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.

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Found problems: 966

1942 Putnam, B2
Tags: Putnam , conics , parabola
For the family of parabolas $$y= \frac{ a^3 x^{2}}{3}+ \frac{ a^2 x}{2}-2a$$ (i) find the locus of vertices, (ii) find the envelope, (iii) sketch the envelope and two typical curves of the family.

1951 Putnam, A1
Tags: Putnam
Show that the determinant: \[ \begin{vmatrix} 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0 & f \\ -c & -e & -f & 0 \end{vmatrix} \] is non-negative, if its elements $a, b, c,$ etc., are real.

1950 Putnam, A1
Tags: Putnam
For what values of the ratio $a/b$ is the limaçon $r = a - b \cos \theta$ a convex curve? $(a > b > 0)$

1940 Putnam, B7
Tags: Putnam , inequalities
Which is greater $$\sqrt{n}^{\sqrt{n+1}} \;\; \; \text{or}\;\;\; \sqrt{n+1}^{\sqrt{n}}$$ where $n>8?$

1942 Putnam, A5
Tags: Putnam , ratio , Torus
A circle of radius $a$ is revolved through $180^{\circ}$ about a line in its plane, distant $b$ from the center of the circle, where $b>a$. For what value of the ratio $\frac{b}{a}$ does the center of gravity of the solid thus generated lie on the surface of the solid?

1975 Putnam, B1
Tags: Putnam , group theory , abstract algebra
Consider the additive group $\mathbb{Z}^{2}$. Let $H$ be the smallest subgroup containing $(3,8), (4,-1)$ and $(5,4)$. Let $H_{xy}$ be the smallest subgroup containing $(0,x)$ and $(1,y)$. Find some pair $(x,y)$ with $x>0$ such that $H=H_{xy}$.

2021 Putnam, A1
Tags: Putnam , Putnam 2021
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$?

1965 Putnam, A2
Tags: Putnam , binomial coefficients
Show that, for any positive integer $n$, \[ \sum_{r=0}^{[(n-1)/2]}\left\{\frac{n-2r}n\binom nr\right\}^2 = \frac 1n\binom{2n-2}{n-1}, \] where $[x]$ means the greatest integer not exceeding $x$, and $\textstyle\binom nr$ is the binomial coefficient "$n$ choose $r$", with the convention $\textstyle\binom n0 = 1$.

1956 Putnam, B4
Tags: Putnam , trigonometry , inequalities
Prove that if $A,B,$ and $C$ are angles of a triangle measured in radians then $A \cos B +\sin A \cos C >0.$

2009 Putnam, B2
Tags: Putnam , integration , function , calculus , algebra , polynomial , limit
A game involves jumping to the right on the real number line. If $ a$ and $ b$ are real numbers and $ b>a,$ the cost of jumping from $ a$ to $ b$ is $ b^3\minus{}ab^2.$ For what real numbers $ c$ can one travel from $ 0$ to $ 1$ in a finite number of jumps with total cost exactly $ c?$

2020 Putnam, B2
Tags: Putnam , Putnam 2020
Let $k$ and $n$ be integers with $1\leq k<n$. Alice and Bob play a game with $k$ pegs in a line of $n$ holes. At the beginning of the game, the pegs occupy the $k$ leftmost holes. A legal move consists of moving a single peg to any vacant hole that is further to the right. The players alternate moves, with Alice playing first. The game ends when the pegs are in the $k$ rightmost holes, so whoever is next to play cannot move and therefore loses. For what values of $n$ and $k$ does Alice have a winning strategy?

1990 Putnam, B2
Tags: induction , Putnam , college contests
Prove that for $ |x| < 1 $, $ |z| > 1 $, \[ 1 + \displaystyle\sum_{j=1}^{\infty} \left( 1 + x^j \right) P_j = 0, \]where $P_j$ is \[ \dfrac {(1-z)(1-zx)(1-zx^2) \cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3)\cdots(z-x^j)}. \]

1974 Putnam, A1
Tags: Putnam , Integers , relatively prime
Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$?

2019 Putnam, B2
Tags: Putnam , Putnam 2019 , limit
For all $n\ge 1$, let $a_n=\sum_{k=1}^{n-1}\frac{\sin(\frac{(2k-1)\pi}{2n})}{\cos^2(\frac{(k-1)\pi}{2n})\cos^2(\frac{k\pi}{2n})}$. Determine $\lim_{n\rightarrow \infty}\frac{a_n}{n^3}$.

