Olympiad problems

2005 Today's Calculation Of Integral, 60

Let $a_n=\int_0^{\frac{\pi}{2}} \sin 2t\ (1-\sin t)^{\frac{n-1}{2}}dt\ (n=1,2,\cdots)$ Evaluate \[\sum_{n=1}^{\infty} (n+1)(a_n-a_{n+1})\]

2019 Romania National Olympiad, 1

Tags: integration , concave function , definite integral

Let $a>0$ and $\mathcal{F} = \{f:[0,1] \to \mathbb{R} : f \text{ is concave and } f(0)=1 \}.$ Determine $$\min_{f \in \mathcal{F}} \bigg\{ \left( \int_0^1 f(x)dx\right)^2 - (a+1) \int_0^1 x^{2a}f(x)dx \bigg\}.$$

2017 Azerbaijan EGMO TST, 4

Tags: calculus , integration , induction , modular arithmetic , number theory

Find all natural numbers a, b such that $ a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1}$ where c and n are naturals.

2005 Today's Calculation Of Integral, 81

Tags: calculus , integration , trigonometry , inequalities , derivative , function , logarithm

Prove the following inequality. \[\frac{1}{12}(\pi -6+2\sqrt{3})\leq \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \ln (1+\cos 2x) dx\leq \frac{1}{4}(2-\sqrt{3})\]

2020 Jozsef Wildt International Math Competition, W23

Tags: integration , calculus

Prove that $$\int^{\frac\pi3}_{\frac\pi6}\frac u{\sin u}du=\frac83\sum_{k=0}^\infty\frac{(-1)^k}{3^k(2k+1)^2}+\frac{\pi\ln3}{3\sqrt3}-\frac{4C}3+\frac\pi6\ln\left(2+\sqrt3\right)-\operatorname{Im}\left(\frac2{\sqrt3}\operatorname{Li}_2\left(\frac{1-i\sqrt3}2\right)-\frac2{\sqrt3}\operatorname{Li}_2\left(\frac{\sqrt3-i}{2\sqrt3}\right)\right)$$ where as usual $$\operatorname{Li}_2(z)=-\int^z_0\frac{\ln(1-t)}tdt,z\in\mathbb C\setminus[1,\infty)$$ and $C=\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^2}$ is the Catalan constant. [i]Proposed by Paolo Perfetti[/i]

1976 Miklós Schweitzer, 4

Tags: calculus , integration , algebra , function , domain , polynomial , ring theory

Let $ \mathbb{Z}$ be the ring of rational integers. Construct an integral domain $ I$ satisfying the following conditions: a)$ \mathbb{Z} \varsubsetneqq I$; b) no element of $ I \minus{} \mathbb{Z}$ (only in $ I$) is algebraic over $ \mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $ \mathbb{Z}$); c) $ I$ only has trivial endomorphisms. [i]E. Fried[/i]

2009 Today's Calculation Of Integral, 519

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^2 \frac{1}{\sqrt {1 \plus{} x^3}}\ dx$.

2007 Harvard-MIT Mathematics Tournament, 3

Tags: calculus , integration

Let $a$ be a positive real number. Find the value of $a$ such that the definite integral \[\int_a^{a^2} \dfrac{dx}{x+\sqrt{x}}\] achieves its smallest possible value.

2007 Moldova National Olympiad, 12.4

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

If the function $f\colon [1,2]\to R$ is such that $\int_{1}^{2}f(x) dx=\frac{73}{24}$, then show that there exists a $x_{0}\in (1;2)$ such that \[x_{0}^{2}<f(x_{0})<x_{0}^{3}\] [Edit: $f$ is continuous]

2007 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , integration

Compute \[\int_0^\infty \dfrac{e^{-x}\sin(x)}{x}dx\]

Today's calculation of integrals, 900

Tags: calculus , integration , trigonometry , calculus computations

Find $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\ _n C_k.$

1997 China Team Selection Test, 3

Tags: calculus , integration , algebra

Prove that there exists $m \in \mathbb{N}$ such that there exists an integral sequence $\lbrace a_n \rbrace$ which satisfies: [b]I.[/b] $a_0 = 1, a_1 = 337$; [b]II.[/b] $(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall$ $n \geq 1$; [b]III. [/b]$\frac{1}{6}(a_n + 1)(2a_n + 1)$ is a perfect square $\forall$ $n \geq 1$.

