Found problems: 966
1953 Putnam, A1
Prove that for every positive integer $n$
$$ \frac{2}{3} n \sqrt{n} < \sqrt{1} + \sqrt{2} +\ldots +\sqrt{n} < \frac{4n+3}{6} \sqrt{n}.$$
1961 Putnam, A7
Let $S$ be a nonempty closed set in the euclidean plane for which there is a closed disk $D$ containing $S$ such that $D$ is a subset of every closed disk that contains $S$. Prove that every point inside $D$ is the midpoint of a segment joining two points of $S.$
2003 Putnam, 5
Let $A$, $B$ and $C$ be equidistant points on the circumference of a circle of unit radius centered at $O$, and let $P$ be any point in the circle's interior. Let $a$, $b$, $c$ be the distances from $P$ to $A$, $B$, $C$ respectively. Show that there is a triangle with side lengths $a$, $b$, $c$, and that the area of this triangle depends only on the distance from $P$ to $O$.
1958 February Putnam, A4
If $a_1 ,a_2 ,\ldots, a_n$ are complex numbers such that
$$ |a_1| =|a_2 | =\cdots = |a_n| =r \ne 0,$$
and if $T_s$ denotes the sum of all products of these $n$ numbers taken $s$ at a time, prove that
$$ \left| \frac{T_s }{T_{n-s}}\right| =r^{2s-n}$$
whenever the denominator of the left-hand side is different from $0$.
1988 Putnam, A5
Prove that there exists a [i]unique[/i] function $f$ from the set $\mathrm{R}^+$ of positive real numbers to $\mathrm{R}^+$ such that\[
f(f(x)) = 6x-f(x)
\]and\[
f(x)>0
\]for all $x>0$.
2003 Putnam, 2
Let $n$ be a positive integer. Starting with the sequence $1,\frac{1}{2}, \frac{1}{3} , \cdots , \frac{1}{n}$, form a new sequence of $n -1$ entries $\frac{3}{4}, \frac{5}{12},\cdots ,\frac{2n -1}{2n(n -1)}$, by taking the averages of two consecutive entries in the first sequence. Repeat the averaging of neighbors on the second sequence to obtain a third sequence of $n -2$ entries and continue until the final sequence consists of a single number $x_n$. Show that $x_n < \frac{2}{n}$.
1990 Putnam, B4
Let $G$ be a finite group of order $n$ generated by $a$ and $b$. Prove or disprove: there is a sequence \[ g_1, g_2, g_3, \cdots, g_{2n} \] such that:
$(1)$ every element of $G$ occurs exactly twice, and
$(2)$ $g_{i+1}$ equals $g_{i}a$ or $g_ib$ for $ i = 1, 2, \cdots, 2n $. (interpret $g_{2n+1}$ as $g_1$.)
2023 Putnam, B1
Consider an $m$-by-$n$ grid of unit squares, indexed by $(i, j)$ with $1 \leq i \leq m$ and $1 \leq j \leq n$. There are $(m-1)(n-1)$ coins, which are initially placed in the squares $(i, j)$ with $1 \leq i \leq m-1$ and $1 \leq j \leq n-1$. If a coin occupies the square $(i, j)$ with $i \leq m-1$ and $j \leq n-1$ and the squares $(i+1, j),(i, j+1)$, and $(i+1, j+1)$ are unoccupied, then a legal move is to slide the coin from $(i, j)$ to $(i+1, j+1)$. How many distinct configurations of coins can be reached starting from the initial configuration by a (possibly empty) sequence of legal moves?
1988 Putnam, B3
For every $n$ in the set $\mathrm{N} = \{1,2,\dots \}$ of positive integers, let $r_n$ be the minimum value of $|c-d\sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for all $n \in \mathbb{N}$.
1967 Putnam, A6
Given real numbers $(a_i)$ and $(b_i)$ (for $i=1,2,3,4$) such that $a_1 b _2 \ne a_2 b_1 .$ Consider the set of all solutions $(x_1 ,x_2 ,x_3 , x_4)$ of the simultaneous equations
$$ a_1 x_1 +a _2 x_2 +a_3 x_3 +a_4 x_4 =0 \;\; \text{and}\;\; b_1 x_1 +b_2 x_2 +b_3 x_3 +b_4 x_4 =0 $$
for which no $x_i$ is zero. Each such solution generates a $4$-tuple of plus and minus signs (by considering the sign of $x_i$).
[list=a]
[*] Determine, with proof, the maximum number of distinct $4$-tuples possible.
[*] Investigate necessary and sufficient conditions on $(a_i)$ and $(b_i)$ such that the above maximum of distinct $4$-tuples is attained.
