Olympiad problems

2004 Nicolae Coculescu, 2

Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admits bounded primitives. Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\left\{ \begin{matrix} x, & \quad x\le 0 \\ f(1/x)\cdot\ln x ,& \quad x>0 \end{matrix}\right. $$ admits primitives. [i]Florian Dumitrel[/i]

2006 Petru Moroșan-Trident, 3

Tags: primitive , indefinite integral , calculus

Determine the primitives of: [b]1)[/b] $ (0,\pi /2)\ni x\mapsto\frac{x^2}{-x+\tan x} $ [b]2)[/b] $ 1<x\mapsto \frac{-1+\ln x}{x^2-\ln^2 x} $ [i]Ion Nedelcu[/i]

2008 District Olympiad, 2

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

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. $

2015 NIMO Summer Contest, 14

Tags: residue , primitive , order , number theory

We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2\equiv a \pmod n$, $x^3 \equiv a \pmod n$, or $x^5 \equiv a \pmod n$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of residues modulo $n$. How many integers $1 \le a \le 2015$ are primitive residues modulo $2015$? [i] Proposed by Michael Ren [/i]

2005 Grigore Moisil Urziceni, 2

Tags: primitive , real analysis , calculus , function

Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{\ge 0} $ that admits primitives and such that $ \lim_{x\to 0 } \frac{f(x)}{x} =0. $ Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} , $ defined as $$ g(x)=\left\{ \begin{matrix} f(x)/x ,&\quad x\neq 0\\ 0,& \quad x=0 \end{matrix} \right. , $$ is primitivable.

2007 Mathematics for Its Sake, 3

Tags: integration , primitive , indefinite integral , function

Let be three positive real numbers $ a,b,c, $ a natural number $ n, $ and the functions $ f:\mathbb{R}\longrightarrow\mathbb{R} ,g:(0,\infty )\longrightarrow\mathbb{R} $ defined as: $$ f(x)=\frac{2(n+1)x^n(x^{n+1}-a) +nx^{n+1} +2a^2x+a}{x^{2n+2}-2ax^{n+1} +a^2x^2+a^2} , $$ $$ g(x)=\frac{a+bx^n}{x+cx^{2n+1}} $$ Calculate the antiderivatives of $ f $ and $ g. $ [i]Nicolae Sanda[/i]

2004 Nicolae Coculescu, 4

Tags: real analysis , primitive , limit , function

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a continuous function having a primitive $ F $ having the property that $ f-F $ is positive globally. Calculate $ \lim_{x\to\infty } f(x) . $ [i]Florian Dumitrel[/i]

2007 Gheorghe Vranceanu, 1

Tags: function , group theory , real analysis , primitive

Let $ M $ denote the set of the primitives of a function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]ii)[/b] Show that $ M $ along with the operation $ *:M^2\longrightarrow M $ defined as $ F*G=F+G(2007) $ form a commutative group. [b]iii)[/b] Show that $ M $ is isomorphic with the additive group of real numbers.

2000 District Olympiad (Hunedoara), 3

Tags: real analysis , function , wilkosz , primitive

Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that: $ \text{(i)}\quad f(0)=0 $ $ \text{(ii)}\quad f'(x)\neq 0,\quad\forall x\in\mathbb{R} $ $ \text{(iii)}\quad \left. f''\right|_{\mathbb{R}}\text{ exists and it's continuous} $ Demonstrate that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ g(x)=\left\{\begin{matrix}\cos\frac{1}{f(x)},\quad x\neq 0\\ 0,\quad x=0\end{matrix}\right. $$ is primitivable.

2006 Grigore Moisil Urziceni, 3

Tags: real analysis , function , rolle , lagrange , primitivability , primitive

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that admits a primitive $ F. $ [b]a)[/b] Show that there exists a real number $ c $ such that $ f(c)-F(c)>1 $ if $ \lim_{x\to\infty } \frac{1+F(x)}{e^x} =-\infty . $ [b]b)[/b] Prove that there exists a real number $ c' $ such that $ f(c') -(F(c'))^2<1. $ [i]Cristinel Mortici[/i]

2006 Grigore Moisil Urziceni, 2

Tags: function , real analysis , calculus , primitive , primitivability

Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admits primitives. Prove that: $ \text{(i)} $ Every term (function) of the sequence functions $ \left( h_n\right)_{n\ge 2}:\mathbb{R}\longrightarrow\mathbb{R} $ defined, for any natural number $ n $ as $ h_n(x)=x^nf\left( x^3 \right) , $ is primitivable. $ \text{(ii)} $ The function $ \phi :\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ \phi (x) =\left\{ \begin{matrix} e^{-1/x^2} f(x),& \quad x\neq 0 \\ 0,& \quad x=0 \end{matrix} \right. $$ is primitivable. [i]Cristinel Mortici[/i]

2006 Petru Moroșan-Trident, 3

Tags: function , calculus , primitive , differentiation

Let be a differentiable function $ f:\mathbb{R}_{> 0}\longrightarrow\mathbb{R}_{> 0} , $ and a primitive $ F:\mathbb{R}_{> 0}\longrightarrow\mathbb{R}_{> 0} $ of it such that $ F=f+f\cdot f. $ Show that: [b]a)[/b] $ f $ is nondecreasing. [b]b)[/b] $ \lim_{x\to\infty } f(x)/x =1/2 $ [i]Vasile Solovăstru[/i]

2007 Nicolae Coculescu, 3

Tags: function , real analysis , primitive

Let $ F:\mathbb{R}\longrightarrow\mathbb{R} $ be a primitive with $ F(0)=0 $ of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $ f(x)=\sin (x^2) , $ and let be a sequence $ \left( a_n \right)_{n\ge 0} $ with $ a_0\in (0,1) $ and defined as $ a_{n}=a_{n-1}-F\left( a_{n-1} \right) . $ Calculate $ \lim_{n\to\infty } a_n\sqrt{n} . $ [i]Florian Dumitrel[/i]

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. $

2008 Gheorghe Vranceanu, 1

Tags: function , inverse calculus , calculus , primitive

Find the $ \mathcal{C}^1 $ class functions $ f:[0,1]\longrightarrow\mathbb{R} $ satisfying the following three clauses: $ \text{(i) } f(0)=0 $ $ \text{(ii) } \text{Im} f'\subset (0,1] $ $ \text{(iii) }F(1)-\frac{\left( f(1) \right)^3}{3} =F(0)=0, $ where $ F $ is a primitive of $ f. $

2007 Nicolae Păun, 2

Tags: function , inequalities , real analysis , monotone , primitivability , convexity , primitive

Consider a sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ and a primitivable function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]a)[/b] Prove that $ f $ is monotonic and continuous if for any natural numbers $ n $ and real numbers $ x, $ the inequality $$ f\left( x+x_n \right)\geqslant f(x) $$ is true. [b]b)[/b] Show that $ f $ is convex if for any natural numbers $ n $ and real numbers $ x, $ the inequality $$ f\left( x+2x_n \right) +f(x)\geqslant 2f\left( x+x_n \right) $$ is true. [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

1986 Traian Lălescu, 2.1

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

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