Olympiad problems

1999 Miklós Schweitzer, 6

Show that for every real function f in 1-period $L^2(0, 1)$ there exist three functions $g_1, g_2, g_3$ with the same properties and constants $c_0, c_1, c_2, c_3$ satisfying $$f(x)=c_0+\sum_{i=1}^3(g_i(x+c_i)-g_i(x))$$

2001 Romania National Olympiad, 4

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

Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$. a) Show that: \[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\] b) Show that: \[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]

2011 Laurențiu Duican, 3

Tags: real analysis , function , continuity , fractional part

Let be two continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R} $ satisfying the following equations: $$ \lim_{x\to\infty } f(x) =\infty =\lim_{x\to\infty } g(x) $$ Prove that there exists a divergent sequence $ \left( k_n \right)_{n\ge 1} $ of nonnegative integers which has the property that each term (function) of the sequence of functions $ \left( h_{n} \right)_{n\ge 1} :[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ h_{n} (x) =f\left( k_n+g(x) -\left\lfloor g(x) \right\rfloor \right) , $$ doesn't have limit at $ \infty . $ [i]Romeo Ilie[/i]

1950 Miklós Schweitzer, 9

Tags: real analysis , calculus

Find the sum of the series $ x\plus{}\frac{x^3}{1\cdot 3}\plus{}\frac{x^5}{1\cdot 3\cdot 5}\plus{}\cdots\plus{}\frac{x^{2n\plus{}1}}{1\cdot 3\cdot 5\cdot \cdots \cdot (2n\plus{}1)}\plus{}\cdots$

2003 District Olympiad, 2

Tags: real analysis , function

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]

2007 Grigore Moisil Intercounty, 2

Tags: function , real analysis

Prove that $ |f(x)|\le |f(0)| +\int_0^x |f(t) +f'(t)|dt , $ for any nonnegative real numbers $ x, $ and functions $f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ of class $ \mathcal{C}^1. $

2021 Simon Marais Mathematical Competition, A4

Tags: calculus , real analysis

For each positive real number $r$, define $a_0(r) = 1$ and $a_{n+1}(r) = \lfloor ra_n(r) \rfloor$ for all integers $n \ge 0$. (a) Prove that for each positive real number $r$, the limit \[ L(r) = \lim_{n \to \infty} \frac{a_n(r)}{r^n} \] exists. (b) Determine all possible values of $L(r)$ as $r$ varies over the set of positive real numbers. [i]Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.[/i]

2011 N.N. Mihăileanu Individual, 4

Tags: real analysis , integration , calculus

Let be the sequence $ \left( I_n \right)_{n\ge 1} $ defined as $ I_n=\int_0^1 \frac{x^n}{\sqrt{x^{2n} +1}} dx . $ [b]a)[/b] Show that $ \left( I_n \right)_{n\ge 1} $ converges to $ 0. $ [b]b)[/b] Calculate $ \lim_{m\to\infty } m\cdot I_m. $ [b]c)[/b] Prove that the sequence $ \left( n\left( -n\cdot I_n +\lim_{m\to\infty } m\cdot I_m \right) \right)_{n\ge 1} $ is convergent.

2000 Romania National Olympiad, 4

Tags: real analysis , function , limit , darboux , epsilon-delta , second iterate

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that satisfies the conditions: $ \text{(i)}\quad \lim_{x\to\infty} (f\circ f) (x) =\infty =-\lim_{x\to -\infty} (f\circ f) (x) $ $ \text{(ii)}\quad f $ has Darboux’s property [b]a)[/b] Prove that the limits of $ f $ at $ \pm\infty $ exist. [b]b)[/b] Is possible for the limits from [b]a)[/b] to be finite?

2014 Contests, 4

Tags: real analysis , trigonometry

Let $f,g$ are defined in $(a,b)$ such that $f(x),g(x)\in\mathcal{C}^2$ and non-decreasing in an interval $(a,b)$ . Also suppose $f^{\prime \prime}(x)=g(x),g^{\prime \prime}(x)=f(x)$. Also it is given that $f(x)g(x)$ is linear in $(a,b)$. Show that $f\equiv 0 \text{ and } g\equiv 0$ in $(a,b)$.

