Found problems: 966
2023 Putnam, B3
A sequence $y_1, y_2, \ldots, y_k$ of real numbers is called $\textit{zigzag}$ if $k=1$, or if $y_2-y_1, y_3-y_2, \ldots, y_k-y_{k-1}$ are nonzero and alternate in sign. Let $X_1, X_2, \ldots, X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a\left(X_1, X_2, \ldots, X_n\right)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1, i_2, \ldots, i_k$ such that $X_{i_1}, X_{i_2}, \ldots X_{i_k}$ is zigzag. Find the expected value of $a\left(X_1, X_2, \ldots, X_n\right)$ for $n \geq 2$.
2015 Putnam, A3
Compute \[\log_2\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}\left(1+e^{2\pi iab/2015}\right)\right)\] Here $i$ is the imaginary unit (that is, $i^2=-1$).
1996 Putnam, 5
Given a finite binary string $S$ of symbols $X,O$ we define $\Delta(S)=n(X)-n(O)$ where $n(X),n(O)$ respectively denote number of $X$'s and $O$'s in a string. For example $\Delta(XOOXOOX)=3-4=-1$. We call a string $S$ $\emph{balanced}$ if every substring $T$ of $S$ has $-2\le \Delta(T)\le 2$. Find number of balanced strings of length $n$.
1954 Putnam, B7
Let $a>0$. Show that
$$ \lim_{n \to \infty} \sum_{s=1}^{n} \left( \frac{a+s}{n} \right)^{n}$$
lies between $e^a$ and $e^{a+1}.$
1972 Putnam, A6
Let $ f$ be an integrable real-valued function on the closed interval $ [0, 1]$ such that
$$\int_{0}^{1} x^{m}f(x) dx=\begin{cases} 0 \;\; \text{for}\; m=0,1,\ldots,n-1;\\
1\;\; \text{for}\; m=n. \end{cases} $$
Show that $|f(x)|\geq2^{n}(n+1)$ on a set of positive measure.
1996 Putnam, 1
Define a $\emph{selfish}$ set to be a set which has its own cardinality as its element. And a set is a $\emph{minimal }\text{ selfish}$ set if none of its proper subsets are $\emph{selfish}$. Find with proof the number of $\text{minimal selfish}$ subsets of $\{1,2,\cdots ,n\}$.
1957 Putnam, A2
Let $a>1.$ A uniform wire is bent into a form coinciding with the portion of the curve $y=e^x$ for $x\in [0,a]$, and the line segment $y=e^a$ for $x\in [a-1,a].$ The wire is then suspended from the point $(a-1, e^a)$ and a horizontal force $F$ is applied to the point $(0,1)$ to hold the wire in coincidence with the curve and segment. Show that the force $F$ is directed to the right.
2014 Putnam, 5
Let $P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.$ Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j\ne k.$
1988 Putnam, B1
A [i]composite[/i] (positive integer) is a product $ab$ with $a$ and $b$ not necessarily distinct integers in $\{2,3,4,\dots\}$. Show that every composite is expressible as $xy+xz+yz+1$, with $x,y,z$ positive integers.
2024 Putnam, A6
Let $c_0,\,c_1,\,c_2,\,\ldots$ be a sequence defined so that
\[
\frac{1-3x-\sqrt{1-14x+9x^2}}{4}=\sum_{k=0}^\infty c_kx^k
\]
for sufficiently small $x$. For a positive integer $n$, let $A$ be the $n$-by-$n$ matrix with $i,j$-entry $c_{i+j-1}$ for $i$ and $j$ in $\{1,\,\ldots,\,n\}$. Find the determinant of $A$.
1954 Putnam, A6
Suppose that $u_0 , u_1 ,\ldots$ is a sequence of real numbers such that
$$u_n = \sum_{k=1}^{\infty} u_{n+k}^{2}\;\;\; \text{for} \; n=0,1,2,\ldots$$
Prove that if $\sum u_n$ converges, then $u_k=0$ for all $k$.
1948 Putnam, A6
Answer either (i) or (ii):
(i) A force acts on the element $ds$ of a closed plane curve. The magnitude of this force is $r^{-1} ds$ where $r$ is the radius of curvature at the point considered, and the direction of the force is perpendicular to the curve, it points to the convex side. Show that the system of such forces acting on all elements of the curve keep it in equilibrium.
