Found problems: 966
1972 Putnam, A2
Let $S$ be a set with a binary operation $\ast$ such that
1) $a \ast(a\ast b)=b$ for all $a,b\in S$.
2) $(a\ast b)\ast b=a$ for all $a,b\in S$.
Show that $\ast$ is commutative and give an example where $\ast$ is not associative.
1993 Putnam, B1
What is the smallest integer $n > 0$ such that for any integer m in the range $1, 2, 3, ... , 1992$ we can always find an integral multiple of $\frac{1}{n}$ in the open interval $(\frac{m}{1993}, \frac{m+1}{1994})$?
1992 Putnam, A1
Find all functions $ f : Z\rightarrow Z$ for which we have $ f (0) \equal{} 1$ and $ f ( f (n)) \equal{} f ( f (n\plus{}2)\plus{}2) \equal{} n$, for every natural number $ n$.
2018 Putnam, A1
Find all ordered pairs $(a, b)$ of positive integers for which
\[\frac{1}{a} + \frac{1}{b} = \frac{3}{2018}.\]
1985 Putnam, B2
Define polynomials $f_{n}(x)$ for $n \geq 0$ by $f_{0}(x)=1, f_{n}(0)=0$ for $n \geq 1,$ and
$$
\frac{d}{d x} f_{n+1}(x)=(n+1) f_{n}(x+1)
$$
for $n \geq 0 .$ Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes.
2007 Putnam, 1
Let $ f$ be a polynomial with positive integer coefficients. Prove that if $ n$ is a positive integer, then $ f(n)$ divides $ f(f(n)\plus{}1)$ if and only if $ n\equal{}1.$
1997 Putnam, 6
For a positive integer $n$ and any real number $c$, define $x_k$ recursively by :
\[ x_0=0,x_1=1 \text{ and for }k\ge 0, \;x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1} \]
Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$. Find $x_k$ in terms of $n$ and $k,\; 1\le k\le n$.
2003 Putnam, 3
Find the minimum value of \[|\sin{x} + \cos{x} + \tan{x} + \cot{x} + \sec{x} + \csc{x}|\] for real numbers $x$.
2016 Bundeswettbewerb Mathematik, 3
Find all functions $f$ that is defined on all reals but $\tfrac13$ and $- \tfrac13$ and satisfies \[ f \left(\frac{x+1}{1-3x} \right) + f(x) = x \] for all $x \in \mathbb{R} \setminus \{ \pm \tfrac13 \}$.
1995 Iran MO (2nd round), 1
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, $23=9+8+6.)$
1950 Putnam, A5
A function $D(n)$ of the positive integral variable $n$ is defined by the following properties: $D(1) = 0, D(p) = 1$ if $p$ is a prime, $D(uv) = u D(v) + v D(u)$ for any two positive integers $u$ and $v.$ Answer all three parts below.
(i) Show that these properties are compatible and determine uniquely $D(n).$ (Derive a formula for $D(n) /n,$ assuming that $n = p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_k^{\alpha_k}$ where $p_1, p_2, \ldots, p_k$ are different primes.)
(ii) For what values of $n$ is $D(n) = n?$
(iii) Define $D^2 (n) = D[D(n)],$ etc., and find the limit of $D^m (63)$ as $m$ tends to $\infty.$
1955 Putnam, B3
Prove that there exists no distance-preserving map of a spherical cap into the plane. (Distances on the sphere are to be measured along great circles on the surface.)
2004 District Olympiad, 1
Let $(x_n)_{n\ge 0}$ a sequence of real numbers defined by $x_0>0$ and $x_{n+1}=x_n+\frac{1}{\sqrt{x_n}}$. Compute $\lim_{n\to \infty}x_n$ and $\lim_{n\to \infty} \frac{x_n^3}{n^2}$.
