Olympiad problems

2006 Cezar Ivănescu, 1

Tags: factorial , harmonic series , newton s binom , limit , integral , calculus

[b]a)[/b] $ \lim_{n\to\infty } \frac{1}{n^2}\sum_{i=0}^n\sqrt{\binom{n+i}{2}} $ [b]b)[/b] $ \lim_{n\to\infty } \frac{a^{H_n}}{1+n} ,\quad a>0 $

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 $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.

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 $a \leq x \leq b.$

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.

1985 Traian Lălescu, 2.2

Tags: young s inequality , real analysis , inequalities , integral inequality , bijective , integral , calculus

Let $ a,b,c\in\mathbb{R}_+^*, $ and $ f:[0,a]\longrightarrow [0,b] $ bijective and non-decreasing. Prove that: $$ \frac{1}{b}\int_0^a f^2 (x)dx +\frac{1}{a}\int_0^b \left( f^{-1} (x)\right)^2dx\le ab. $$

2017 Korea USCM, 5

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

Evaluate the following limit. \[\lim_{n\to\infty} \sqrt{n} \int_0^\pi \sin^n x dx\]

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 $n$.

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 $$\lim_{x \to \infty} \int\limits_0^x (1+t^a)^{-b} \mathrm{d}t=1.$$

1967 Putnam, A4

Tags: function , integral , functional equation

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: $$u(x)= 1+ \lambda \int_{x}^{1} u(y) u(y-x) \; dy.$$

2023 District Olympiad, P3

Tags: real analysis , integral , limit

Let $f:[0,1]\to\mathbb{R}$ be a continuous function. Prove that \[\lim_{n\to\infty}\int_0^1 f(x^n) \ dx=f(0).\]Furthermore, if $f(0)=0$ and $f$ is right-differentiable in $0{}$, prove that the limits \[\lim_{\varepsilon\to0}\int_\varepsilon^1\frac{f(x)}{x} \ dx\quad\text{and}\quad\lim_{n\to\infty}\left(n\int_0^1f(x^n) \ dx\right)\]exist, are finite and are equal.

1996 Romania National Olympiad, 4

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

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.

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 District Olympiad, P3

Tags: integral

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

1986 Traian Lălescu, 2.3

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

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 $$ \left| \int_0^2 f(t) dt\right| >1. $$

1999 Estonia National Olympiad, 2

Tags: calculus , integration , integral

Find the value of the integral $\int_{-1}^{1} ln \left(x +\sqrt{1 + x^2}\right) dx$.

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.)

2010 Gheorghe Vranceanu, 2

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

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.

2018 Romania National Olympiad, 3

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

Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$ $\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$ $\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$ [i]Nicolae Bourbacut[/i]

2003 District Olympiad, 4

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

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

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]