(ii) Show that
$$x+ \frac{2}{3}x^{3}+ \frac{2\cdot 4}{3\cdot 5} x^5 +\frac{2\cdot 4\cdot 6}{3\cdot 5\cdot 7} x^7 + \ldots= \frac{ \arcsin x}{\sqrt{1-x^{2}}}.$$
1968 Putnam, A5
Find the smallest possible $\alpha\in \mathbb{R}$ such that if $P(x)=ax^2+bx+c$ satisfies $|P(x)|\leq1 $ for $x\in [0,1]$ , then we also have $|P'(0)|\leq \alpha$.
Putnam 1939, B6
Do either $(1)$ or $(2)$:
$(1)$ $f$ is continuous on the closed interval $[a, b]$ and twice differentiable on the open interval $(a, b).$ Given $x_0 \in (a, b),$ prove that we can find $\xi \in (a, b)$ such that
$\dfrac{ ( \dfrac{(f(x_0) - f(a))}{(x_0 - a)} - \dfrac{(f(b) - f(a))}{(b - a)} )}{(x_0 - b)} = \dfrac{f''(\xi)}{2}.$
$(2)$ $AB$ and $CD$ are identical uniform rods, each with mass $m$ and length $2a.$ They are placed a distance $b$ apart, so that $ABCD$ is a rectangle. Calculate the gravitational attraction between them. What is the limiting value as a tends to zero?
1949 Putnam, B1
Each rational number $\frac{p}{q}$ with $p,q$ coprime of the open interval $(0,1)$ is covered by the closed interval $\left[\frac{p}{q}-\frac{1}{4q^{2}}, \frac{p}{q}+\frac{1}{4q^{2}}\right]$. Prove that $\frac{\sqrt{2}}{2}$ is not covered by any of the above closed intervals.
2009 Kazakhstan National Olympiad, 6
Is there exist four points on plane, such that distance between any two of them is integer odd number?
May be it is geometry or number theory or combinatoric, I don't know, so it here :blush:
1985 Putnam, B3
Let $$\begin{array}{cccc}{a_{1,1}} & {a_{1,2}} & {a_{1,3}} & {\dots} \\ {a_{2,1}} & {a_{2,2}} & {a_{2,3}} & {\cdots} \\ {a_{3,1}} & {a_{3,2}} & {a_{3,3}} & {\cdots} \\ {\vdots} & {\vdots} & {\vdots} & {\ddots}\end{array}$$ be a doubly infinite array of positive integers, and suppose each positive integer appears exactly eight times in the array. Prove that $a_{m, n}>m n$ for some pair of positive integers $(m, n) .$
2019 Putnam, B1
Denote by $\mathbb Z^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n\geq 0$, let $P_n$ be the subset of $\mathbb Z^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k\leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.
1997 Putnam, 1
For all reals $x$ define $\{x\}$ to be the difference between $x$ and the closest integer to $x$. For each positive integer $n$ evaluate :
\[ S_n=\sum_{m=1}^{6n-1}\min \left(\left\{\frac{m}{6n}\right\},\left\{\frac{m}{3n}\right\}\right) \]
2010 Putnam, B6
Let $A$ be an $n\times n$ matrix of real numbers for some $n\ge 1.$ For each positive integer $k,$ let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k=A^{[k]}$ for $k=1,2,\cdots,n+1,$ then $A^k=A^{[k]}$ for all $k\ge 1.$
2016 Putnam, B1
Let $x_0,x_1,x_2,\dots$ be the sequence such that $x_0=1$ and for $n\ge 0,$
\[x_{n+1}=\ln(e^{x_n}-x_n)\]
(as usual, the function $\ln$ is the natural logarithm). Show that the infinite series
\[x_0+x_1+x_2+\cdots\]
converges and find its sum.
1959 Putnam, A3
Find all complex-valued functions $f$ of a complex variable such that $$f(z)+zf(1-z)=1+z$$
for all $z\in \mathbb{C}$.
1957 Putnam, A5
Given $n$ points in the plane, show that the largest distance determined by these points cannot occur more than $n$ times.
1998 Putnam, 6
Prove that, for any integers $a,b,c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer.
1946 Putnam, B1
Let $K$ denote the circumference of a circular disk of radius $1$, and let $k$ denote a circular arc that joins two points
$a,b$ on $K$ and lies otherwise in the given circular disc. Suppose that $k$ divides the circular disk into two parts of equal area. Prove that the length of $k$ exceeds $2.$