Olympiad problems

2005 Today's Calculation Of Integral, 54

evaluate \[\int_{-1}^0 \sqrt{\frac{1+x}{1-x}}dx\]

2008 Harvard-MIT Mathematics Tournament, 14

Tags: algebra , partial fractions

Evaluate the infinite sum $ \sum_{n\equal{}1}^{\infty}\frac{n}{n^4\plus{}4}$.

2010 USAMO, 5

Tags: partial fractions , modular arithmetic

Let $q = \frac{3p-5}{2}$ where $p$ is an odd prime, and let\[ S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7} + \cdots + \frac{1}{q(q+1)(q+2)} \]Prove that if $\frac{1}{p}-2S_q = \frac{m}{n}$ for integers $m$ and $n$, then $m - n$ is divisible by $p$.

2005 Today's Calculation Of Integral, 30

Tags: calculus , integration , partial fractions , calculus computations , algebra

A sequence $\{a_n\}$ is defined by $a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)$ Find the constant number $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$

2015 Harvard-MIT Mathematics Tournament, 2

Tags: algebra , partial fractions

The fraction $\tfrac1{2015}$ has a unique "(restricted) partial fraction decomposition'' of the form \[\dfrac1{2015}=\dfrac a5+\dfrac b{13}+\dfrac c{31},\] where $a$, $b$, and $c$ are integers with $0\leq a<5$ and $0\leq b<13$. Find $a+b$.

2009 Today's Calculation Of Integral, 509

Tags: algebra , calculus , integration , partial fractions , calculus computations , logarithm , trigonometry

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\tan x}{1\plus{}\sin x}\ dx$.

2007 Today's Calculation Of Integral, 195

Tags: calculus computations , integration , algebra , real analysis , partial fractions , calculus , function

Find continuous functions $x(t),\ y(t)$ such that $\ \ \ \ \ \ \ \ \ x(t)=1+\int_{0}^{t}e^{-2(t-s)}x(s)ds$ $\ \ \ \ \ \ \ \ \ y(t)=\int_{0}^{t}e^{-2(t-s)}\{2x(s)+3y(s)\}ds$

2007 Today's Calculation Of Integral, 170

Tags: algebra , calculus computations , calculus , integration , partial fractions , logarithm

Let $a,\ b$ be constant numbers such that $a^{2}\geq b.$ Find the following definite integrals. (1) $I=\int \frac{dx}{x^{2}+2ax+b}$ (2) $J=\int \frac{dx}{(x^{2}+2ax+b)^{2}}$

2012 Online Math Open Problems, 32

Tags: algebra , partial fractions

The sequence $\{a_n\}$ satisfies $a_0=1, a_1=2011,$ and $a_n=2a_{n-1}+a_{n-2}$ for all $n \geq 2$. Let \[ S = \sum_{i=1}^{\infty} \frac{a_{i-1}}{a_i^2-a_{i-1}^2} \] What is $\frac{1}{S}$? [i]Author: Ray Li[/i]

2019 AMC 10, 24

Tags: algebra , partial fractions

Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\] for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$? $\textbf{(A) }243\qquad\textbf{(B) }244\qquad\textbf{(C) }245\qquad\textbf{(D) }246\qquad\textbf{(E) } 247$