Olympiad problems

2017 Korea USCM, 5

Tags: Integral , real analysis , limit , college contests , calculus , integration

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

2003 District Olympiad, 2

Tags: function , Sequences , Integral , calculus , integration , ratio

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.

1992 Putnam, A2

Tags: Putnam , 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 $$\int_{0}^{1} \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\, dy.$$

2022 Romania National Olympiad, P1

Tags: function , Integral , romania , 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]

1967 Putnam, B3

Tags: Putnam , function , periodic function , Integral

If $f$ and $g$ are continuous and periodic functions with period $1$ on the real line, then $$\lim_{n\to \infty} \int_{0}^{1} f(x)g (nx)\; dx =\left( \int_{0}^{1} f(x)\; dx\right)\left( \int_{0}^{1} g(x)\; dx\right).$$

2011 District Olympiad, 1

Tags: function , Integral , primitives , FTC , calculus

Prove the rationality of the number $ \frac{1}{\pi }\int_{\sin\frac{\pi }{13}}^{\cos\frac{\pi }{13}} \sqrt{1-x^2} dx. $

1979 Spain Mathematical Olympiad, 5

Tags: trigonometry , integration , Integral , calculus

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

2003 District Olympiad, 4

Tags: function , limits , 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 , college contests , romania

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

2023 District Olympiad, P1

Tags: real analysis , Integral , 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 \[\int_{-\pi/2}^{\pi/2}f(x)\cdot \sin(x) \ dx\geqslant \pi-2.\]

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

2006 Cezar Ivănescu, 1

Tags: harmonic series , Newton s binom , fatorial , 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 $

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.

2024 District Olympiad, P4

Tags: Integral , inequalities , real analysis

Let $f:[0,\infty)\to\mathbb{R}$ be a differentiable function, with a continous derivative. Given that $f(0)=0$ and $0\leqslant f'(x)\leqslant 1$ for every $x>0$ prove that\[\frac{1}{n+1}\int_0^af(t)^{2n+1}\mathrm{d}t\leqslant\left(\int_0^af(t)^n\mathrm{d}t\right)^2,\]for any positive integer $n{}$ and real number $a>0.$

2017 District Olympiad, 1

Tags: function , Integral , inequalities , 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 $$ \left| \int_0^1 f(x)dx \right| \ge 1\text{ or } \left| \int_0^1 g(x)dx \right| \ge 1. $$