This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 966

1983 Putnam, B4
Tags: number theory , Putnam , function , Sequence
[b]Problem.[/b] Let $f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ be a function defined as $$f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+.$$ Prove that for any positive integer $m,$ the sequence $$m,f(m),f(f(m)),f(f(f(m))),\ldots$$ contains a perfect square.

2016 Putnam, A5
Tags: Putnam , Putnam 2016 , Putnam algebra
Suppose that $G$ is a finite group generated by the two elements $g$ and $h,$ where the order of $g$ is odd. Show that every element of $G$ can be written in the form \[g^{m_1}h^{n_1}g^{m_2}h^{n_2}\cdots g^{m_r}h^{n_r}\] with $1\le r\le |G|$ and $m_n,n_1,m_2,n_2,\dots,m_r,n_r\in\{1,-1\}.$ (Here $|G|$ is the number of elements of $G.$)

1986 Putnam, B4
Tags: Putnam
For a positive real number $r$, let $G(r)$ be the minimum value of $|r - \sqrt{m^2+2n^2}|$ for all integers $m$ and $n$. Prove or disprove the assertion that $\lim_{r\to \infty}G(r)$ exists and equals $0.$

1951 Putnam, B7
Tags: Putnam , Volume , sphere , four dimensions
Find the volume of the four-dimensional hypersphere $x^2 +y^2 +z^2 +t^2 =r^2$ and the hypervolume of its interior $x^2 +y^2 +z^2 +t^2 <r^2$

2014 Contests, 2
Tags: Putnam , linear algebra , matrix , induction , college contests , Putnam 2014 , Putnam matrices
Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$

2014 Putnam, 1
Tags: Putnam , function , relatively prime , prime factorization , Putnam calculus , Putnam 2014
Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.

2015 Putnam, B3
Tags: Putnam , Putnam 2015 , Putnam matrices
Let $S$ be the set of all $2\times 2$ real matrices \[M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\] whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S.$

Putnam 1938, A2
Tags: Putnam
A solid has a cylindrical middle with a conical cap at each end. The height of each cap equals the length of the middle. For a given surface area, what shape maximizes the volume?

2023 Putnam, A3
Tags: Putnam , Putnam 2023
Determine the smallest positive real number $r$ such that there exist differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ satisfying (a) $f(0)>0$, (b) $g(0)=0$, (c) $\left|f^{\prime}(x)\right| \leq|g(x)|$ for all $x$, (d) $\left|g^{\prime}(x)\right| \leq|f(x)|$ for all $x$, and (e) $f(r)=0$.

1964 Putnam, A6
Tags: Putnam , ratio , geometry
Let $S$ be a finite subset of a straight line. Say that $S$ has the [i]repeated distance property [/i] if every value of the distance between two points of $S$ (except the longest) occurs at least twice. Show that if $S$ has the [i]repeated distance property [/i] then the ratio of any two distances between two points of $S$ is rational.

1963 Putnam, A2
Tags: Putnam , function , number theory , relatively prime , functional equation
Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a strictly increasing function such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every pair of relatively prime positive integers $m$ and $n$. Prove that $f(n)=n$ for every positive integer $n$.

1987 Putnam, A1
Tags: Putnam
Curves $A,B,C$ and $D$ are defined in the plane as follows: \begin{align*} A &= \left\{ (x,y): x^2-y^2 = \frac{x}{x^2+y^2} \right\}, \\ B &= \left\{ (x,y): 2xy + \frac{y}{x^2+y^2} = 3 \right\}, \\ C &= \left\{ (x,y): x^3-3xy^2+3y=1 \right\}, \\ D &= \left\{ (x,y): 3x^2 y - 3x - y^3 = 0\right\}. \end{align*} Prove that $A \cap B = C \cap D$.

1949 Putnam, A5
Tags: Putnam , roots
How many roots of the equation $z^6 +6z +10=0$ lie in each quadrant of the complex plane?

1958 February Putnam, A1
Tags: Putnam , roots , polynomial
If $a_0 , a_1 ,\ldots, a_n$ are real number satisfying $$ \frac{a_0 }{1} + \frac{a_1 }{2} + \ldots + \frac{a_n }{n+1}=0,$$ show that the equation $a_n x^n + \ldots +a_1 x+a_0 =0$ has at least one real root.

2003 Putnam, 2
Tags: Putnam , inequalities , college contests , Putnam inequalities
Let $a_1, a_2, \cdots , a_n$ and $b_1, b_2,\cdots, b_n$ be nonnegative real numbers. Show that \[(a_1a_2 \cdots a_n)^{1/n}+ (b_1b_2 \cdots b_n)^{1/n} \le ((a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n}\]

1952 Putnam, B1
Tags: Putnam
A mathematical moron is given two sides and the included angle of a triangle and attempts to use the Law of Cosines: $a^2 = b^2 + c^2 - 2bc \cos A,$ to find the third side $a.$ He uses logarithms as follows. He finds $\log b$ and doubles it; adds to that the double of $\log c;$ subtracts the sum of the logarithms of $2, b, c,$ and $\cos A;$ divides the result by $2;$ and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?

1965 Putnam, B3
Tags: Putnam
Prove that there are exactly three right-angled triangles whose sides are integers while the area is numerically equal to twice the perimeter.

1980 Putnam, B6
Tags: Putnam
An infinite array of rational numbers $G(d, n)$ is defined for integers $d$ and $ n$ with $1\leq d \leq n$ as follows: $$G(1, n)= \frac{1}{n}, \;\;\; G(d,n)= \frac{d}{n} \sum_{i=d}^{n} G(d-1, i-1) \; \text{for} \; d>1.$$ For $1 < d < p$ and $p$ prime, prove that $G(d, p)$ is expressible as a quotient $s\slash t$ of integers $s$ and $t$ with $t$ not divisible by $p.$

1981 Putnam, A1
Tags: Putnam , Convergence
Let $E(n)$ denote the largest integer $k$ such that $5^k$ divides $1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.$ Calculate $$\lim_{n\to \infty} \frac{E(n)}{n^2 }.$$

1942 Putnam, A2
Tags: Putnam , algebra , polynomial
If a polynomial $f(x)$ is divided by $(x-a)^{2} (x-b)$, where $a\ne b$, derive a formula for the remainder.

1975 Putnam, A1
Tags: Putnam , number theory
Show that a positive integer $m$ is a sum of two triangular numbers if and only if $4m+1$ is a sum of two squares.

1994 Putnam, 3
Tags: Putnam , college contests
Show that if the points of an isosceles right triangle of side length $1$ are each colored with one of four colors, then there must be two points of the same color which are at least a distance $2-\sqrt 2$ apart.

2008 Putnam, B4
Tags: Putnam , algebra , polynomial , modular arithmetic , function , group theory , binomial theorem
Let $ p$ be a prime number. Let $ h(x)$ be a polynomial with integer coefficients such that $ h(0),h(1),\dots, h(p^2\minus{}1)$ are distinct modulo $ p^2.$ Show that $ h(0),h(1),\dots, h(p^3\minus{}1)$ are distinct modulo $ p^3.$

1954 Putnam, A3
Tags: Putnam , differential equation
Prove that if the family of integral curves of the differential equation $$ \frac{dy}{dx} +p(x) y= q(x),$$ where $p(x) q(x) \ne 0$, is cut by the line $x=k$ the tangents at the points of intersection are concurrent.

1974 Putnam, A4
Tags: Putnam , expected value , coin tosses
An unbiased coin is tossed $n$ times. What is the expected value of $|H-T|$, where $H$ is the number of heads and $T$ is the number of tails?