Olympiad problems

2007 Gheorghe Vranceanu, 2

Let be a function $ f:(0,\infty )\longrightarrow\mathbb{R} $ satisfying the following two properties: $ \text{(i) } 2\lfloor x \rfloor \le f(x) \le 2 \lfloor x \rfloor +2,\quad\forall x\in (0,\infty ) $ $ \text{(ii) } f\circ f $ is monotone Can $ f $ be non-monotone? Justify.

2024 Middle European Mathematical Olympiad, 1

Tags: algebra , monotone , construction , functional inequality , function , sequence

Let $\mathbb{N}_0$ denote the set of non-negative integers. Determine all non-negative integers $k$ for which there exists a function $f: \mathbb{N}_0 \to \mathbb{N}_0$ such that $f(2024) = k$ and $f(f(n)) \leq f(n+1) - f(n)$ for all non-negative integers $n$.

2018 Ramnicean Hope, 1

Tags: algebra , equation , monotone

Solve in the real numbers the equation $ \sqrt[5]{2^x-2^{-1}} -\sqrt[5]{2^x+2^{-1}} =-1. $ [i]Mihai Neagu[/i]

2012 Centers of Excellency of Suceava, 4

Tags: cartesian circle , algebra , equation , monotone

Solve in the reals the following system. $$ \left\{ \begin{matrix} \log_2|x|\cdot\log_2|y| =3/2 \\x^2+y^2=12 \end{matrix} \right. $$ [i]Gheorghe Marchitan[/i]

2012 Grigore Moisil Intercounty, 3

Tags: monotone

Solve in the real numbers the equation $ (n+1)^x+(n+3)^x+\left( n^2+2n\right)^x=n^x+(n+2)^x+\left( n^2+4n+3\right)^x, $ wher $ n\ge 2 $ is a fixed natural number.

2010 Laurențiu Panaitopol, Tulcea, 3

Tags: function , real analysis , calculus , derivative , monotone

Let be a twice-differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the properties that: $ \text{(i) supp} f''=f\left(\mathbb{R}\right) $ $ \text{(ii)}\exists g:\mathbb{R}\longrightarrow\mathbb{R}\quad\forall x\in\mathbb{R}\quad f(x+1)=f(x)+f'\left( g(x)\right)\text{ and } f'(x+1)=f'(x)+f''\left( g(x)\right) $ Prove that: [b]a)[/b] any such $ g $ is injective. [b]b)[/b] $ f $ is of class $ C^{\infty } , $ and for any natural number $ n, $ any real number $ x $ and any such $ g, $ $$f^{(n)}(x+1)=f^{(n)}(x)+f^{(n+1)}\left( g(x)\right) . $$ [i]Laurențiu Panaitopol[/i]

2004 Unirea, 3

Tags: real analysis , inequalities , sequence , limit , monotone

[b]a)[/b] Prove that for any natural numbers $ n, $ the inequality $$ e^{2-1/n} >\prod_{k=1}^n (1+1/k^2) $$ holds. [b]b)[/b] Prove that the sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined by the recursive relation $ a_{n+1}=\frac{2}{n^2}\sum_{k=1}^n ka_k $ is nondecreasing. Is it convergent?

2006 Cezar Ivănescu, 1

Tags: equation , quadratic formula , cristinel mortici , algebra , monotone

Solve the equation [b]a)[/b] $ \log_2^2 +(x-1)\log_2 x =6-2x $ in $ \mathbb{R} . $ [b]b)[/b] $ 2^{x+1}+3^{x+1} +2^{1/x^2}+3^{1/x^2}=18 $ in $ (0,\infty ) . $ [i]Cristinel Mortici[/i]

2004 Gheorghe Vranceanu, 2

Tags: inequalities , equation , monotone

Solve in $ \mathbb{R}^2 $ the following equation. $$ 9^{\sqrt x} +9^{\sqrt{y}} +9^{1/\sqrt{xy}} =\frac{81}{\sqrt{x} +\sqrt{y} +1/\sqrt{xy}} $$ [i]O. Trofin[/i]

2019 Ramnicean Hope, 1

Tags: algebra , equation , monotone

Solve in the reals the equation $ \sqrt[3]{x^2-3x+4} +\sqrt[3]{-2x+2} +\sqrt[3]{-x^2+5x+2} =2. $ [i]Ovidiu Țâțan[/i]

