Found problems: 966
2024 Putnam, A1
Determine all positive integers $n$ for which there exists positive integers $a$, $b$, and $c$ satisfying
\[
2a^n+3b^n=4c^n.
\]
2015 Putnam, A2
Let $a_0=1,a_1=2,$ and $a_n=4a_{n-1}-a_{n-2}$ for $n\ge 2.$
Find an odd prime factor of $a_{2015}.$
2020 Putnam, A2
Let $k$ be a nonnegative integer. Evaluate
\[ \sum_{j=0}^k 2^{k-j} \binom{k+j}{j}. \]
2012 Putnam, 5
Let $\mathbb{F}_p$ denote the field of integers modulo a prime $p,$ and let $n$ be a positive integer. Let $v$ be a fixed vector in $\mathbb{F}_p^n,$ let $M$ be an $n\times n$ matrix with entries in $\mathbb{F}_p,$ and define $G:\mathbb{F}_p^n\to \mathbb{F}_p^n$ by $G(x)=v+Mx.$ Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x)=G(x)$ and $G^{(k+1)}(x)=G(G^{(k)}(x)).$ Determine all pairs $p,n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0),$ $k=1,2,\dots,p^n$ are distinct.
1968 Putnam, A6
Find all polynomials whose coefficients are all $\pm1$ and whose roots are all real.
1953 Putnam, B6
Let $P$ and $Q$ be any points inside a circle $C$ with center $O$ such that $OP=OQ.$ Determine the location of a point $Z$ on $C$ such that $PZ+QZ$ is minimal.
1986 Putnam, A4
A [i]transversal[/i] of an $n\times n$ matrix $A$ consists of $n$ entries of $A$, no two in the same row or column. Let $f(n)$ be the number of $n \times n$ matrices $A$ satisfying the following two conditions:
(a) Each entry $\alpha_{i,j}$ of $A$ is in the set $\{-1,0,1\}$.
(b) The sum of the $n$ entries of a transversal is the same for all transversals of $A$.
An example of such a matrix $A$ is
\[
A = \left( \begin{array}{ccc} -1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 1 & 0
\end{array}
\right).
\]
Determine with proof a formula for $f(n)$ of the form
\[
f(n) = a_1 b_1^n + a_2 b_2^n + a_3 b_3^n + a_4,
\]
where the $a_i$'s and $b_i$'s are rational numbers.
1997 Putnam, 4
Let $a_{m,n}$ denote the coefficient of $x^n$ in the expansion $(1+x+x^2)^n$. Prove the inequality for all integers $k\ge 0$ :
\[ 0\le \sum_{\ell=0}^{\left\lfloor{\frac{2k}{3}}\right\rfloor} (-1)^{\ell} a_{k-\ell,\ell}\le 1 \]
1961 Putnam, A4
Let $\Omega(n)$ be the number of prime factors of $n$. Define $f(1)=1$ and $f(n)=(-1)^{\Omega(n)}.$ Furthermore, let
$$F(n)=\sum_{d|n} f(d).$$
Prove that $F(n)=0,1$ for all positive integers $n$. For which integers $n$ is $F(n)=1?$
1999 Putnam, 2
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P^{\prime\prime}(x)$, where $Q(x)$ is a quadratic polynomial and $P^{\prime\prime}(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.
2017 Putnam, A3
Let $a$ and $b$ be real numbers with $a<b,$ and let $f$ and $g$ be continuous functions from $[a,b]$ to $(0,\infty)$ such that $\int_a^b f(x)\,dx=\int_a^b g(x)\,dx$ but $f\ne g.$ For every positive integer $n,$ define
\[I_n=\int_a^b\frac{(f(x))^{n+1}}{(g(x))^n}\,dx.\]
Show that $I_1,I_2,I_3,\dots$ is an increasing sequence with $\displaystyle\lim_{n\to\infty}I_n=\infty.$
1940 Putnam, A3
Let $a$ be a real number. Find all real-valued functions $f$ such that
$$\int f(x)^{a} dx=\left( \int f(x) dx \right)^{a}$$
when constants of integration are suitably chosen.
2018 Putnam, A3
Determine the greatest possible value of $\sum_{i = 1}^{10} \cos(3x_i)$ for real numbers $x_1, x_2, \dots, x_{10}$ satisfying $\sum_{i = 1}^{10} \cos(x_i) = 0$.
