Olympiad problems

2012 Today's Calculation Of Integral, 856

Tags: integration , geometry , calculus , analytic geometry , trigonometry , calculus computations

On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.

2010 Today's Calculation Of Integral, 606

Tags: geometry , calculus , integration , calculus computations

Find the area of the part bounded by two curves $y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $x$-axis. 1956 Tokyo Institute of Technology entrance exam

1996 China National Olympiad, 2

Tags: inequalities , trigonometry , derivative , integration , calculus

Let $n$ be a natural number. Suppose that $x_0=0$ and that $x_i>0$ for all $i\in\{1,2,\ldots ,n\}$. If $\sum_{i=1}^nx_i=1$ , prove that \[1\leq\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < \frac{\pi}{2} \]

2006 ISI B.Math Entrance Exam, 5

Tags: calculus , rectangle , integration , geometry

A domino is a $2$ by $1$ rectangle . For what integers $m$ and $n$ can we cover an $m*n$ rectangle with non-overlapping dominoes???

2024-25 IOQM India, 10

Tags: calculus , integration

Determine the number of positive integral values of $p$ for which there exists a triangle with sides $a,b,$ and $c$ which satisfy $$a^2 + (p^2 + 9)b^2 + 9c^2 - 6ab - 6pbc = 0.$$

2024 Romania National Olympiad, 1

Tags: integration , function , calculus , real analysis

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x)+\sin(f(x)) \ge x,$ for all $x \in \mathbb{R}.$ Prove that $$\int\limits_0^{\pi} f(x) \mathrm{d}x \ge \frac{\pi^2}{2}-2.$$

2007 Today's Calculation Of Integral, 212

Tags: integration , trigonometry , calculus , calculus computations

For integers $k\ (0\leq k\leq 5)$, positive numbers $m,\ n$ and real numbers $a,\ b$, let $f(k)=\int_{-\pi}^{\pi}(\sin kx-a\sin mx-b\sin nx)^{2}\ dx$, $p(k)=\frac{5!}{k!(5-k)!}\left(\frac{1}{2}\right)^{5}, \ E=\sum_{k=0}^{5}p(k)f(k)$. Find the values of $m,\ n,\ a,\ b$ for which $E$ is minimized.

2010 Today's Calculation Of Integral, 545

Tags: function , calculus , calculus computations , integration

(1) Evaluate $ \int_0^1 xe^{x^2}dx$. (2) Let $ I_n\equal{}\int_0^1 x^{2n\minus{}1}e^{x^2}dx$. Express $ I_{n\plus{}1}$ in terms of $ I_n$.

2001 USA Team Selection Test, 2

Tags: combinatorics , calculus , function , integration , algebra , binomial theorem

Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.

2007 IMC, 6

Tags: integration , induction , function , polynomial , algebra

Let $ f \ne 0$ be a polynomial with real coefficients. Define the sequence $ f_{0}, f_{1}, f_{2}, \ldots$ of polynomials by $ f_{0}= f$ and $ f_{n+1}= f_{n}+f_{n}'$ for every $ n \ge 0$. Prove that there exists a number $ N$ such that for every $ n \ge N$, all roots of $ f_{n}$ are real.

2012 Uzbekistan National Olympiad, 2

Tags: calculus , number theory , modular arithmetic , integration

For any positive integers $n$ and $m$ satisfying the equation $n^3+(n+1)^3+(n+2)^3=m^3$, prove that $4\mid n+1$.

2008 Harvard-MIT Mathematics Tournament, 10

Tags: logarithm , integration , calculus computations , calculus

([b]8[/b]) Evaluate the integral $ \int_0^1\ln x \ln(1\minus{}x)\ dx$.

PEN R Problems, 12

Tags: integration , calculus , analytic geometry , the geometry of numbers

Find coordinates of a set of eight non-collinear planar points so that each has an integral distance from others.

2012 Today's Calculation Of Integral, 789

Tags: function , calculus computations , calculus , integration

Find the non-constant function $f(x)$ such that $f(x)=x^2-\int_0^1 (f(t)+x)^2dt.$

2021 Alibaba Global Math Competition, 15

Tags: integration , calculus , topology , analysis , manifold , differential geometry

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $n \ge 2$. Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$, where $\text{Ric}$ is the Ricci tensor of $(M,g)$. Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$. Prove that \[\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.\] Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$.

2005 Today's Calculation Of Integral, 64

Tags: trigonometry , integration , algebra , calculus , polynomial , calculus computations

Let $f(t)$ be the cubic polynomial for $t$ such that $\cos 3x=f(\cos x)$ holds for all real number $x$. Evaluate \[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\]

2006 IMO Shortlist, 2

Tags: integration , calculus , inequalities , sequence , summation , algebra

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

2006 Victor Vâlcovici, 1

Tags: function , integration , monotone , real analysis , ftc , definite integral , calculus

Let be an even natural number $ n $ and a function $ f:[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ f(x)=\int_0^x \prod_{k=0}^n (s-k) ds. $$ Show that [b]a)[/b] $ f(n)=0. $ [b]b)[/b] $ f $ is globally nonnegative. [i]Gheorghe Grigore[/i]

2005 VJIMC, Problem 3

Tags: integration , limit , calculus

Let $f:[0,1]\times[0,1]\to\mathbb R$ be a continuous function. Find the limit $$\lim_{n\to\infty}\left(\frac{(2n+1)!}{(n!)^2}\right)^2\int^1_0\int^1_0(xy(1-x)(1-y))^nf(x,y)\text dx\text dy.$$

2004 Unirea, 3

Tags: polynomial , integration , real analysis , trigonometry , algebra

Hello, I've been trying to solve this for a while now, but no success! I mean, I have an expression for this but not a closed one. I derived something in terms of Tchebychev Polynomials : cos(nx) = P_n(cos(x)). Any help is appreciated. Compute the following primitive: \[ \int \frac{x\sin\left(2004 x\right)}{\tan x}\ dx\]