Olympiad problems

2013 Today's Calculation Of Integral, 876

Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition : 1) $f(-1)\geq f(1).$ 2) $x+f(x)$ is non decreasing function. 3) $\int_{-1}^ 1 f(x)\ dx=0.$ Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

2019 CMI B.Sc. Entrance Exam, 6

Tags: definite integral , lebinitz rule , calculus , logarithm

$(a)$ Compute - \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg] \end{align*} $(b)$ For $x > 0 $ define $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t . $\\ \\$1.$ Determine the open interval(s) (if any) where $F ( x )$ is decreasing and all the open interval(s) (if any) where $F ( x )$ is increasing.\\ \\$2.$ Determine all the local minima of $F ( x )$ (if any) and all the local maxima of $F ( x )$ (if any) $.$

2019 CMI B.Sc. Entrance Exam, 3

Tags: recursion of integrals , definite integral , algebra

Evaluate $\int_{ 0 }^{ \infty } ( 1 + x^2 )^{-( m + 1 )} \mathrm{d}x$ where $m \in \mathbb{N} $

2013 Today's Calculation Of Integral, 881

Tags: calculus computations , definite integral , trigonometry , integration , calculus

Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

2009 Moldova National Olympiad, 12.1

Tags: definite integral , cosinus , integration , calculus

Calculate $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{cos(x)^7}{e^x+1} dx$.

2013 Today's Calculation Of Integral, 877

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

Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$ Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$

2018 Ramnicean Hope, 1

Tags: definite integral , parity , calculus , function , fe , integration , real analysis

Let be two nonzero real numbers $ a,b $ such that $ |a|\neq |b| $ and let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function satisfying the functional relation $$ af(x)+bf(-x)=(x^3+x)^5+\sin^5 x . $$ Calculate $ \int_{-2019}^{2019}f(x)dx . $ [i]Constantin Rusu[/i]

2017 Romania National Olympiad, 1

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

[b]a)[/b] Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation. $$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$ [b]b)[/b] Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.

2011 Laurențiu Duican, 2

Tags: squeeze theorem , definite integral , limit , integration , calculus , real analysis

$ \lim_{n\to\infty } \int_{\pi }^{2\pi } \frac{|\sin (nx) +\cos (nx)|}{ x} dx ? $ [i]Gabriela Boeriu[/i]

Today's calculation of integrals, 876

Tags: function , definite integral , integration , calculus computations , calculus

Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition : 1) $f(-1)\geq f(1).$ 2) $x+f(x)$ is non decreasing function. 3) $\int_{-1}^ 1 f(x)\ dx=0.$ Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

Today's calculation of integrals, 881

Tags: calculus computations , trigonometry , integration , definite integral , calculus

Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

2012 Centers of Excellency of Suceava, 3

Tags: real analysis , factorial , series , definite integral , integral sequence

Consider the sequence $ \left( I_n \right)_{n\ge 1} , $ where $ I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, $ for any natural number $ n. $ [b]a)[/b] Find a relation between any two consecutive terms of $ I_n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } nI_n. $ [i]c)[/i] Show that $ \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx. $ [i]Cătălin Țigăeru[/i]

Today's calculation of integrals, 871

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

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

2012 Grigore Moisil Intercounty, 3

Tags: real analysis , limit , riemann sum , definite integral , claculus

$ \lim_{n\to\infty } \frac{1}{n}\sum_{i,j=1}^n \frac{i+j}{i^2+j^2} $

2004 Alexandru Myller, 4

Tags: limit , analysis , real analysis , definite integral , factorial , function

For any natural number $ m, \quad\lim_{n\to\infty } n^{1+m} \int_{0}^1 e^{-nx}\ln \left( 1+x^m \right) dx =m! . $ [i]Gheorghe Iurea[/i]

2010 N.N. Mihăileanu Individual, 1

Tags: calculus , real analysis , inequalities , lipschitz , integral inequalities , definite integral , integration

Let $ m:[0,1]\longrightarrow\mathbb{R} $ be a metric map. [b]a)[/b] Prove that $ -\text{identity} +m $ is continuous and nonincreasing. [b]b)[/b] Show that $ \int_0^1\int_0^x (-t+m(t))dtdx=\int_0^1 (x-1)(x-m(x))dx. $ [b]c)[/b] Demonstrate that $ \int_0^1\int_0^x m(t)dtdx -\frac{1}{2}\int_0^1 m(x)dx\ge -\frac{1}{12} . $ [i]Gabriela Constantinescu[/i] and [i]Nelu Chichirim[/i]

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]

2010 Moldova National Olympiad, 12.8

Tags: integration , calculus , definite integral , trigonometry

Find all $t\in \mathbb R$, such that $\int_{0}^{\frac{\pi}{2}}\mid \sin x+t\cos x\mid dx=1$ .

2012 Centers of Excellency of Suceava, 4

Tags: real analysis , calculus , integration , sequence of integrals , definite integral

Let be the sequence $ \left( J_n \right)_{n\ge 1} , $ where $ J_n=\int_{(1+n)^2}^{1+(1+n)^2} \sqrt{\frac{x-1-n-n^2}{x-1}} dx. $ [b]a)[/b] Study its monotony. [b]b)[/b] Calculate $ \lim_{n\to\infty } J_n\sqrt{n} . $ [i]Ion Bursuc[/i]

Today's calculation of integrals, 877

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

Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$ Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$

2016 Bangladesh Mathematical Olympiad, 9

Tags: calculus , definite integral , integer , combinatorics

Consider the integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$. [b](a)[/b] Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$, where $j$ is not a function of $x$, is $Z(j)=e^{j^{2}/4a} Z(0)$. [b](b)[/b] Show that $$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n},$$ where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times\cdots\times3\times 1$. [b](c)[/b] What is the number of ways to form $n$ pairs from $2n$ distinct objects? Interpret the previous part of the problem in term of this answer.

2012 Today's Calculation Of Integral, 795

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

Evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2+\sin x}{1+\cos x}\ dx.$

2012 Grigore Moisil Intercounty, 2

Tags: real analysis , integration , definite integral , calculus

$ \int_0^{\pi^2/4} \frac{dx}{1+\sin\sqrt x +\cos\sqrt x} $

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 $$\min_{f \in \mathcal{F}} \bigg\{ \left( \int_0^1 f(x)dx\right)^2 - (a+1) \int_0^1 x^{2a}f(x)dx \bigg\}.$$

2013 Today's Calculation Of Integral, 871

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

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$