1998 Putnam, 3
Let $f$ be a real function on the real line with continuous third derivative. Prove that there exists a point $a$ such that \[f(a)\cdot f^\prime(a)\cdot f^{\prime\prime}(a)\cdot f^{\prime\prime\prime}(a)\geq 0.\]
1956 Putnam, B6
Given $T_1 =2, T_{n+1}= T_{n}^{2} -T_n +1$ for $n>0.$ Prove:
(i) If $m \ne n,$ $T_m$ and $T_n$ have no common factor greater than $1.$
(ii) $\sum_{i=1}^{\infty} \frac{1}{T_i }=1.$
1993 Putnam, B4
$K(x, y), f(x)$ and $g(x)$ are positive and continuous for $x, y \subseteq [0, 1]$. $\int_{0}^{1} f(y) K(x, y) dy = g(x)$ and $\int_{0}^{1} g(y) K(x, y) dy = f(x)$ for all $x \subseteq [0, 1]$. Show that $f = g$ on $[0, 1]$.
2008 Putnam, B6
Let $ n$ and $ k$ be positive integers. Say that a permutation $ \sigma$ of $ \{1,2,\dots n\}$ is $ k$-[i]limited[/i] if $ |\sigma(i)\minus{}i|\le k$ for all $ i.$ Prove that the number of $ k$-limited permutations of $ \{1,2,\dots n\}$ is odd if and only if $ n\equiv 0$ or $ 1\pmod{2k\plus{}1}.$
2013 Online Math Open Problems, 42
Find the remainder when \[\prod_{i=0}^{100}(1-i^2+i^4)\] is divided by $101$.
[i]Victor Wang[/i]
1962 Putnam, B1
Let $x^{(n)}=x(x-1)\cdots (x-n+1)$ for $n$ a positive integer and let $x^{(0)}=1.$ Prove that
$$(x+y)^{(n)}= \sum_{k=0}^{n} \binom{n}{k} x^{(k)} y^{(n-k)}.$$
1959 Putnam, A4
If $f$ and $g$ are real-valued functions of one real variable, show that there exist $x$ and $y$ in $[0,1]$ such that $$|xy-f(x)-g(y)|\geq \frac{1}{4}.$$
2002 Putnam, 4
In Determinant Tic-Tac-Toe, Player $1$ enters a $1$ in an empty $3 \times 3$ matrix. Player $0$ counters with a $0$ in a vacant position and play continues in turn intil the $ 3 \times 3 $ matrix is completed with five $1$’s and four $0$’s. Player $0$ wins if the determinant is $0$ and player $1$ wins otherwise. Assuming both players pursue optimal strategies, who will win and how?
2023 Putnam, B4
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0, t_1, \ldots, t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties:
(a) $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1, \ldots, t_n$;
(b) $f\left(t_0\right)=1 / 2$;
(c) $\lim _{t \rightarrow t_k^{+}} f^{\prime}(t)=0$ for $0 \leq k \leq n$;
(d) For $0 \leq k \leq n-1$, we have $f^{\prime \prime}(t)=k+1$ when $t_k<t<t_{k+1}$, and $f^{\prime \prime}(t)=n+1$ when $t>t_n$.
Considering all choices of $n$ and $t_0, t_1, \ldots, t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f\left(t_0+T\right)=2023$?
1986 Putnam, B1
Inscribe a rectangle of base $b$ and height $h$ in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of $h$ do the rectangle and triangle have the same area?
2012 Putnam, 4
Let $q$ and $r$ be integers with $q>0,$ and let $A$ and $B$ be intervals on the real line. Let $T$ be the set of all $b+mq$ where $b$ and $m$ are integers with $b$ in $B,$ and let $S$ be the set of all integers $a$ in $A$ such that $ra$ is in $T.$ Show that if the product of the lengths of $A$ and $B$ is less than $q,$ then $S$ is the intersection of $A$ with some arithmetic progression.
2002 Putnam, 6
Let $p$ be a prime number. Prove that the determinant of the matrix \[ \begin{bmatrix}x & y & z\\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{bmatrix} \] is congruent modulo $p$ to a product of polynomials of the form $ax+by+cz$, where $a$, $b$, and $c$ are integers. (We say two integer polynomials are congruent modulo $p$ if corresponding coefficients are congruent modulo $p$.)
1957 Putnam, B1
Consider the determinant of the matrix $(a_{ij})_{ij}$ with $1\leq i,j \leq 100$ and $a_{ij}=ij.$ Prove that if the absolute value of each of the $100!$ terms in the expansion of this determinant is divided by $101,$ then the remainder is always $1.$
2012 Putnam, 5
Prove that, for any two bounded functions $g_1,g_2 : \mathbb{R}\to[1,\infty),$ there exist functions $h_1,h_2 : \mathbb{R}\to\mathbb{R}$ such that for every $x\in\mathbb{R},$\[\sup_{s\in\mathbb{R}}\left(g_1(s)^xg_2(s)\right)=\max_{t\in\mathbb{R}}\left(xh_1(t)+h_2(t)\right).\]
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 $n$.