1981 Putnam, A6
Suppose that each of the vertices of $ABC$ is a lattice point in the $xy$-plane and that there is exactly one lattice
point $P$ in the interior of the triangle. The line $AP$ is extended to meet $BC$ at $E$. Determine the largest possible value for the ratio of lengths of segments
$$\frac{|AP|}{|PE|}.$$
1996 Putnam, 1
Find the least number $A$ such that for any two squares of combined area $1$, a rectangle of area $A$ exists such that the two squares can be packed in the rectangle (without the interiors of the squares overlapping) . You may assume the sides of the squares will be parallel to the sides of the rectangle.
1978 Putnam, A2
Let $a,b, p_1 ,p_2, \ldots, p_n$ be real numbers with $a \ne b$. Define $f(x)= (p_1 -x) (p_2 -x) \cdots (p_n -x)$. Show that
$$ \text{det} \begin{pmatrix} p_1 & a& a & \cdots & a \\
b & p_2 & a & \cdots & a\\
b & b & p_3 & \cdots & a\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
b & b& b &\cdots &p_n
\end{pmatrix}= \frac{bf(a) -af(b)}{b-a}.$$
2023 Putnam, B5
Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$ such that $\pi(\pi(k)) \equiv m k(\bmod n)$ for all $k \in\{1,2, \ldots, n\}$.
2005 Putnam, A2
Let $S=\{(a,b)|a=1,2,\dots,n,b=1,2,3\}$. A [i]rook tour[/i] of $S$ is a polygonal path made up of line segments connecting points $p_1,p_2,\dots,p_{3n}$ is sequence such that
(i) $p_i\in S,$
(ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1\le i<3n,$
(iii) for each $p\in S$ there is a unique $i$ such that $p_i=p.$
How many rook tours are there that begin at $(1,1)$ and end at $(n,1)?$
(The official statement includes a picture depicting an example of a rook tour for $n=5.$ This example consists of line segments with vertices at which there is a change of direction at the following points, in order: $(1,1),(2,1),(2,2),(1,2), (1,3),(3,3),(3,1),(4,1), (4,3),(5,3),(5,1).$)
1955 Putnam, B2
Suppose that $f$ is a function with two continuous derivatives 2and $f(0) = 0.$ Prove that the function $g,$ defined by $g(0) = f '(0), g(x) = f(x) / x$ for $x \ne 0, $ has a continuous derivative.
1999 Putnam, 1
Find polynomials $f(x)$, $g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x)=\begin{cases}-1 & \text{if }x<-1\\3x+2 &\text{if }-1\leq x\leq 0\\-2x+2 & \text{if }x>0.\end{cases}\]
1981 Putnam, B1
Find
$$\lim_{n\to \infty} \frac{1}{n^5 } \sum_{h=1}^{n} \sum_{k=1}^{n} (5h^4 -18h^2 k^2 +5k^4).$$
1983 Putnam, B6
Let $ k$ be a positive integer, let $ m\equal{}2^k\plus{}1$, and let $ r\neq 1$ be a complex root of $ z^m\minus{}1\equal{}0$. Prove that there exist polynomials $ P(z)$ and $ Q(z)$ with integer coefficients such that $ (P(r))^2\plus{}(Q(r))^2\equal{}\minus{}1$.
1964 Putnam, A3
Let $P_1 , P_2 , \ldots$ be a sequence of distinct points which is dense in the interval $(0,1)$. The points $P_1 , \ldots , P_{n-1}$ decompose the interval into $n$ parts, and $P_n$ decomposes one of these into two parts. Let $a_n$ and $b_n$ be the length of these two intervals. Prove that
$$\sum_{n=1}^{\infty} a_n b_n (a_n +b_n) =1 \slash 3.$$
1941 Putnam, B3
Let $y_1$ and $y_2$ be two linearly independent solutions of the equation
$$y''+P(x)y'+Q(x)=0.$$
Find the differential equation satisfied by the product $z=y_1 y_2$.
2005 Iran MO (3rd Round), 4
a) Year 1872 Texas
3 gold miners found a peice of gold. They have a coin that with possibility of $\frac 12$ it will come each side, and they want to give the piece of gold to one of themselves depending on how the coin will come. Design a fair method (It means that each of the 3 miners will win the piece of gold with possibility of $\frac 13$) for the miners.
b) Year 2005, faculty of Mathematics, Sharif university of Technolgy
Suppose $0<\alpha<1$ and we want to find a way for people name $A$ and $B$ that the possibity of winning of $A$ is $\alpha$. Is it possible to find this way?
c) Year 2005 Ahvaz, Takhti Stadium
Two soccer teams have a contest. And we want to choose each player's side with the coin, But we don't know that our coin is fair or not. Find a way to find that coin is fair or not?
d) Year 2005,summer
In the National mathematical Oympiad in Iran. Each student has a coin and must find a way that the possibility of coin being TAIL is $\alpha$ or no. Find a way for the student.