Olympiad problems

2010 Today's Calculation Of Integral, 600

Evaluate $\int_{-a}^a \left(x+\frac{1}{\sin x+\frac{1}{e^x-e^{-x}}}\right)dx\ (a>0)$. created by kunny

2010 Today's Calculation Of Integral, 656

Tags: calculus , integration , limit , trigonometry , function , calculus computations

Find $\lim_{n\to\infty} n\int_0^{\frac{\pi}{2}} \frac{1}{(1+\cos x)^n}dx\ (n=1,\ 2,\ \cdots).$

2008 Harvard-MIT Mathematics Tournament, 10

Tags: function , calculus , integration

Evaluate the infinite sum \[\sum_{n \equal{} 0}^\infty \binom{2n}{n}\frac {1}{5^n}.\]

2009 Purple Comet Problems, 8

Tags: calculus , integration , inequalities , triangle inequality

Find the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length $11$.

2012 Today's Calculation Of Integral, 827

Tags: calculus , integration , limit , trigonometry , calculus computations

Find $\lim_{n\to\infty}\sum_{k=0}^{\infty} \int_{2k\pi}^{(2k+1)\pi} xe^{-x}\sin x\ dx.$

2007 Today's Calculation Of Integral, 173

Tags: calculus , integration , function , trigonometry , calculus computations

Find the function $f(x)$ such that $f(x)=\cos (2mx)+\int_{0}^{\pi}f(t)|\cos t|\ dt$ for positive inetger $m.$

2007 Today's Calculation Of Integral, 178

Tags: calculus , integration , function , trigonometry , limit , calculus computations

Let $f(x)$ be a differentiable function such that $f'(x)+f(x)=4xe^{-x}\sin 2x,\ \ f(0)=0.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}f(k\pi).$

2012 NIMO Problems, 8

Tags: geometry , geometric transformation , reflection , integration , trigonometry , probability , expected value

Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$. [i]Proposed by Lewis Chen[/i]

2010 Romania Team Selection Test, 1

Tags: calculus , integration , algebra , polynomial , modular arithmetic , induction , arithmetic sequence

A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$. [i]AMM Magazine[/i]

2011 Today's Calculation Of Integral, 751

Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm

Find $\lim_{n\to\infty}\left(\frac{1}{n}\int_0^n (\sin ^ 2 \pi x)\ln (x+n)dx-\frac 12\ln n\right).$

1994 Cono Sur Olympiad, 2

Tags: quadratic , number theory , greatest common divisor , calculus , integration , analytic geometry , conic

Solve the following equation in integers with gcd (x, y) = 1 $x^2 + y^2 = 2 z^2$

Today's calculation of integrals, 765

Tags: calculus , integration , function , limit , calculus computations

Define two functions $g(x),\ f(x)\ (x\geq 0)$ by $g(x)=\int_0^x e^{-t^2}dt,\ f(x)=\int_0^1 \frac{e^{-(1+s^2)x}}{1+s^2}ds.$ Now we know that $f'(x)=-\int_0^1 e^{-(1+s^2)x}ds.$ (1) Find $f(0).$ (2) Show that $f(x)\leq \frac{\pi}{4}e^{-x}\ (x\geq 0).$ (3) Let $h(x)=\{g(\sqrt{x})\}^2$. Show that $f'(x)=-h'(x).$ (4) Find $\lim_{x\rightarrow +\infty} g(x)$ Please solve the problem without using Double Integral or Jacobian for those Japanese High School Students who don't study them.