denote the set of rational numbers different from $ \{ -1, 0, 1 \} $. Define $f: S \rightarrow S $ by $f(x)=x-1/x$. Prove or disprove that \[ \cap_{n=1}^{\infty} f^{(n)} (S) = \emptyset \] where $f^{(n)}$ denotes $f$ composed with itself $n$ times.
1949 Putnam, B5
let $(a_{n})$ be an arbitrary sequence of positive numbers. Show that
$$\limsup_{n\to \infty} \left(\frac{a_1 +a_{n+1}}{a_{n}}\right)^{n} \geq e.$$
2019 Putnam, A3
Given real numbers $b_0,b_1,\ldots, b_{2019}$ with $b_{2019}\neq 0$, let $z_1,z_2,\ldots, z_{2019}$ be the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019}b_kz^k.
\]
Let $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots, z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots, b_{2019}$ that satisfy
\[
1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\]
2002 Putnam, 2
Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.
2007 IMC, 3
Let $ C$ be a nonempty closed bounded subset of the real line and $ f: C\to C$ be a nondecreasing continuous function. Show that there exists a point $ p\in C$ such that $ f(p) \equal{} p$.
(A set is closed if its complement is a union of open intervals. A function $ g$ is nondecreasing if $ g(x)\le g(y)$ for all $ x\le y$.)
1988 Putnam, A2
A not uncommon calculus mistake is to believe that the product rule for derivatives says that $(fg)' = f'g'$. If $f(x)=e^{x^2}$, determine, with proof, whether there exists an open interval $(a,b)$ and a nonzero function $g$ defined on $(a,b)$ such that this wrong product rule is true for $x$ in $(a,b)$.
1948 Putnam, A3
Let $(a_n)$ be a decreasing sequence of positive numbers with limit $0$ such that
$$b_n = a_n -2 a_{n+1}+a_{n+2} \geq 0$$
for all $n.$ Prove that
$$\sum_{n=1}^{\infty} n b_n =a_1.$$
1987 Putnam, B6
Let $F$ be the field of $p^2$ elements, where $p$ is an odd prime. Suppose $S$ is a set of $(p^2-1)/2$ distinct nonzero elements of $F$ with the property that for each $a\neq 0$ in $F$, exactly one of $a$ and $-a$ is in $S$. Let $N$ be the number of elements in the intersection $S \cap \{2a: a \in S\}$. Prove that $N$ is even.
1960 Putnam, B7
Let $g(t)$ and $h(t)$ be real, continuous functions for $t\geq 0.$ Show that any function $v(t)$ satisfying the differential inequality
$$\frac{dv}{dt}+g(t)v \geq h(t),\;\; v(t)=c,$$
satisfies the further inequality $v(t)\geq u(t),$ where
$$\frac{du}{dt}+g(t)u = h(t),\;\; u(t)=c.$$
From this, conclude that for sufficiently small $t>0,$ the solution of
$$\frac{dv}{dt}+g(t)v = v^2 ,\;\; v(t)=c$$
may be written
$$v=\max_{w(t)} \left( c e^{- \int_{0}^{t} |g(s)-2w(s)| \, ds} -\int_{0}^{t} e^{-\int_{0}^{t} |g(s')-2w(s')| \, ds'} w(s)^{2} ds \right),$$
where the maximum is over all continuous functions $w(t)$ defined over some $t$-interval $[0,t_0 ].$