Found problems: 966
2005 Putnam, A1
Show that every positive integer is a sum of one or more numbers of the form $2^r3^s,$ where $r$ and $s$ are nonnegative integers and no summand divides another.
(For example, $23=9+8+6.)$
1955 Putnam, B6
Prove: If $f(x) > 0$ for all $x$ and $f(x) \rightarrow 0$ as $x \rightarrow \infty,$ then there exists at most a finite number of solutions of \[ f(m) + f(n) + f(p) = 1 \] in positive integers $m, n,$ and $p.$
Putnam 1938, B2
Find all solutions of the differential equation $zz" - 2z'z' = 0$ which pass through the point $x=1, z=1.$
2007 Putnam, 6
For each positive integer $ n,$ let $ f(n)$ be the number of ways to make $ n!$ cents using an unordered collection of coins, each worth $ k!$ cents for some $ k,\ 1\le k\le n.$ Prove that for some constant $ C,$ independent of $ n,$
\[ n^{n^2/2\minus{}Cn}e^{\minus{}n^2/4}\le f(n)\le n^{n^2/2\plus{}Cn}e^{\minus{}n^2/4}.\]
1978 Putnam, A5
Let $0 < x_i < \pi$ for $i=1,2,\ldots, n$ and set
$$x= \frac{ x_1 +x_2 + \ldots+ x_n }{n}.$$
Prove that
$$ \prod_{i=1}^{n} \frac{ \sin x_i }{x_i } \leq \left( \frac{ \sin x}{x}\right)^{n}.$$
1953 Putnam, A5
S is a parabola with focus F and axis L. Three distinct normals to S pass through P. Show that the sum of the angles which these make with L less the angle which PF makes with L is a multiple of π.
1987 Putnam, A6
For each positive integer $n$, let $a(n)$ be the number of zeroes in the base 3 representation of $n$. For which positive real numbers $x$ does the series\[
\sum_{n=1}^\infty \frac{x^{a(n)}}{n^3}
\]converge?
1981 Putnam, B6
Let $C$ be a fixed unit circle in the cartesian plane. For any convex polygon $P$ , each of whose sides is tangent to $C$, let $N( P, h, k)$ be the number of points common to $P$ and the unit circle with center at $(h, k).$ Let $H(P)$ be the region of all points $(x, y)$ for which $N(P, x, y) \geq 1$ and $F(P)$ be the area of $H(P).$ Find the smallest number $u$ with
$$ \frac{1}{F(P)} \int \int N(P,x,y)\;dx \;dy <u$$
for all polygons $P$, where the double integral is taken over $H(P).$
1989 IberoAmerican, 2
Let the function $f$ be defined on the set $\mathbb{N}$ such that
$\text{(i)}\ \ \quad f(1)=1$
$\text{(ii)}\ \quad f(2n+1)=f(2n)+1$
$\text{(iii)}\quad f(2n)=3f(n)$
Determine the set of values taken $f$.
1956 Putnam, A1
Evaluate
$$ \lim_{x\to \infty} \left( \frac{a^x -1}{x(a-1)} \right)^{1\slash x},$$
where $a>0$ and $a\ne 1.$
1989 Putnam, B1
A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equall likely to be hit, find the probability that hte point hit is nearer to the center than any edge.
2014 Putnam, 3
Let $A$ be an $m\times n$ matrix with rational entries. Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A.$ Show that the rank of $A$ is at least $2.$
1987 Putnam, B2
Let $r,s$ and $t$ be integers with $0 \leq r$, $0 \leq s$ and $r+s \leq t$. Prove that
\[
\frac{\binom s0}{\binom tr}
+ \frac{\binom s1}{\binom{t}{r+1}} + \cdots
+ \frac{\binom ss}{\binom{t}{r+s}}
= \frac{t+1}{(t+1-s)\binom{t-s}{r}}.
\]
2007 India IMO Training Camp, 3
Given a finite string $S$ of symbols $X$ and $O$, we denote $\Delta(s)$ as the number of$X'$s in $S$ minus the number of $O'$s (For example, $\Delta(XOOXOOX)=-1$). We call a string $S$ [b]balanced[/b] if every sub-string $T$ of (consecutive symbols) $S$ has the property $-1\leq \Delta(T)\leq 2.$ (Thus $XOOXOOX$ is not balanced, since it contains the sub-string $OOXOO$ whose $\Delta$ value is $-3.$ Find, with proof, the number of balanced strings of length $n$.
1997 Putnam, 6
The dissection of the $3-4-5$ triangle shown below has diameter $5/2$.
