Found problems: 966
1951 Putnam, B6
Assuming that all of the roots of the cubic equation $x^3 + ax^2 +bx + c = 0$ are real, show that the difference between the greatest and the least roots is not less than $(a^2 - 3b)^{1/2}$ or greater than $2 (a^2 - 3b)^{1/2} / 3^{1/2}.$
2018 Putnam, A4
Let $m$ and $n$ be positive integers with $\gcd(m, n) = 1$, and let
\[a_k = \left\lfloor \frac{mk}{n} \right\rfloor - \left\lfloor \frac{m(k-1)}{n} \right\rfloor\]
for $k = 1, 2, \dots, n$. Suppose that $g$ and $h$ are elements in a group $G$ and that
\[gh^{a_1} gh^{a_2} \cdots gh^{a_n} = e,\]
where $e$ is the identity element. Show that $gh = hg$. (As usual, $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
1958 November Putnam, A2
Let $R_1 =1$ and $R_{n+1}= 1+ n\slash R_n$ for $n\geq 1.$ Show that for $n\geq 1,$
$$ \sqrt{n} \leq R_n \leq \sqrt{n} +1.$$
1993 Putnam, A1
Let $O$ be the origin. $y = c$ intersects the curve $y = 2x - 3x^3$ at $P$ and $Q$ in the first quadrant and cuts the y-axis at $R$. Find $c$ so that the region $OPR$ bounded by the y-axis, the line $y = c$ and the curve has the same area as the region between $P$ and $Q$ under the curve and above the line $y = c$.
2002 Putnam, 6
Fix an integer $ b \geq 2$. Let $ f(1) \equal{} 1$, $ f(2) \equal{} 2$, and for each $ n \geq 3$, define $ f(n) \equal{} n f(d)$, where $ d$ is the number of base-$ b$ digits of $ n$. For which values of $ b$ does
\[ \sum_{n\equal{}1}^\infty \frac{1}{f(n)}
\]
converge?
1999 Putnam, 5
For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos(j\theta+k\theta)$ for all $j,k$.
1999 Putnam, 4
Sum the series \[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]
1952 Putnam, B3
Develop necessary and sufficient conditions that the equation \[ \begin{vmatrix} 0 & a_1 - x & a_2 - x \\ -a_1 - x & 0 & a_3 - x \\ -a_2 - x & -a_3 - x & 0\end{vmatrix} = 0 \qquad (a_i \neq 0) \] shall have a multiple root.
1948 Putnam, A4
Let $D$ be a plane region bounded by a circle of radius $r.$ Let $(x,y)$ be a point of $D$ and consider a circle of radius $\delta$ and center at $(x,y).$ Denote by $l(x,y)$ the length of that arc of the circle which is outside $D.$ Find
$$\lim_{\delta \to 0} \frac{1}{\delta^{2}} \int_{D} l(x,y)\; dx\; dy.$$
1987 Putnam, B4
Let $(x_1,y_1) = (0.8, 0.6)$ and let $x_{n+1} = x_n \cos y_n - y_n \sin y_n$ and $y_{n+1}= x_n \sin y_n + y_n \cos y_n$ for $n=1,2,3,\dots$. For each of $\lim_{n\to \infty} x_n$ and $\lim_{n \to \infty} y_n$, prove that the limit exists and find it or prove that the limit does not exist.
1952 Putnam, A4
The flag of the United Nations consists of a polar map of the world, with the North Pole as its center, extending to approximately $45^\circ$ South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude $30^\circ$ S.In the very close vicinity of this parallel how much are East and West distances exaggerated as compared to North and South distances?
1988 Putnam, A4
(a) If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart?
(b) What if "three'' is replaced by "nine''?
1987 Putnam, B5
Let $O_n$ be the $n$-dimensional vector $(0,0,\cdots, 0)$. Let $M$ be a $2n \times n$ matrix of complex numbers such that whenever $(z_1, z_2, \dots, z_{2n})M = O_n$, with complex $z_i$, not all zero, then at least one of the $z_i$ is not real. Prove that for arbitrary real numbers $r_1, r_2, \dots, r_{2n}$, there are complex numbers $w_1, w_2, \dots, w_n$ such that
\[
\mathrm{re}\left[ M \left( \begin{array}{c} w_1 \\ \vdots \\ w_n \end{array}
\right) \right] = \left( \begin{array}{c} r_1 \\ \vdots \\ r_n
\end{array} \right).
\]
(Note: if $C$ is a matrix of complex numbers, $\mathrm{re}(C)$ is the matrix whose entries are the real parts of the entries of $C$.)
