Olympiad problems

1998 Putnam, 2

Tags: Putnam , geometry , calculus , trigonometry , integration , college contests , Putnam calculus

Let $s$ be any arc of the unit circle lying entirely in the first quadrant. Let $A$ be the area of the region lying below $s$ and above the $x$-axis and let $B$ be the area of the region lying to the right of the $y$-axis and to the left of $s$. Prove that $A+B$ depends only on the arc length, and not on the position, of $s$.

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?

2002 Putnam, 6

Tags: Putnam , geometry , integration , logarithms , function , college contests , Putnam calculus

Fix an integer $ b \geq 2$. Let $ f(1) \equal{} 1$, $ f(2) \equal{} 2$, and for each $ n \geq 3$, define $ f(n) \equal{} n f(d)$, where $ d$ is the number of base-$ b$ digits of $ n$. For which values of $ b$ does \[ \sum_{n\equal{}1}^\infty \frac{1}{f(n)} \] converge?

2017 Putnam, A3

Tags: Putnam , Putnam 2017 , Putnam calculus , Putnam analysis

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 $\displaystyle\lim_{n\to\infty}I_n=\infty.$

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.

2017 Putnam, B4

Tags: Putnam , Putnam 2017 , Putnam calculus

Evaluate the sum \[\sum_{k=0}^{\infty}\left(3\cdot\frac{\ln(4k+2)}{4k+2}-\frac{\ln(4k+3)}{4k+3}-\frac{\ln(4k+4)}{4k+4}-\frac{\ln(4k+5)}{4k+5}\right)\] \[=3\cdot\frac{\ln 2}2-\frac{\ln 3}3-\frac{\ln 4}4-\frac{\ln 5}5+3\cdot\frac{\ln 6}6-\frac{\ln 7}7-\frac{\ln 8}8-\frac{\ln 9}9+3\cdot\frac{\ln 10}{10}-\cdots.\] (As usual, $\ln x$ denotes the natural logarithm of $x.$)

2016 Putnam, A6

Tags: Putnam , Putnam 2016 , Putnam calculus , Putnam analysis

Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree $3$ that has a root in the interval $[0,1],$ \[\int_0^1|P(x)|\,dx\le C\max_{x\in[0,1]}|P(x)|.\]

2015 Putnam, A1

Tags: Putnam , Putnam 2015 , Putnam calculus

Let $A$ and $B$ be points on the same branch of the hyperbola $xy=1.$ Suppose that $P$ is a point lying between $A$ and $B$ on this hyperbola, such that the area of the triangle $APB$ is as large as possible. Show that the region bounded by the hyperbola and the chord $AP$ has the same area as the region bounded by the hyperbola and the chord $PB.$

2016 Putnam, A3

Tags: Putnam , Putnam 2016 , Putnam calculus , mapping

Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that \[f(x)+f\left(1-\frac1x\right)=\arctan x\] for all real $x\ne 0.$ (As usual, $y=\arctan x$ means $-\pi/2<y<\pi/2$ and $\tan y=x.$) Find \[\int_0^1f(x)\,dx.\]

2016 Putnam, B6

Tags: Putnam , Putnam 2016 , Putnam calculus

Evaluate \[\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=0}^{\infty}\frac{1}{k2^n+1}.\]

1998 Putnam, 3

Tags: Putnam , function , integration , college contests , Putnam calculus

Let $f$ be a real function on the real line with continuous third derivative. Prove that there exists a point $a$ such that \[f(a)\cdot f^\prime(a)\cdot f^{\prime\prime}(a)\cdot f^{\prime\prime\prime}(a)\geq 0.\]

1999 Putnam, 3

Tags: Putnam , limit , algebra , polynomial , modular arithmetic , college contests , Putnam calculus

Let $A=\{(x,y): 0\le x,y < 1\}.$ For $(x,y)\in A,$ let \[S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).\]