Olympiad problems

1973 Putnam, B4

(a) On $[0, 1]$, let $f(x)$ have a continuous derivative satisfying $0 <f'(x) \leq1$. Also suppose that $f(0) = 0.$ Prove that $$ \left( \int_{0}^{1} f(x)\; dx \right)^{2} \geq \int_{0}^{1} f(x)^{3}\; dx.$$ (b) Show an example in which equality occurs.

2023 CIIM, 1

Tags: fractional part , integral , infinite series , integration , calculus

Determine all the pairs of positive real numbers $(a, b)$ with $a < b$ such that the following series $$\sum_{k=1}^{\infty} \int_a^b\{x\}^k dx =\int_a^b\{x\} dx + \int_a^b\{x\}^2 dx + \int_a^b\{x\}^3 dx + \cdots$$ is convergent and determine its value in function of $a$ and $b$. [b]Note: [/b] $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.

2011 District Olympiad, 3

Tags: integral , riemann sum , real analysis , function

Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous and nondecreasing function. [b]a)[/b] Show that the sequence $ \left( \frac{1}{2^n}\sum_{i=1}^{2^n} f\left(\frac{i}{2^n}\right) \right)_{n\ge 1} $ is nonincreasing. [b]b)[/b] Prove that, if there exists some natural index at which the sequence above is equal to $ \int_0^1 f(x)dx, $ then $ f $ is constant.

2022 Romania National Olympiad, P1

Tags: function , integral , calculus

Let $\mathcal{F}$ be the set of functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(2x)=f(x)$ for all $x\in\mathbb{R}.$ [list=a] [*]Determine all functions $f\in\mathcal{F}$ which admit antiderivatives on $\mathbb{R}.$ [*]Give an example of a non-constant function $f\in\mathcal{F}$ which is integrable on any interval $[a,b]\subset\mathbb{R}$ and satisfies \[\int_a^bf(x) \ dx=0\]for all real numbers $a$ and $b.$ [/list][i]Mihai Piticari and Sorin Rădulescu[/i]

2019 District Olympiad, 4

Tags: improper integral , integral , calculus , limit

Let $a$ be a real number, $a>1.$ Find the real numbers $b \ge 1$ such that $$\lim_{x \to \infty} \int\limits_0^x (1+t^a)^{-b} \mathrm{d}t=1.$$

1958 February Putnam, B7

Tags: continuity , integral

Prove that if $f(x)$ is continuous for $a\leq x \leq b$ and $$\int_{a}^{b} x^n f(x) \, dx =0$$ for $n=0,1,2, \ldots,$ then $f(x)$ is identically zero on $a \leq x \leq b.$

Gheorghe Țițeica 2024, P1

Tags: real analysis , integral

Let $a>1$ and $b>1$ be rational numbers. Denote by $\mathcal{F}_{a,b}$ the set of functions $f:[0,\infty)\rightarrow\mathbb{R}$ such that $$f(ax)=bf(x), \text{ for all }x\geq 0.$$ a) Prove that the set $\mathcal{F}_{a,b}$ contains both Riemann integrable functions on any interval and functions that are not Riemann integrable on any interval. b) If $f\in\mathcal{F}_{a,b}$ is Riemann integrable on $[0,\infty)$ and $\int_{\frac{1}{a}}^{a}f(x)dx=1$, calculate $$\int_a^{a^2} f(x)dx\text{ and }\int_0^1 f(x)dx.$$ [i]Vasile Pop[/i]

1996 Romania National Olympiad, 4

Tags: real analysis , integral , monotone functions , limit , function , calculus

Let $f:[0,1) \to \mathbb{R}$ be a monotonic function. Prove that the limits [center]$\lim_{x \nearrow 1} \int_0^x f(t) \mathrm{d}t$ and $\lim_{n \to \infty} \frac{1}{n} \left[ f(0) + f \left(\frac{1}{n}\right) + \ldots + f \left( \frac{n-1}{n} \right) \right]$[/center] exist and are equal.

2013 Today's Calculation Of Integral, 879

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

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$

2010 Laurențiu Panaitopol, Tulcea, 1

Tags: differentiability , primitivableness , antiderivative , integral , indefinite integral , real analysis , function

Let be two real numbers $ a<b $ and a function $ f:[a,b]\longrightarrow\mathbb{R} $ having the property that if the sequence $ \left(f\left( x_n \right)\right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. [b]a)[/b] Prove that if $ f $ admits antiderivatives, then $ f $ is integrable. [b]b)[/b] Is the converse of [b]a)[/b] true? [i]Marcelina Popa[/i]

II Soros Olympiad 1995 - 96 (Russia), 11.1

Tags: integral , integration , algebra , calculus

Find some antiderivative of the function $y = 1/x^3$, the graph of which has exactly three common points with the graph of the function $y = |x|$.

