Olympiad problems

2004 Olympic Revenge, 5

$a_0 = a_1 = 1$ and ${a_{n+1} . a_{n-1}} = a_n . (a_n + 1)$ for all positive integers n. prove that $a_n$ is one integer for all positive integers n.

2010 Today's Calculation Of Integral, 628

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

(1) Evaluate the following definite integrals. (a) $\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx$ (b) $\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx$ (c) $\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx$ (2) Let $a$ be a positive constant. Find the area of the cross section cut by the plane $z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ of the solid such that \[x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0\] , then find the volume of the solid. [i]1984 Yamanashi Medical University entrance exam[/i] Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them. Thanks in advance. kunny

2011 Today's Calculation Of Integral, 723

Tags: calculus , integration , logarithm , calculus computations

Evaluate $\int_1^e \frac{\{1-(x-1)e^{x}\}\ln x}{(1+e^x)^2}dx.$

2010 Today's Calculation Of Integral, 556

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

Prove the following inequality. \[ \sqrt[3]{\int_0^{\frac {\pi}{4}} \frac {x}{\cos ^ 2 x\cos ^ 2 (\tan x)\cos ^ 2(\tan (\tan x))\cos ^ 2(\tan (\tan (\tan x)))}dx}<\frac{4}{\pi}\] Last Edited. Sorry, I have changed the problem. kunny

1984 Vietnam National Olympiad, 1

Tags: algebra , quadratic , integration , number theory , calculus

$(a)$ Let $x, y$ be integers, not both zero. Find the minimum possible value of $|5x^2 + 11xy - 5y^2|$. $(b)$ Find all positive real numbers $t$ such that $\frac{9t}{10}=\frac{[t]}{t - [t]}$.

2014 Contests, 2

Tags: integration , calculus , number theory

For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

2019 Jozsef Wildt International Math Competition, W. 10

Tags: calculus , trigonometry , exponent , integration

If ${si}(x) =- \int \limits_{x}^{\infty}\left(\frac{\sin t}{t}\right)dt; x>0$ then $$\int \limits_{e}^{e^2} \left(\frac{1}{x}\left(si\left(e^4x\right)-si\left(e^3x\right)\right)\right)\,dx=\int \limits_{3}^{e^4} \left(\frac{1}{x}\left(\operatorname{si}\left(e^2x\right)-si\left(ex\right)\right)\right)dx$$

2011 Today's Calculation Of Integral, 679

Tags: calculus computations , calculus , integration

Find $\sum_{k=1}^{3n} \frac{1}{\int_0^1 x(1-x)^k\ dx}$. [i]2011 Hosei University entrance exam/Design and Enginerring[/i]

2009 Today's Calculation Of Integral, 508

Tags: calculus , trigonometry , integration , calculus computations

Compare the size of the definite integrals? \[ \int_0^{\frac {\pi}{4}} x^{2008}\tan ^{2008}x\ dx,\ \int_0^{\frac {\pi}{4}} x^{2009}\tan ^{2009}x\ dx,\ \int_0^{\frac {\pi}{4}} x^{2010}\tan ^{2010}x\ dx\]

2012 Bogdan Stan, 2

Tags: integration , find all functions , function , calculus

Find the continuous functions $ f:\left[ 0,\frac{1}{3} \right] \longrightarrow (0,\infty ) $ that satisfy the functional relation $$ 54\int_0^{1/3} f(x)dx +32\int_0^{1/3} \frac{dx}{\sqrt{x+f(x)}} =21. $$ [i]Cristinel Mortici[/i]

2011 Today's Calculation Of Integral, 738

Tags: integration , trigonometry , calculus , calculus computations

Answer the following questions: (1) Find the value of $a$ for which $S=\int_{-\pi}^{\pi} (x-a\sin 3x)^2dx$ is minimized, then find the minimum value. (2) Find the vlues of $p,\ q$ for which $T=\int_{-\pi}^{\pi} (\sin 3x-px-qx^2)^2dx$ is minimized, then find the minimum value.

2001 Vietnam National Olympiad, 3

Tags: trigonometry , integration , calculus , calculus computations

For real $a, b$ define the sequence $x_{0}, x_{1}, x_{2}, ...$ by $x_{0}= a, x_{n+1}= x_{n}+b \sin x_{n}$. If $b = 1$, show that the sequence converges to a finite limit for all $a$. If $b > 2$, show that the sequence diverges for some $a$.

