Olympiad problems

2008 Brazil Undergrad MO, 3

Prove that there are real numbers $ a_1, a_2, ..$ such that: i) For all real numbers x, the serie $ f(x) \equal{} \sum_{n \equal{} 1}^\infty a_nx^n$ converge; ii) f is a bijection of R to R; iii) f'(x) >0; iv) f(Q) = A, where Q is the set of rational numbers and A is the set of algebraic numbers.

2010 N.N. Mihăileanu Individual, 1

Tags: calculus , real analysis , inequalities , lipschitz , integral inequalities , definite integral , integration

Let $ m:[0,1]\longrightarrow\mathbb{R} $ be a metric map. [b]a)[/b] Prove that $ -\text{identity} +m $ is continuous and nonincreasing. [b]b)[/b] Show that $ \int_0^1\int_0^x (-t+m(t))dtdx=\int_0^1 (x-1)(x-m(x))dx. $ [b]c)[/b] Demonstrate that $ \int_0^1\int_0^x m(t)dtdx -\frac{1}{2}\int_0^1 m(x)dx\ge -\frac{1}{12} . $ [i]Gabriela Constantinescu[/i] and [i]Nelu Chichirim[/i]

2023 USEMO, 2

Tags: algebra , real analysis

Each point in the plane is labeled with a real number. Show that there exist two distinct points $P$ and $Q$ whose labels differ by less than the distance from $P$ to $Q$. [i]Holden Mui[/i]

2012 Gheorghe Vranceanu, 1

Tags: real analysis , function

Prove that the function $ f:\mathbb{R}\longrightarrow\mathbb{R} , f(x)=\text{arcsin} \frac{2x}{1+x^2} $ admits primitives and describe a primitive of it.

2006 Victor Vâlcovici, 1

Tags: real analysis , lipschitz , fixed point , short map , nonexpansive map , composition , function

Let be a nondegenerate and closed interval $ I $ of real numbers, a short map $ m:I\longrightarrow I, $ and a sequence of functions $ \left( x_n \right)_{n\ge 1} :I\longrightarrow\mathbb{R} $ such that $ x_1 $ is the identity map and $$ 2x_{n+1}=x_n+m\circ x_n , $$ for any natural numbers $ n. $ Prove that: [b]a)[/b] there exists a nondegenerate interval having the property that any point of it is a fixed point for $ m. $ [b]b)[/b] $ \left( x_n \right)_{n\ge 1} $ is pointwise convergent and that its limit function is a short map.

1975 Miklós Schweitzer, 4

Tags: real analysis , number theory , function

Prove that the set of rational-valued, multiplicative arithmetical functions and the set of complex rational-valued, multiplicative arithmetical functions form isomorphic groups with the convolution operation $ f \circ g$ defined by \[{ (f \circ g)(n)= %Error. "displatmath" is a bad command. \sum_{d|n} f(d)g(\frac nd}).\] (We call a complex number $ \textit{complex rational}$, if its real and imaginary parts are both rational.) [i]B. Csakany[/i]

1973 Putnam, A4

Tags: function , real analysis , root

How many zeroes does the function $f(x)=2^x -1 -x^2 $ have on the real line?

2003 Romania National Olympiad, 2

Tags: function , integration , real analysis , definite integral , titu andreescu

Let be an odd natural number $ n\ge 3. $ Find all continuous functions $ f:[0,1]\longrightarrow\mathbb{R} $ that satisfy the following equalities. $$ \int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n,\quad\forall k\in\{ 1,2,\ldots ,n-1\} $$ [i]Titu Andreescu[/i]

2003 IMC, 6

Tags: real analysis , convergence , limit

Let $(a_{n})$ be the sequence defined by $a_{0}=1,a_{n+1}=\sum_{k=0}^{n}\dfrac{a_k}{n-k+2}$. Find the limit \[\lim_{n \rightarrow \infty} \sum_{k=0}^{n}\dfrac{a_{k}}{2^{k}},\] if it exists.

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]

2022 Miklós Schweitzer, 5

Tags: real analysis , topology

Is it possible to select a non-degenerate segment from each line of the plane such that any two selected segments are disjoint?

2020 Putnam, A3

Tags: real analysis

Let $a_0=\pi /2$, and let $a_n=\sin (a_{n-1})$ for $n\ge 1$. Determine whether \[ \sum_{n=1}^{\infty}a_n^2 \] converges.

1986 Traian Lălescu, 2.4

Tags: function , real analysis , continuity

Prove that, if a continuous function has limits at $ \pm\infty , $ and these are equal, then it touches its maximum or minimum at one point.

