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: 15

2014 Putnam, 6
Tags: Putnam , function , slope , integration , college contests , real analysis , Putnam 2014
Let $f:[0,1]\to\mathbb{R}$ be a function for which there exists a constant $K>0$ such that $|f(x)-f(y)|\le K|x-y|$ for all $x,y\in [0,1].$ Suppose also that for each rational number $r\in [0,1],$ there exist integers $a$ and $b$ such that $f(r)=a+br.$ Prove that there exist finitely many intervals $I_1,\dots,I_n$ such that $f$ is a linear function on each $I_i$ and $[0,1]=\bigcup_{i=1}^nI_i.$

2014 Putnam, 4
Tags: Putnam , algebra , polynomial , inequalities , calculus , Putnam 2014
Show that for each positive integer $n,$ all the roots of the polynomial \[\sum_{k=0}^n 2^{k(n-k)}x^k\] are real numbers.

2014 Putnam, 2
Tags: Putnam , function , integration , real analysis , absolute value , Putnam calculus , Putnam 2014
Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be?

2014 Contests, 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.

2014 Putnam, 4
Tags: Putnam , probability , inequalities , algebra , polynomial , expected value , Putnam 2014
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$

2014 Putnam, 3
Tags: Putnam , pigeonhole principle , vector , Putnam matrices , Putnam 2014
Let $A$ be an $m\times n$ matrix with rational entries. Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A.$ Show that the rank of $A$ is at least $2.$

2014 Putnam, 5
Tags: Putnam , abstract algebra , group theory , college contests , Putnam 2014 , Putnam matrices
In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible $n\times n$ matrices with entries in the field $\mathbb{Z}/p\mathbb{Z}$ of integers modulo $p,$ where $n$ is a fixed positive integer and $p$ is a fixed prime number. The rules of the game are: (1) A player cannot choose an element that has been chosen by either player on any previous turn. (2) A player can only choose an element that commutes with all previously chosen elements. (3) A player who cannot choose an element on his/her turn loses the game. Patniss takes the first turn. Which player has a winning strategy?

2014 Putnam, 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 , modular arithmetic , induction , college contests , Putnam 2014
A [i]base[/i] 10 [i]over-expansion[/i] of a positive integer $N$ is an expression of the form $N=d_k10^k+d_{k-1}10^{k-1}+\cdots+d_0 10^0$ with $d_k\ne 0$ and $d_i\in\{0,1,2,\dots,10\}$ for all $i.$ For instance, the integer $N=10$ has two base 10 over-expansions: $10=10\cdot 10^0$ and the usual base 10 expansion $10=1\cdot 10^1+0\cdot 10^0.$ Which positive integers have a unique base 10 over-expansion?

2014 Contests, 3
Tags: Putnam , limit , induction , inequalities , algebra , difference of squares , Putnam 2014
Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.

2014 Putnam, 3
Tags: Putnam , limit , induction , inequalities , algebra , difference of squares , Putnam 2014
Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.

2014 Putnam, 6
Tags: Putnam , algebra , polynomial , vector , function , Putnam matrices , Putnam 2014
Let $n$ be a positive integer. What is the largest $k$ for which there exist $n\times n$ matrices $M_1,\dots,M_k$ and $N_1,\dots,N_k$ with real entries such that for all $i$ and $j,$ the matrix product $M_iN_j$ has a zero entry somewhere on its diagonal if and only if $i\ne j?$

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.

2014 Putnam, 5
Tags: Putnam , algebra , polynomial , MIT , inequalities , Putnam 2014
Let $P_n(x)=1+2x+3x^2+\cdots+nx^{n-1}.$ Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j\ne k.$