Olympiad problems

2008 Moldova National Olympiad, 12.8

Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

1986 Traian Lălescu, 2.3

Tags: limit , calculus , real analysis , continuity

Discuss $ \lim_{x\to 0}\frac{\lambda +\sin\frac{1}{x} \pm\cos\frac{1}{x}}{x} . $

2001 Romania National Olympiad, 1

Tags: function , calculus , derivative , real analysis

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function, derivable on $R\backslash\{x_0\}$, having finite side derivatives in $x_0$. Show that there exists a derivable function $g:\mathbb{R}\rightarrow\mathbb{R}$, a linear function $h:\mathbb{R}\rightarrow\mathbb{R}$ and $\alpha\in\{-1,0,1\}$ such that: \[ f(x)=g(x)+\alpha |h(x)|,\ \forall x\in\mathbb{R} \]

1999 Romania National Olympiad, 1

Tags: function , limit , inequalities , integration , logarithm , real analysis

Find all continuous functions $ f: \mathbb{R}\to [1,\infty)$ for wich there exists $ a\in\mathbb{R}$ and a positive integer $ k$ such that \[ f(x)f(2x)\cdot...\cdot f(nx)\leq an^k\] for all real $ x$ and all positive integers $ n$. [i]author :Radu Gologan[/i]

2007 Gheorghe Vranceanu, 2

Tags: function , darboux , real analysis , continuity , primitivability

Let be areal number $ r, $ a nonconstant and continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with period $ T $ and $ F $ be its primitive having $ F(0)=0. $ Define the funtion $ g:\mathbb{R}\longrightarrow\mathbb{R} $ as $$ g(x)=\left\{\begin{matrix} f(1/x), & x\neq 0 \\ r, & x=0 \end{matrix}\right. $$ Prove that: [b]a)[/b] the image of $ f $ is closed. [b]b)[/b] $ g $ has the intermediate value property if and only if $ r\in f\left(\mathbb{R}\right) . $ [b]c)[/b] $ g $ is primitivable if and only if $ r=\frac{F(T)}{T} . $

2015 VJIMC, 2

Tags: number theory , modular arithmetic , real analysis

[b]Problem 2[/b] Determine all pairs $(n, m)$ of positive integers satisfying the equation $$5^n = 6m^2 + 1\ . $$

2025 District Olympiad, P4

Tags: function , real analysis

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Define $\tilde{f}:[0,1]\rightarrow\mathbb{R}$ as $$\tilde{f}(x)=\begin{dcases} \frac{1}{x}\cdot\int_0^x f(t)dt &\text{ for } x>0;\\ f(0) &\text{ for }x=0.\end{dcases}$$ Show that: [list=a] [*] $\tilde{f}$ is continuous in $0$ and differentiable on $(0,1]$. [*] the following equality takes place: $$\int_0^1 f^2(x)dx = \left(\int_0^1 f(x)dx\right)^2 + \int_0^1 \left(f(x)-\tilde{f}(x)\right)^2dx.$$

2003 District Olympiad, 2

Tags: function , real analysis

Let $f:[0,1]\rightarrow [0,1]$ a continuous function in $0$ and in $1$, which has one-side limits in any point and $f(x-0)\le f(x)\le f(x+0),\ (\forall)x\in (0,1)$. Prove that: a)for the set $A=\{x\in [0,1]\ |\ f(x)\ge x\}$, we have $\sup A\in A$. b)there is $x_0\in [0,1]$ such that $f(x_0)=x_0$. [i]Mihai Piticari[/i]

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 Brazil Undergrad MO, 4

Tags: series , real analysis

Let $\alpha, c > 0$, define $x_1 = c$ and let $x_{n + 1} = x_n e^{-x_n^\alpha}$ for $n \geq 1$. For which values of $\beta$ does $\sum_{i = 1}^{\infty} x_n^\beta$ converge?

2003 Alexandru Myller, 3

Tags: inequalities , real analysis

Let be a nonnegative integer $ n. $ Prove that there exists an increasing and finite sequence of positive real numbers, $ \left( a_k \right)_{0\le k\le n} , $ that satisfy the equality $$ a_0/0! +a_1/1! +a_2/2! +\cdots +a_n/n! =1/n! , $$ and the inequality $$ a_0+a_1+a_2+\cdots +a_n<\frac{3}{2^n} . $$ [i]Dorin Andrica[/i]

1953 Miklós Schweitzer, 4

Tags: geometry , real analysis

[b]4.[/b] Show that every closed curve c of length less than $ 2\pi $ on the surface of the unit sphere lies entirely on the surface of some hemisphere of the unit sphere. [b](G. 8)[/b]

1963 Miklós Schweitzer, 8

Tags: trigonometry , function , integration , search , real analysis

Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]

