Olympiad problems

1967 Putnam, A4

Show that if $\lambda > \frac{1}{2}$ there does not exist a real-valued function $u(x)$ such that for all $x$ in the closed interval $[0,1]$ the following holds: $$u(x)= 1+ \lambda \int_{x}^{1} u(y) u(y-x) \; dy.$$

1985 Traian Lălescu, 2.1

Tags: function , real analysis , integral , lagrange

Let $ f:[-1,1]\longrightarrow\mathbb{R} $ a derivable function and a non-negative integer $ n. $ Show that there is a $ c\in [-1,1] $ so that: $$ \int_{-1}^1 x^{2n+1} f(x)dx =\frac{2}{2n+3}f'(c). $$

2015 District Olympiad, 2

Tags: integral , real analysis , calculus , integration

[b]a)[/b] Calculate $ \int_{0}^1 x\sin\left( \pi x^2\right) dx. $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx. $ [i]Florin Stănescu[/i]

2009 District Olympiad, 1

Tags: function , inequalities , integral , real analysis

Let $ f:[0,\infty )\longrightarrow [0,\infty ) $ a nonincreasing function that satisfies the inequality: $$ \int_0^x f(t)dt <1,\quad\forall x\ge 0. $$ Prove the following affirmations: [b]a)[/b] $ \exists \lim_{x\to\infty} \int_0^x f(t)dt \in\mathbb{R} . $ [b]b)[/b] $ \lim_{x\to\infty} xf(x) =0. $

2021 Romania National Olympiad, 1

Tags: integral

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

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]

2019 District Olympiad, 4

Tags: improper integral , integral , limit , calculus

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

II Soros Olympiad 1995 - 96 (Russia), 11.1

Tags: calculus , algebra , integration , integral

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 $y = |x|$.

2008 District Olympiad, 2

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

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. $

2004 District Olympiad, 4

Tags: integral , real analysis , calculus , integration

Let $ a,b\in (0,1) $ and a continuous function $ f:[0,1]\longrightarrow\mathbb{R} $ with the property that $$ \int_0^x f(t)dt=\int_0^{ax} f(t)dt +\int_0^{bx} f(t)dt,\quad\forall x\in [0,1] . $$ [b]a)[/b] Show that if $ a+b<1, $ then $ f=0. $ [b]b)[/b] Show that if $ a+b=1, $ 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]

2016 VJIMC, 4

Tags: function , integral , bounded variation , differentiable 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. $$\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.$$

1996 Romania National Olympiad, 4

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

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.

2007 Gheorghe Vranceanu, 4

Tags: function , ftc , inverse function , primitivability , integral , real analysis , newton-leibniz

Let $ F $ be the primitive of a continuous function $ f:\mathbb{R}\longrightarrow (0,\infty ), $ with $ F(0)=0. $ Determine for which values of $ \lambda \in (0,1) $ the function $ \left( F^{-1}\circ \lambda F \right)/\text{id.} $ has limit at $ 0, $ and calculate it.

1992 Putnam, A2

Tags: integral , power series

Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x = 0$ of $(1 + x)^{\alpha}$ . Evaluate $$\int_{0}^{1} \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\, dy.$$

1986 Traian Lălescu, 2.3

Tags: integral , function , calculus , derivative , ftc , rolle theorem , inequalities

Let $ f:[0,2]\longrightarrow \mathbb{R} $ a differentiable function having a continuous derivative and satisfying $ f(0)=f(2)=1 $ and $ |f’|\le 1. $ Show that $$ \left| \int_0^2 f(t) dt\right| >1. $$

1995 IMC, 2

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

Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$, we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.

2003 District Olympiad, 2

Tags: function , integral , calculus , integration , ratio , sequence

Let be two distinct continuous functions $ f,g:[0,1]\longrightarrow (0,\infty ) $ corelated by the equality $ \int_0^1 f(x)dx =\int_0^1 g(x)dx , $ and define the sequence $ \left( x_n \right)_{n\ge 0} $ as $$ x_n=\int_0^1 \frac{\left( f(x) \right)^{n+1}}{\left( g(x) \right)^n} dx . $$ [b]a)[/b] Show that $ \infty =\lim_{n\to\infty} x_n. $ [b]b)[/b] Demonstrate that the sequence $ \left( x_n \right)_{n\ge 0} $ is monotone.

1979 Spain Mathematical Olympiad, 5

Tags: trigonometry , integration , integral , calculus

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

2024 District Olympiad, P2

Tags: integral , derivative , 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]

2003 District Olympiad, 4

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

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: $$ \lim_{n \to \infty} \frac{1}{n}\int_0^n f(x) g\left( \frac{x}{n} \right) dx =\int_0^1 g(x)dx\cdot\lim_{x\to\infty} f(x) . $$

2022 District Olympiad, P3

Tags: integral

Find all values of $n\in\mathbb{N}^*$ for which \[I_n:=\int_0^\pi\cos(x)\cdot\cos(2x)\cdot\ldots\cdot\cos(nx) \ dx=0.\]

2016 Korea USCM, 7

Tags: inequalities , holder inequality , integral , 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.)

2022 CIIM, 1

Tags: quadratic , function , integral , calculus

Given the function $f(x) = x^2$, the sector of $f$ from $a$ to $b$ is defined as the limited region between the graph of $y = f(x)$ and the straight line segment that joins the points $(a, f(a))$ and $(b, f(b))$. Define the increasing sequence $x_0$, $x_1, \cdots$ with $x_0 = 0$ and $x_1 = 1$, such that the area of the sector of $f$ from $x_n$ to $x_{n+1}$ is constant for $n \geq 0$. Determine the value of $x_n$ in function of $n$.

1981 Spain Mathematical Olympiad, 4

Tags: calculus , integration , integral

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