2016 Putnam, A4
Consider a $(2m-1)\times(2n-1)$ rectangular region, where $m$ and $n$ are integers such that $m,n\ge 4.$ The region is to be tiled using tiles of the two types shown:
\[
\begin{picture}(140,40)
\put(0,0){\line(0,1){40}}
\put(0,0){\line(1,0){20}}
\put(0,40){\line(1,0){40}}
\put(20,0){\line(0,1){20}}
\put(20,20){\line(1,0){20}}
\put(40,20){\line(0,1){20}}
\multiput(0,20)(5,0){4}{\line(1,0){3}}
\multiput(20,20)(0,5){4}{\line(0,1){3}}
\put(80,0){\line(1,0){40}}
\put(120,0){\line(0,1){20}}
\put(120,20){\line(1,0){20}}
\put(140,20){\line(0,1){20}}
\put(80,0){\line(0,1){20}}
\put(80,20){\line(1,0){20}}
\put(100,20){\line(0,1){20}}
\put(100,40){\line(1,0){40}}
\multiput(100,0)(0,5){4}{\line(0,1){3}}
\multiput(100,20)(5,0){4}{\line(1,0){3}}
\multiput(120,20)(0,5){4}{\line(0,1){3}}
\end{picture}
\]
(The dotted lines divide the tiles into $1\times 1$ squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping.
What is the minimum number of tiles required to tile the region?
1940 Putnam, B6
Prove that the determinant of the matrix
$$\begin{pmatrix}
a_{1}^{2}+k & a_1 a_2 & a_1 a_3 &\ldots & a_1 a_n\\
a_2 a_1 & a_{2}^{2}+k & a_2 a_3 &\ldots & a_2 a_n\\
\ldots & \ldots & \ldots & \ldots & \ldots \\
a_n a_1& a_n a_2 & a_n a_3 & \ldots & a_{n}^{2}+k
\end{pmatrix}$$
is divisible by $k^{n-1}$ and find its other factor.
2020 Putnam, A1
How many positive integers $N$ satisfy all of the following three conditions?\\
(i) $N$ is divisible by $2020$.\\
(ii) $N$ has at most $2020$ decimal digits.\\
(iii) The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros.
1940 Putnam, B1
A projectile, thrown with initial velocity $v_0$ in a direction making angle $\alpha$ with the horizontal, is acted on by no force except gravity. Find the lenght of its path until it strikes a horizontal plane through the starting point. Show that the flight is longest when
$$\sin \alpha \log(\sec \alpha+ \tan \alpha)=1.$$
1961 Putnam, B1
Let $a_1 , a_2 , a_3 ,\ldots$ be a sequence of positive real numbers, define $s_n = \frac{a_1 +a_2 +\ldots+a_n }{n}$ and $r_n = \frac{a_{1}^{-1} +a_{2}^{-1} +\ldots+a_{n}^{-1} }{n}.$ Given that $\lim_{n\to \infty} s_n $ and $\lim_{n\to \infty} r_n $ exist, prove that the product of these limits is not less than $1.$
2024 Putnam, B4
Let $n$ be a positive integer. Set $a_{n,0}=1$. For $k\geq 0$, choose an integer $m_{n,k}$ uniformly at random from the set $\{1,\,\ldots,\,n\}$, and let
\[
a_{n,k+1}=
\begin{cases}
a_{n,k}+1, & \text{if $m_{n,k}>a_{n,k}$;}\\
a_{n,k}, & \text{if $m_{n,k}=a_{n,k}$;}\\
a_{n,k}-1, & \text{if $m_{n,k}<a_{n,k}$.}
\end{cases}
\]
Let $E(n)$ be the expected value of $a_{n,n}$. Determine $\lim_{n\to\infty}E(n)/n$.
2009 Putnam, A6
Let $ f: [0,1]^2\to\mathbb{R}$ be a continuous function on the closed unit square such that $ \frac{\partial f}{\partial x}$ and $ \frac{\partial f}{\partial y}$ exist and are continuous on the interior of $ (0,1)^2.$ Let $ a\equal{}\int_0^1f(0,y)\,dy,\ b\equal{}\int_0^1f(1,y)\,dy,\ c\equal{}\int_0^1f(x,0)\,dx$ and $ d\equal{}\int_0^1f(x,1)\,dx.$ Prove or disprove: There must be a point $ (x_0,y_0)$ in $ (0,1)^2$ such that
$ \frac{\partial f}{\partial x}(x_0,y_0)\equal{}b\minus{}a$ and $ \frac{\partial f}{\partial y}(x_0,y_0)\equal{}d\minus{}c.$
2006 Putnam, A5
Let $n$ be a positive odd integer and let $\theta$ be a real number such that $\theta/\pi$ is irrational. Set $a_{k}=\tan(\theta+k\pi/n),\ k=1,2\dots,n.$ Prove that
\[\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1}a_{2}\cdots a_{n}}\]
is an integer, and determine its value.
1972 Putnam, B5
Let $A,B,C$ and $D$ be non-coplanar points such that $\angle ABC=\angle ADC$ and $\angle BAD=\angle BCD$.
Show that $AB=CD$ and $AD=BC$.
Russian TST 2017, P2
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\le y\le x^3,$ then $(f(x))^2\le f(y) \le (f(x))^3.$
1956 Putnam, B1
Show that if the differential equation
$$M(x,y)\, dx +N(x,y) \, dy =0$$
is both homogeneous and exact, then the solution $y=y(x)$ satisfies that $xM(x,y)+yN(x,y)$ is constant.
1956 Putnam, A6
i) A transformation of the plane into itself preserves all rational distances. Prove that it preserves all distances.
ii) Show that the corresponding statement for the line is false.