1966 Miklós Schweitzer, 6

Tags: function , real analysis

A sentence of the following type if often heard in Hungarian weather reports: "Last night's minimum temperatures took all values between $ \minus{}3$ degrees and $ \plus{}5$ degrees." Show that it would suffice to say, "Both $ \minus{}3$ degrees and $ \plus{}5$ degrees occurred among last night's minimum temperatures." (Assume that temperature as a two-variable function of place and time is continuous.) [i]A.Csaszar[/i]

2018 Ramnicean Hope, 2

Tags: seqeunce , real analysis , limit

Let be a sequence $ \left( x_n \right)_{n\ge 0} $ with $ x_0\in (0,1) $ and defined as $$ 2x_n=x_{n-1}+\sqrt{3-3x_{n-1}^2} . $$ Prove that this sequence is bounded and periodic. Moreover, find $ x_0 $ for which this sequence is convergent. [i]Ovidiu Țâțan[/i]

2014 VJIMC, Problem 1

Tags: real analysis , limit

Let $f:(0,\infty)\to\mathbb R$ be a differentiable function. Assume that $$\lim_{x\to\infty}\left(f(x)+\frac{f'(x)}x\right)=0.$$Prove that $$\lim_{x\to\infty}f(x)=0.$$

2023 Miklós Schweitzer, 9

Tags: polynomial , analysis , functional analysis , real analysis

Let $C[-1,1]$ be the space of continuous real functions on the interval $[-1,1]$ with the usual supremum norm, and let $V{}$ be a closed, finite-codimensional subspace of $C[-1,1].$ Prove that there exists a polynomial $p\in V$ with norm at most one, which satisfies $p'(0)>2023.$

1986 Traian Lălescu, 1.4

Tags: real analysis , calculus , integration , set theory , function

Let be a parametric set: $$ \mathcal{F}_{\lambda } =\left\{ f:[1,\infty)\longrightarrow\mathbb{R}\bigg| x\in(1,\infty )\implies \int_{x}^{x^2+\lambda^2 x} f\left( \xi\right) d\xi =1\right\} . $$ [b]a)[/b] Show that $ \mathcal{F}_0 =\emptyset . $ [b]b)[/b] Prove that $ \lambda\neq 0 $ implies $ \mathcal{F}_{\lambda }\neq\emptyset . $

2011 District Olympiad, 4

Tags: function , analytic geometry , graphing lines , slope , inequalities , triangle inequality , real analysis

Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have: \[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\] for all $x,y\in [0,1]$.

2004 VJIMC, Problem 4

Tags: algebra , polynomial , real analysis

Let $f:\mathbb R\to\mathbb R$ be an infinitely differentiable function. Assume that for every $x\in\mathbb R$ there is an $n\in\mathbb N$ (depending on $x$) such that $$f^{(n)}(x)=0.$$Prove that $f$ is a polynomial.

2018 IMC, 1

Tags: real analysis , sequence

Let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be two sequences of positive numbers. Show that the following statements are equivalent: [list=1] [*]There is a sequence $(c_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}{\frac{a_n}{c_n}}$ and $\sum_{n=1}^{\infty}{\frac{c_n}{b_n}}$ both converge;[/*] [*]$\sum_{n=1}^{\infty}{\sqrt{\frac{a_n}{b_n}}}$ converges.[/*] [/list] [i]Proposed by Tomáš Bárta, Charles University, Prague[/i]

2019 IMC, 3

Tags: function , real analysis

Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that $$2f’(x)+xf''(x)\geqslant 1 \quad \text{ for } x\in (-1,1).$$ Prove that $$\int_{-1}^{1}xf(x)dx\geqslant \frac{1}{3}.$$ [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan[/i]

1997 Romania National Olympiad, 4

Tags: function , induction , real analysis

Let two bijective and continuous functions$f,g: \mathbb{R}\to\mathbb{R}$ such that : $\left(f\circ g^{-1}\right)(x)+\left(g\circ f^{-1}\right)(x)=2x$ for any real $x$. Show that If we have a value $x_{0}\in\mathbb{R}$ such that $f(x_{0})=g(x_{0})$, then $f=g$.

1986 Traian Lălescu, 2.1

Tags: calculus , integration , function , real analysis , algebra , functional equation , primitive

Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds: $$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$

2009 Romania National Olympiad, 4

Tags: function , calculus , integration , real analysis , invariant , algebra

Find all functions $ f:[0,1]\longrightarrow [0,1] $ that are bijective, continuous and have the property that, for any continuous function $ g:[0,1]\longrightarrow\mathbb{R} , $ the following equality holds. $$ \int_0^1 g\left( f(x) \right) dx =\int_0^1 g(x) dx $$