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(23cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = 1.42, xmax = 24.42, ymin = 3.8, ymax = 15.54; /* image dimensions */
Label laxis; laxis.p = fontsize(10);
xaxis(xmin, xmax,defaultpen+black, Ticks(laxis, Step = 1, Size = 2, NoZero), Arrows(6), above = true);
yaxis(ymin, ymax,defaultpen+black, Ticks(laxis, Step = 1, Size = 2, NoZero), Arrows(6), above = true); /* draws axes; NoZero hides '0' label */
/* draw figures */
draw((9.44,8.52)--(12.44,8.52));
draw((9.44,12.52)--(9.44,8.52));
draw((9.44,12.52)--(12.44,8.52));
draw((9.44,10.52)--(10.94,10.52));
draw((10.94,10.52)--(10.94,8.52));
draw((9.44,8.52)--(10.94,10.52));
/* dots and labels */
dot((9.44,8.52),dotstyle);
label("$A$", (9.52,8.64), NE * labelscalefactor);
dot((12.44,8.52),dotstyle);
label("$B$", (12.52,8.64), NE * labelscalefactor);
dot((9.44,12.52),dotstyle);
label("$C$", (9.52,12.64), NE * labelscalefactor);
dot((10.94,8.52),dotstyle);
label("$D$", (11.02,8.64), NE * labelscalefactor);
dot((9.44,10.52),dotstyle);
label("$E$", (9.52,10.64), NE * labelscalefactor);
dot((10.94,10.52),dotstyle);
label("$F$", (11.02,10.64), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]
Find the least diameter of this triangle into four parts. (The diameter of a dissection is the least upper bound of the distances between pairs of points belonging to the same part.)
2001 Putnam, 2
For each $k$, $\mathcal{C}_k$ is biased so that, when tossed, it has probability $\tfrac{1}{(2k+1)}$ of falling heads. If the $n$ coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function $n$.
1994 Putnam, 3
Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f^{\prime}(x) > f(x)$ for all $x,$ there is some number $N$ such that $f(x) > e^{kx}$ for all $x > N.$
1988 Putnam, B2
Prove or disprove: If $x$ and $y$ are real numbers with $y\geq0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1)\leq x^2$.
2011 Putnam, A4
For which positive integers $n$ is there an $n\times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?
2009 Putnam, B4
Say that a polynomial with real coefficients in two variable, $ x,y,$ is [i]balanced[/i] if the average value of the polynomial on each circle centered at the origin is $ 0.$ The balanced polynomials of degree at most $ 2009$ form a vector space $ V$ over $ \mathbb{R}.$ Find the dimension of $ V.$
2009 Putnam, B3
Call a subset $ S$ of $ \{1,2,\dots,n\}$ [i]mediocre[/i] if it has the following property: Whenever $ a$ and $ b$ are elements of $ S$ whose average is an integer, that average is also an element of $ S.$ Let $ A(n)$ be the number of mediocre subsets of $ \{1,2,\dots,n\}.$ [For instance, every subset of $ \{1,2,3\}$ except $ \{1,3\}$ is mediocre, so $ A(3)\equal{}7.$] Find all positive integers $ n$ such that $ A(n\plus{}2)\minus{}2A(n\plus{}1)\plus{}A(n)\equal{}1.$
2007 Putnam, 5
Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$
\[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\]
($ \lfloor a\rfloor$ means the largest integer $ \le a.$)
1941 Putnam, A6
If the $x$-coordinate $\overline{x}$ of the center of mass of the area lying between the $x$-axis and the curve $y=f(x)$
with $f(x)>0$, and between the lines $x=0$ and $x=a$ is given by
$$\overline{x}=g(a),$$
show that
$$f(x)=A\cdot \frac{g'(x)}{(x-g(x))^{2}} \cdot e^{\int \frac{1}{t-g(t)} dt},$$
where $A$ is a positive constant.
2022 Putnam, A3
Let $p$ be a prime number greater than 5. Let $f(p)$ denote the number of infinite sequences $a_1, a_2, a_3,\ldots$ such that $a_n \in \{1, 2,\ldots, p-1\}$ and $a_na_{n+2}\equiv1+a_{n+1}$ (mod $p$) for all $n\geq 1.$ Prove that $f(p)$ is congruent to 0 or 2 (mod 5).
2013 Putnam, 6
Define a function $w:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ as follows. For $|a|,|b|\le 2,$ let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0.$
\[\begin{array}{|lr|rrrrr|}\hline &&&&b&&\\
&w(a,b)&-2&-1&0&1&2\\ \hline
&-2&-1&-2&2&-2&-1\\
&-1&-2&4&-4&4&-2\\
a&0&2&-4&12&-4&2\\
&1&-2&4&-4&4&-2\\
&2&-1&-2&2&-2&-1\\ \hline\end{array}\]
For every finite subset $S$ of $\mathbb{Z}\times\mathbb{Z},$ define \[A(S)=\sum_{(\mathbf{s},\mathbf{s'})\in S\times S} w(\mathbf{s}-\mathbf{s'}).\] Prove that if $S$ is any finite nonempty subset of $\mathbb{Z}\times\mathbb{Z},$ then $A(S)>0.$ (For example, if $S=\{(0,1),(0,2),(2,0),(3,1)\},$ then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4.$)