2005 Grigore Moisil Urziceni, 1

Tags: inequalities , monotone

Prove that $ 5^x+6^x\le 4^x+8^x, $ for any nonegative real numbers $ x. $

2007 Nicolae Păun, 4

Tags: function , topology , darboux s property , real analysis , pathological , monotone

Construct a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following properties: $ \text{(i)} f $ is not monotonic on any real interval. $ \text{(ii)} f $ has Darboux property (intermediate value property) on any real interval. $ \text{(iii)} f(x)\leqslant f\left( x+1/n \right) ,\quad \forall x\in\mathbb{R} ,\quad \forall n\in\mathbb{N} $ [i]Alexandru Cioba[/i]

2007 Nicolae Păun, 2

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

2004 Nicolae Coculescu, 1

Tags: equation , system of equations , monotone , algebra

Solve in the real numbers the system: $$ \left\{ \begin{matrix} x^2+7^x=y^3\\x^2+3=2^y \end{matrix} \right. $$ [i]Eduard Buzdugan[/i]

2006 Petru Moroșan-Trident, 2

Tags: system of equations , algebra , equation , monotone

Solve in the positive real numbers the following system. $$ \left\{\begin{matrix} x^y=2^3\\y^z=3^4\\z^x=2^4 \end{matrix}\right. $$ [i]Aurel Ene[/i]

2004 Nicolae Coculescu, 2

Tags: quadratic , algebra , trigonometry , logarithm , equation , monotone

Solve in the real numbers the equation: $$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$ [i]Gheorghe Mihai[/i]

2005 Czech And Slovak Olympiad III A, 6

Tags: coloring , monotone , sequence , combinatorics , permutation

Decide whether for every arrangement of the numbers $1,2,3, . . . ,15$ in a sequence one can color these numbers with at most four different colors in such a way that the numbers of each color form a monotone subsequence.

2006 Victor Vâlcovici, 1

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

Let be an even natural number $ n $ and a function $ f:[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ f(x)=\int_0^x \prod_{k=0}^n (s-k) ds. $$ Show that [b]a)[/b] $ f(n)=0. $ [b]b)[/b] $ f $ is globally nonnegative. [i]Gheorghe Grigore[/i]

2007 Gheorghe Vranceanu, 3

Tags: function , sequence , algebra , monotone

Given a function $ f:\mathbb{N}\longrightarrow\mathbb{N} , $ find the necessary and sufficient condition that makes the sequence $$ \left(\left( 1+\frac{(-1)^{f(n)}}{n+1} \right)^{(-1)^{-f(n+1)}\cdot(n+2)}\right)_{n\ge 1} $$ to be monotone.

2010 Gheorghe Vranceanu, 1

Tags: geometry , analytic geometry , function , monotone

Let be a number $ x $ and three positive numbers $ a,b,c $ such that $ a^x+b^x=c^x. $ Prove that $ a^y,b^y,c^y $ are the lenghts of the sides of an obtuse triangle if and only if $ y<x<2y. $

2008 Grigore Moisil Intercounty, 1

Tags: function , find all functions , trigonometry , algebra , monotone

Find all monotonic functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with the property that $$ (f(\sin x))^2-3f(x)=-2, $$ for any real numbers $ x. $ [i]Dorin Andrica[/i] and [i]Mihai Piticari[/i]

2006 Grigore Moisil Urziceni, 3

Tags: algebra , system of equations , equation , monotone

Solve in $ \mathbb{R}^3 $ the system: $$ \left\{ \begin{matrix} 3^x+4^x=5^y \\8^y+15^y=17^z \\ 20^z+21^z=29^x \end{matrix} \right. $$ [i]Cristinel Mortici[/i]

2004 Alexandru Myller, 4

Tags: continuity , function , darboux s property , darboux , real analysis , monotone

Let be a real function that has the intermediate value property and is monotone on the irrationals. Show that it's continuous. [i]Mihai Piticari[/i]

2018 Ramnicean Hope, 2

Tags: trigonometry , algebra , equation , monotone

Solve in the real numbers the equation $ \arctan\sqrt{3^{1-2x}} +\arctan {3^x} =\frac{7\pi }{12} . $ [i]Ovidiu Țâțan[/i]

2004 Nicolae Coculescu, 4

Tags: function , algebra , cauchy functional equation , monotone

Let be a function satisfying [url=http://mathworld.wolfram.com/CauchyFunctionalEquation.html]Cauchy's functional equation,[/url] and having the property that it's monotonic on a real interval. Prove that this function is globally monotonic. [i]Florian Dumitrel[/i]