1990 Putnam, B6
Let $S$ be a nonempty closed bounded convex set in the plane. Let $K$ be a line and $t$ a positive number. Let $L_1$ and $L_2$ be support lines for $S$ parallel to $K_1$, and let $ \overline {L} $ be the line parallel to $K$ and midway between $L_1$ and $L_2$. Let $B_S(K,t)$ be the band of points whose distance from $\overline{L}$ is at most $ \left( \frac {t}{2} \right) w $, where $w$ is the distance between $L_1$ and $L_2$. What is the smallest $t$ such that \[ S \cap \bigcap_K B_S (K, t) \ne \emptyset \]for all $S$? ($K$ runs over all lines in the plane.)
1970 Putnam, A6
Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i ]$ ($i=1,2,3$). If the distribution of each random number is uniform with respect to the length of the interval it is chosen from, determine the expected value of the smallest number chosen.
1974 Putnam, B6
For a set with $n$ elements, how many subsets are there whose cardinality is respectively $\equiv 0$ (mod $3$), $\equiv 1$ (mod $3$), $ \equiv 2$ (mod $3$)? In other words, calculate
$$s_{i,n}= \sum_{k\equiv i \;(\text{mod} \;3)} \binom{n}{k}$$
for $i=0,1,2$. Your result should be strong enough to permit direct evaluation of the numbers $s_{i,n}$ and to show clearly the relationship of $s_{0,n}, s_{1,n}$ and $s_{2,n}$ to each other for all positive integers $n$. In particular, show the relationships among these three sums for $n = 1000$.
1940 Putnam, A7
If $\sum_{i=1}^{\infty} u_{i}^{2}$ and $\sum_{i=1}^{\infty} v_{i}^{2}$ are convergent series of real numbers, prove that
$$\sum_{i=1}^{\infty}(u_{i}-v_{i})^{p}$$
is convergent, where $p\geq 2$ is an integer.
1987 Putnam, A4
Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that
\[
P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \mbox{for all real $x,y,z,u$},
\]
and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$.
2002 Putnam, 3
Let $N$ be an integer greater than $1$ and let $T_n$ be the number of non empty subsets $S$ of $\{1,2,.....,n\}$ with the property that the average of the elements of $S$ is an integer.Prove
that $T_n - n$ is always even.
2003 Putnam, 4
Let $f(z) = az^4+ bz^3+ cz^2+ dz + e = a(z -r_1)(z -r_2)(z -r_3)(z -r_4)$ where $a, b, c, d, e$ are integers, $a \not= 0$. Show that if $r_1 + r_2$ is a rational number, and if $r_1 + r_2 \neq r_3 + r_4$, then $r_1r_2$ is a rational number.
2012 Putnam, 2
Let $*$ be a commutative and associative binary operation on a set $S.$ Assume that for every $x$ and $y$ in $S,$ there exists $z$ in $S$ such that $x*z=y.$ (This $z$ may depend on $x$ and $y.$) Show that if $a,b,c$ are in $S$ and $a*c=b*c,$ then $a=b.$
1967 Putnam, B5
Show that the sum of the first $n$ terms in the binomial expansion of $(2-1)^{-n}$ is $\frac{1}{2},$ where $n$ is a positive integer.
1941 Putnam, B6
Assuming that $f(x)$ is continuous in the interval $(0,1)$, prove that
$$\int_{x=0}^{x=1} \int_{y=x}^{y=1} \int_{z=x}^{z=y} f(x)f(y)f(z)\;dz dy dx= \frac{1}{6}\left(\int_{0}^{1} f(t)\; dt\right)^{3}.$$
2022 Putnam, B6
Find all continuous functions $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ such that $$f(xf(y))+f(yf(x))=1+f(x+y)$$ for all $x, y>0.$
1974 Putnam, B1
Which configurations of five (not necessarily distinct) points $p_1 ,\ldots, p_5$ on the circle $x^2 +y^2 =1$ maximize the sum of the ten distances
$$\sum_{i<j} d(p_i, p_j)?$$