2005 Today's Calculation Of Integral, 11

Tags: calculus , integration , algebra , function , domain , complex number , logarithm

Calculate the following indefinite integrals. [1] $\int \frac{6x+1}{\sqrt{3x^2+x+4}}dx$ [2] $\int \frac{e^x}{e^x+e^{a-x}}dx$ [3] $\int \frac{(\sqrt{x}+1)^3}{\sqrt{x}}dx$ [4] $\int x\ln (x^2-1)dx$ [5] $\int \frac{2(x+2)}{x^2+4x+1}dx$

2007 Today's Calculation Of Integral, 169

Tags: calculus , integration , function , calculus computations

(1) Let $f(x)$ be the differentiable and increasing function such that $f(0)=0.$Prove that $\int_{0}^{1}f(x)f'(x)dx\geq \frac{1}{2}\left(\int_{0}^{1}f(x)dx\right)^{2}.$ (2) $g_{n}(x)=x^{2n+1}+a_{n}x+b_{n}\ (n=1,\ 2,\ 3,\ \cdots)$ satisfies $\int_{-1}^{1}(px+q)g_{n}(x)dx=0$ for all linear equations $px+q.$ Find $a_{n},\ b_{n}.$

2010 Gheorghe Vranceanu, 2

Tags: limit , function , real analysis , integral , calculus , integration

Let be a natural number $ n, $ a nonzero number $ \alpha, \quad n $ numbers $ a_1,a_2,\ldots ,a_n $ and $ n+1 $ functions $ f_0,f_1,f_2,\ldots ,f_n $ such that $ f_0=\alpha $ and the rest are defined recursively as $$ f_k (x)=a_k+\int_0^x f_{k-1} (x)dx . $$ Prove that if all these functions are everywhere nonnegative, then the sum of all these functions is everywhere nonnegative.

2010 Today's Calculation Of Integral, 638

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

Let $(a,\ b)$ be a point on the curve $y=\frac{x}{1+x}\ (x\geq 0).$ Denote $U$ the volume of the figure enclosed by the curve , the $x$ axis and the line $x=a$, revolved around the the $x$ axis and denote $V$ the volume of the figure enclosed by the curve , the $y$ axis and th line $y=b$, revolved around the $y$ axis. What's the relation of $U$ and $V?$ 1978 Chuo university entrance exam/Science and Technology

2011 Today's Calculation Of Integral, 741

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_0^1 \frac{(x-1)^2(\cos x+1)-(2x-1)\sin x}{(x-1+\sqrt{\sin x})^2}\ dx\]

Today's calculation of integrals, 874

Tags: calculus , integration , parabola , geometry , calculus computations , conic

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

2000 IMC, 6

Tags: function , limit , integration , real analysis

Let $f: \mathbb{R}\rightarrow ]0,+\infty[$ be an increasing differentiable function with $\lim_{x\rightarrow+\infty}f(x)=+\infty$ and $f'$ is bounded, and let $F(x)=\int^x_0 f(t) dt$. Define the sequence $(a_n)$ recursively by $a_0=1,a_{n+1}=a_n+\frac1{f(a_n)}$ Define the sequence $(b_n)$ by $b_n=F^{-1}(n)$. Prove that $\lim_{x\rightarrow+\infty}(a_n-b_n)=0$.

2012 Today's Calculation Of Integral, 838

Tags: calculus , integration , inequalities , calculus computations

Prove that : $\frac{e-1}{e}<\int_0^1 e^{-x^2}dx<\frac{\pi}{4}.$

2010 Today's Calculation Of Integral, 666

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

Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying: (i) $f\left(\frac{\pi}{6}\right)=0$ (ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$. Find $f(x)$. [i]1987 Sapporo Medical University entrance exam[/i]

1970 IMO, 3

Tags: limit , inequality , calculus , integration , area

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

1973 Miklós Schweitzer, 6

Tags: function , integration , search , real analysis

If $ f$ is a nonnegative, continuous, concave function on the closed interval $ [0,1]$ such that $ f(0)=1$, then \[ \int_0^1 xf(x)dx \leq \frac 23 \left[ %Error. "diaplaymath" is a bad command. \int_0^1 f(x)dx \right]^2.\] [i]Z. Daroczy[/i]

1985 Canada National Olympiad, 3

Tags: perimeter , trigonometry , calculus , integration , inequalities , geometry

Let $P_1$ and $P_2$ be regular polygons of 1985 sides and perimeters $x$ and $y$ respectively. Each side of $P_1$ is tangent to a given circle of circumference $c$ and this circle passes through each vertex of $P_2$. Prove $x + y \ge 2c$. (You may assume that $\tan \theta \ge \theta$ for $0 \le \theta < \frac{\pi}{2}$.)

2022 JHMT HS, 1

Tags: calculus , derivative , integration

Compute the value of \[ \frac{d}{dx}\int_{1}^{10} x^3\,dx. \]