1981 Spain Mathematical Olympiad, 4

Tags: integration , integral , calculus

Calculate the integral $$\int \frac{dx}{\sin (x - 1) \sin (x - 2)} .$$ Hint: Change $\tan x = t$ .

2021 Romania National Olympiad, 1

Tags: integral

Find all continuous functions $f:\left[0,1\right]\rightarrow[0,\infty)$ such that: $\int_{0}^{1}f\left(x\right)dx\cdotp\int_{0}^{1}f^{2}\left(x\right)dx\cdotp...\cdotp\int_{0}^{1}f^{2020}\left(x\right)dx=\left(\int_{0}^{1}f^{2021}\left(x\right)dx\right)^{1010}$

2017 District Olympiad, 1

Tags: integral , inequalities , function , calculus

Let $ f,g:[0,1]\longrightarrow{R} $ be two continuous functions such that $ f(x)g(x)\ge 4x^2, $ for all $ x\in [0,1] . $ Prove that $$ \left| \int_0^1 f(x)dx \right| \ge 1\text{ or } \left| \int_0^1 g(x)dx \right| \ge 1. $$

2024 District Olympiad, P2

Tags: derivative , integral , real analysis

Let $f:[0,1]\to(0,\infty)$ be a continous function on $[0,1]$ and let $A=\int_0^1 f(t)\mathrm{d}t.$[list=a] [*]Consider the function $F:[0,1]\to[0,A]$ defined by \[F(x)=\int_0^xf(t)\mathrm{d}t.\]Prove that $F(x)$ has an inverse function, which is differentiable. [*]Prove that there exists a unique function $g:[0,1]\to[0,1]$ for which\[\int_0^xf(t)\mathrm{d}t=\int_{g(x)}^1f(t)\mathrm{d}t\]is satisfied for every $x\in [0,1].$ [*]Prove that there exists $c\in[0,1]$ for which\[\lim_{x\to c}\frac{g(x)-c}{x-c}=-1,\]whre $g$ is the function uniquely determined at b. [/list]

1979 Spain Mathematical Olympiad, 5

Tags: integral , calculus , integration , trigonometry

Calculate the definite integral $$\int_2^4 \sin ((x-3)^3) dx$$

2016 VJIMC, 4

Tags: integral , bounded variation , differentiable function , function , real analysis

Let $f: [0,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $$f(x) = \int_{x - 1}^xf(t)\mathrm{d}t$$ for all $x \geq 1$. Show that $f$ has bounded variation on $[1,\infty)$, i.e. $$\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.$$

2003 District Olympiad, 4

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

Consider the continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R}, g: [0,1]\longrightarrow\mathbb{R} , $ where $ f $ has a finite limit at $ \infty . $ Show that: $$ \lim_{n \to \infty} \frac{1}{n}\int_0^n f(x) g\left( \frac{x}{n} \right) dx =\int_0^1 g(x)dx\cdot\lim_{x\to\infty} f(x) . $$

2000 Romania National Olympiad, 1

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

Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property: $$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$ [b]a)[/b] Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal. [b]b)[/b] Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $

2010 Gheorghe Vranceanu, 2

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

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.

2023 District Olympiad, P1

Tags: integral , real analysis , inequalities

Let $f:[-\pi/2,\pi/2]\to\mathbb{R}$ be a twice differentiable function which satisfies \[\left(f''(x)-f(x)\right)\cdot\tan(x)+2f'(x)\geqslant 1,\]for all $x\in(-\pi/2,\pi/2)$. Prove that \[\int_{-\pi/2}^{\pi/2}f(x)\cdot \sin(x) \ dx\geqslant \pi-2.\]

2008 District Olympiad, 2

Tags: analysis , primitive , integral , periodic , function , real analysis

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a countinuous and periodic function, of period $ T. $ If $ F $ is a primitive of $ f, $ show that: [b]a)[/b] the function $ G:\mathbb{R}\longrightarrow\mathbb{R}, G(x)=F(x)-\frac{x}{T}\int_0^T f(t)dt $ is periodic. [b]b)[/b] $ \lim_{n\to\infty}\sum_{i=1}^n\frac{F(i)}{n^2+i^2} =\frac{\ln 2}{2T}\int_0^T f(x)dx. $

2016 Korea USCM, 7

Tags: holder inequality , integral , inequalities , real analysis , function

$M$ is a postive real and $f:[0,\infty)\to[0,M]$ is a continuous function such that $$\int_0^\infty (1+x)f(x) dx<\infty$$ Then, prove the following inequality. $$\left(\int_0^\infty f(x) dx \right)^2 \leq 4M \int_0^\infty x f(x) dx$$ (@below, Thank you. I fixed.)

2008 Alexandru Myller, 3

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

Find the nondecreasing functions $ f:[0,1]\rightarrow\mathbb{R} $ that satisfy $$ \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , $$ for any nonnegative integer $ n. $ [i]Mihai Piticari[/i]

Today's calculation of integrals, 879

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

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$