Olympiad problems

Gheorghe Țițeica 2024, P4

Let $f:\mathbb{R}\rightarrow (0,\infty)$ be continuous function of period $1$. Prove that for any $a\in\mathbb{R}$ $$\int_0^1\frac{f(x)}{f(x+a)}dx\geq 1.$$

1985 Traian Lălescu, 2.2

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

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

2015 District Olympiad, 2

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

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. $

1995 IMC, 2

Tags: calculus , function , integral , real analysis , 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 $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.

2007 Gheorghe Vranceanu, 4

Tags: real analysis , function , ftc , inverse function , primitivability , integral , 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.

2008 District Olympiad, 1

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

Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a countinuous function such that $$ \int_0^1 f(x)dx=\int_0^1 xf(x)dx. $$ Show that there is a $ c\in (0,1) $ such that $ f(c)=\int_0^c f(x)dx. $

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

2018 Romania National Olympiad, 3

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

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]

2006 Cezar Ivănescu, 1

Tags: limit , factorial , harmonic series , newton s binom , 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 $

2022 District Olympiad, P3

Tags: integral

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

1999 Estonia National Olympiad, 2

Tags: integration , calculus , integral

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

2017 Korea USCM, 5

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

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

2003 District Olympiad, 2

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

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, 2.1

Tags: real analysis , function , 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: $$ \int_{-1}^1 x^{2n+1} f(x)dx =\frac{2}{2n+3}f'(c). $$

2004 District Olympiad, 4

Tags: integration , real analysis , integral , calculus

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.

2000 Romania National Olympiad, 2

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

For any partition $ P $ of $ [0,1] $ , consider the set $$ \mathcal{A}(P)=\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f’\bigg|_{[0,1]}\right.\wedge\int_0^1 |f(x)|dx =1\wedge \left( y\in P\implies f (y ) =0\right)\right\} . $$ Prove that there exists a partition $ P_0 $ of $ [0,1] $ such that $$ g\in \mathcal{A}\left( P_0\right)\implies \sup_{x\in [0,1]} \big| g’(x)\big| >4\cdot \# P. $$ Here, $ \# D $ denotes the natural number $ d $ such that $ 0=x_0<x_1<\cdots <x_d=1 $ is a partition $ D $ of $ [0,1] . $

2022 District Olympiad, P4

Tags: function , integral

Let $I\subseteq \mathbb{R}$ be an open interval and $f:I\to\mathbb{R}$ a strictly monotonous function. Prove that for all $c\in I$ there exist $a,b\in I$ such that $c\in (a,b)$ and \[\int_a^bf(x) \ dx=f(c)\cdot (b-a).\]

2022 CIIM, 1

Tags: function , quadratic , 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 $n$.

1967 Putnam, A4

Tags: function , functional equation , integral

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 , limit , integral

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.

2011 District Olympiad, 1

Tags: calculus , function , integral , ftc , primitive

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

1967 Putnam, B3

Tags: periodic function , integral , function

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

2024 District Olympiad, P4

Tags: inequalities , real analysis , integral

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 $a>0.$

1986 Traian Lălescu, 2.3

Tags: function , inequalities , calculus , derivative , ftc , rolle theorem , 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 $$ \left| \int_0^2 f(t) dt\right| >1. $$