1947 Putnam, A4
Tags: Putnam , physics
A coast artillery gun can fire at every angle of elevation between $0^{\circ}$ and $90^{\circ}$ in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant ($=v_0 $), determine the set $H$ of points in the plane and above the horizontal which can be hit.

1975 Putnam, A2
Tags: Putnam , analytic geometry , algebra , absolute value
Describe the region $R$ consisting of the points $(a,b)$ of the cartesian plane for which both (possibly complex) roots of the polynomial $z^2+az+b$ have absolute value smaller than $1$.

2018 Putnam, B1
Tags: Putnam , Putnam 2018
Let $\mathcal{P}$ be the set of vectors defined by \[\mathcal{P} = \left\{\begin{pmatrix} a \\ b \end{pmatrix} \, \middle\vert \, 0 \le a \le 2, 0 \le b \le 100, \, \text{and} \, a, b \in \mathbb{Z}\right\}.\] Find all $\mathbf{v} \in \mathcal{P}$ such that the set $\mathcal{P}\setminus\{\mathbf{v}\}$ obtained by omitting vector $\mathbf{v}$ from $\mathcal{P}$ can be partitioned into two sets of equal size and equal sum.

Putnam 1939, A4
Tags: Putnam
Given $4$ lines in Euclidean $3-$space: $L_1: x = 1, y = 0;$ $L_2: y = 1, z = 0;$ $L_3: x = 0, z = 1;$ $L_4: x = y, y = -6z.$ Find the equations of the two lines which both meet all of the $L_i.$

2002 Putnam, 4
Tags: Putnam , modular arithmetic , college contests
An integer $n$, unknown to you, has been randomly chosen in the interval $[1,2002]$ with uniform probability. Your objective is to select $n$ in an ODD number of guess. After each incorrect guess, you are informed whether $n$ is higher or lower, and you $\textbf{must}$ guess an integer on your next turn among the numbers that are still feasibly correct. Show that you have a strategy so that the chance of winning is greater than $\tfrac{2}{3}$.

1999 Putnam, 5
Tags: Putnam , polynomial , integration , calculus , inequalities , college contests , real analysis
Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $1999$, then \[|p(0)|\leq C\int_{-1}^1|p(x)|\,dx.\]

1998 Putnam, 3
Tags: Putnam , geometry , trigonometry , probability , rotation , 3D geometry , sphere
Let $H$ be the unit hemisphere $\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}$, $C$ the unit circle $\{(x,y,0):x^2+y^2=1\}$, and $P$ the regular pentagon inscribed in $C$. Determine the surface area of that portion of $H$ lying over the planar region inside $P$, and write your answer in the form $A \sin\alpha + B \cos\beta$, where $A,B,\alpha,\beta$ are real numbers.

2013 Putnam, 3
Tags: Putnam , function , induction , linear algebra , matrix , college contests , Putnam 2013
Let $P$ be a nonempty collection of subsets of $\{1,\dots,n\}$ such that: (i) if $S,S'\in P,$ then $S\cup S'\in P$ and $S\cap S'\in P,$ and (ii) if $S\in P$ and $S\ne\emptyset,$ then there is a subset $T\subset S$ such that $T\in P$ and $T$ contains exactly one fewer element than $S.$ Suppose that $f:P\to\mathbb{R}$ is a function such that $f(\emptyset)=0$ and \[f(S\cup S')= f(S)+f(S')-f(S\cap S')\text{ for all }S,S'\in P.\] Must there exist real numbers $f_1,\dots,f_n$ such that \[f(S)=\sum_{i\in S}f_i\] for every $S\in P?$

1950 Putnam, A3
Tags: Putnam
The sequence $x_0, x_1, x_2, \ldots$ is defined by the conditions \[ x_0 = a, x_1 = b, x_{n+1} = \frac{x_{n - 1} + (2n - 1) ~x_n}{2n}\] for $n \ge 1,$ where $a$ and $b$ are given numbers. Express $\lim_{n \to \infty} x_n$ concisely in terms of $a$ and $b.$

1956 Putnam, A7
Tags: Putnam , binomial coefficients
Prove that the number of odd binomial coefficients in any finite binomial expansion is a power of $2.$

1947 Putnam, A1
Tags: Putnam , Sequence , limit
If $(a_n)$ is a sequence of real numbers such that for $n \geq 1$ $$(2-a_n )a_{n+1} =1,$$ prove that $\lim_{n\to \infty} a_n =1.$