2005 Today's Calculation Of Integral, 1

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

Calculate the following indefinite integral. [1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$ [2] $\int \sin x\cos 3x dx$ [3] $\int \sin 2x\sin 3x dx$ [4] $\int \frac{dx}{4x^2-12x+9}$ [5] $\int \cos ^4 x dx$

2003 Romania National Olympiad, 2

Tags: function , integration , real analysis , definite integral , titu andreescu

Let be an odd natural number $ n\ge 3. $ Find all continuous functions $ f:[0,1]\longrightarrow\mathbb{R} $ that satisfy the following equalities. $$ \int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n,\quad\forall k\in\{ 1,2,\ldots ,n-1\} $$ [i]Titu Andreescu[/i]

2011 Today's Calculation Of Integral, 691

Tags: geometry , integration , calculus , geometric transformation , rotation , calculus computations , logarithm

Let $a$ be a constant. In the $xy$ palne, the curve $C_1:y=\frac{\ln x}{x}$ touches $C_2:y=ax^2$. Find the volume of the solid generated by a rotation of the part enclosed by $C_1,\ C_2$ and the $x$ axis about the $x$ axis. [i]2011 Yokohama National Universty entrance exam/Engineering[/i]

2010 Today's Calculation Of Integral, 616

Tags: logarithm , integration , calculus computations , calculus

Evaluate $\int_1^3 \frac{\ln (x+1)}{x^2}dx$. [i]2010 Hirosaki University entrance exam[/i]

2019 Jozsef Wildt International Math Competition, W. 52

Tags: periodic function , integration , calculus

Let $f : \mathbb{R} \to \mathbb{R}$ a periodic and continue function with period $T$ and $F : \mathbb{R} \to \mathbb{R}$ antiderivative of $f$. Then $$\int \limits_0^T \left[F(nx)-F(x)-f(x)\frac{(n-1)T}{2}\right]dx=0$$

1991 Arnold's Trivium, 38

Tags: integration , gauss , topology , euler , calculus

Calculate the integral of the Gaussian curvature of the surface \[z^4+(x^2+y^2-1)(2x^2+3y^2-1)=0\]

1991 Arnold's Trivium, 12

Tags: integration , vector , gauss

Find the flux of the vector field $\overrightarrow{r}/r^3$ through the surface \[(x-1)^2+y^2+z^2=2\]

2019 Jozsef Wildt International Math Competition, W. 62

Tags: inequalities , calculus , integration

Prove that $$\int \limits_0^{\frac{\pi}{2}}(\cos x)^{1+\sqrt{2n+1}}dx\leq \frac{2^{n-1}n!\sqrt{\pi}}{\sqrt{2(2n+1)!}}$$for all $n\in \mathbb{N}^*$

2007 Today's Calculation Of Integral, 223

Tags: trigonometry , integration , calculus , calculus computations

Evaluate $ \int_{0}^{\pi}\sqrt{(\cos x\plus{}\cos 2x\plus{}\cos 3x)^{2}\plus{}(\sin x\plus{}\sin 2x\plus{}\sin 3x)^{2}}\ dx$.

2008 Harvard-MIT Mathematics Tournament, 8

Tags: logarithm , function , integration , calculus computations , calculus

Let $ T \equal{} \int_0^{\ln2} \frac {2e^{3x} \plus{} e^{2x} \minus{} 1} {e^{3x} \plus{} e^{2x} \minus{} e^x \plus{} 1}dx$. Evaluate $ e^T$.

1999 National High School Mathematics League, 2

Tags: calculus , integration

The number of intengral points $(x,y)$ that fit $(|x|-1)^2+(|y|-1)^2<2$ is $\text{(A)}16\qquad\text{(B)}17\qquad\text{(C)}18\qquad\text{(D)}25$

2012 Today's Calculation Of Integral, 840

Tags: function , algebra , inequalities , calculus , domain , integration , trigonometry

Let $x,\ y$ be real numbers. For a function $f(t)=x\sin t+y\cos t$, draw the domain of the points $(x,\ y)$ for which the following inequality holds. \[\left|\int_{-\pi}^{\pi} f(t)\cos t\ dt\right|\leq \int_{-\pi}^{\pi} \{f(t)\}^2dt.\]

2009 Romania National Olympiad, 4

Tags: invariant , real analysis , integration , calculus , function , algebra

Find all functions $ f:[0,1]\longrightarrow [0,1] $ that are bijective, continuous and have the property that, for any continuous function $ g:[0,1]\longrightarrow\mathbb{R} , $ the following equality holds. $$ \int_0^1 g\left( f(x) \right) dx =\int_0^1 g(x) dx $$