2016 Brazil Undergrad MO, 1

Tags: real analysis

Let $(a_n)_{n \geq 1}$ s sequence of reals such that $\sum_{n \geq 1}{\frac{a_n}{n}}$ converges. Show that $\lim_{n \rightarrow \infty}{\frac{1}{n} \cdot \sum_{k=1}^{n}{a_k}} = 0$

2010 Miklós Schweitzer, 7

Tags: real analysis

Is there any sequence $(a_n)_{n=1}^{\infty}$ of non-negative numbers, for which $\sum_{n=1}^{\infty} a_n^2<\infty$ , but $\sum_{n=1}^{\infty}\left(\sum_{k=1}^{\infty}\frac{a_{kn}}{k} \right)^2=\infty$ ? [hide=Remark]That contest - Miklos Schweitzer 2010- is missing on the contest page here for now being. The statements of all problems that year can be found [url=http://www.math.u-szeged.hu/~mmaroti/schweitzer/]here[/url], but unfortunately only in Hungarian. I tried google translate but it was a mess. So, it would be wonderful if someone knows Hungarian and wish to translate it. [/hide]

2009 Romania National Olympiad, 4

Tags: invariant , real analysis , integration , calculus , function , 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. $$ \int_0^1 g\left( f(x) \right) dx =\int_0^1 g(x) dx $$

MIPT Undergraduate Contest 2019, 1.2

Tags: real analysis , function

Does there exist a strictly increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$ that takes on only irrational values?

2013 Romania National Olympiad, 1

Tags: real analysis , function , integration , calculus

Determine continuous functions $f:\mathbb{R}\to \mathbb{R}$ such that $\left( {{a}^{2}}+ab+{{b}^{2}} \right)\int\limits_{a}^{b}{f\left( x \right)dx=3\int\limits_{a}^{b}{{{x}^{2}}f\left( x \right)dx,}}$ for every $a,b\in \mathbb{R}$ .

2012 Miklós Schweitzer, 11

Tags: real analysis , probability

Let $X_1,X_2,..$ be independent random variables with the same distribution, and let $S_n=X_1+X_2+...+X_n, n=1,2,...$. For what real numbers $c$ is the following statement true: $$P\left(\left| \frac{S_{2n}}{2n}- c \right| \leqslant \left| \frac{S_n}{n}-c\right| \right)\geqslant \frac{1}{2}$$

2001 Romania National Olympiad, 1

Tags: derivative , real analysis , function , calculus

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: \[ f(x)=g(x)+\alpha |h(x)|,\ \forall x\in\mathbb{R} \]

2004 District Olympiad, 3

Tags: real analysis , function

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a function such that $f\left(\frac{a+b}{2}\right)\in \{f(a),f(b)\},\ (\forall)a,b\in \mathbb{R}$. a) Give an example of a non-constant function that satisfy the hypothesis. b)If $f$ is continuous, prove that $f$ is constant.

2011 Romania National Olympiad, 2

Tags: real analysis , calculus , function , derivative , abstract algebra , inequalities

[color=darkred]Let $u:[a,b]\to\mathbb{R}$ be a continuous function that has finite left-side derivative $u_l^{\prime}(x)$ in any point $x\in (a,b]$ . Prove that the function $u$ is monotonously increasing if and only if $u_l^{\prime}(x)\ge 0$ , for any $x\in (a,b]$ .[/color]

Kvant 2020, M365

Tags: real analysis , convergence , divergence

[list=a] [*]The sum of several numbers is equal to one. Can the sum of their cubes be greater than one? [*]The same question as before, for numbers not exceeding one. [*]Can it happen that the series $a_1+a_2+\cdots$ converges, but the series $a_1^3+a_2^3+\cdots$ diverges? [/list]

2005 Alexandru Myller, 2

Tags: real analysis , calculus , integration , function

Let $f:[0,1]\to\mathbb R$ be an increasing function. Prove that if $\int_0^1f(x)dx=\int_0^1\left(\int_0^xf(t)dt\right)dx=0$ then $f(x)=0,\forall x\in(0,1)$. [i]Mihai Piticari[/i]

2000 Moldova National Olympiad, Problem 2

Tags: real analysis , limit

For $n\in\mathbb N$, define $$a_n=\frac1{\binom n1}+\frac1{\binom n2}+\ldots+\frac1{\binom nn}.$$ (a) Prove that the sequence $b_n=a_n^n$ is convergent and determine the limit. (b) Show that $\lim_{n\to\infty}b_n>\left(\frac32\right)^{\sqrt3+\sqrt2}$.