2017 VJIMC, 2

Tags: function , real analysis

Prove or disprove the following statement. If $g:(0,1) \to (0,1)$ is an increasing function and satisfies $g(x) > x$ for all $x \in (0,1)$, then there exists a continuous function $f:(0,1) \to \mathbb{R}$ satisfying $f(x) < f(g(x)) $ for all $x \in (0,1)$, but $f$ is not an increasing function.

2011 Miklós Schweitzer, 9

Tags: real analysis , differential equation , limit

Let $x: [0, \infty) \to\Bbb R$ be a differentiable function. Prove that if for all t>1 $$x'(t)=-x^3(t)+\frac{t-1}{t}x^3(t-1)$$ then $\lim_{t\to\infty} x(t) = 0$

1970 Miklós Schweitzer, 7

Tags: function , real analysis

Let us use the word $ N$-measure for nonnegative, finitely additive set functions defined on all subsets of the positive integers, equal to $ 0$ on finite sets, and equal to $ 1$ on the whole set. We say that the system $ \Upsilon$ of sets determines the $ N$-measure $ \mu$ if any $ N$-measure coinciding with $ \mu$ on all elements of $ \Upsilon$ is necessarily identical with $ \mu$. Prove the existence of an $ N$-measure $ \mu$ that cannot be determined by a system of cardinality less than continuum. [i]I. Juhasz[/i]

1998 IberoAmerican Olympiad For University Students, 1

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

The definite integrals between $0$ and $1$ of the squares of the continuous real functions $f(x)$ and $g(x)$ are both equal to $1$. Prove that there is a real number $c$ such that \[f(c)+g(c)\leq 2\]

2011 Gheorghe Vranceanu, 2

Tags: real analysis , function , lagrange , darboux , weierstrass , darboux s property

Let $ f:[0,1]\longrightarrow (0,\infty ) $ be a continuous function and $ \left( b_n \right)_{n\ge 1} $ be a sequence of numbers from the interval $ (0,1) $ that converge to $ 0. $ [b]a)[/b] Demonstrate that for any fixed $ n, $ the equation $ F(x)=b_nF(1)+\left( 1-b_n\right) F(0) $ has an unique solution, namely $ x_n, $ where $ F $ is a primitive of $ f. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{x_n}{b_n} . $

2015 VJIMC, 3

Tags: real analysis , summation , convergence , infinite sum

[b]Problem 3[/b] Determine the set of real values of $x$ for which the following series converges, and find its sum: $$\sum_{n=1}^{\infty} \left(\sum_{\substack{k_1, k_2,\ldots , k_n \geq 0\\ 1\cdot k_1 + 2\cdot k_2+\ldots +n\cdot k_n = n}} \frac{(k_1+\ldots+k_n)!}{k_1!\cdot \ldots \cdot k_n!} x^{k_1+\ldots +k_n} \right) \ . $$

2004 Romania National Olympiad, 3

Tags: real analysis , function

Let $f : (a,b) \to \mathbb R$ be a function with the property that for all $x \in (a,b)$ there is a non-degenerated interval $[ a_x,b_x ]$ with $a < a_x \leq x \leq b_x < b$ such that $f$ is constant on $\left[ a_x,b_x \right]$. (a) Prove that $\textrm{Im} \, f$ is finite or numerable. (b) Find all continuous functions which have the property mentioned in the hypothesis.

2007 Romania National Olympiad, 2

Tags: real analysis , integration , calculus

Let $f: [0,1]\rightarrow(0,+\infty)$ be a continuous function. a) Show that for any integer $n\geq 1$, there is a unique division $0=a_{0}<a_{1}<\ldots<a_{n}=1$ such that $\int_{a_{k}}^{a_{k+1}}f(x)\, dx=\frac{1}{n}\int_{0}^{1}f(x)\, dx$ holds for all $k=0,1,\ldots,n-1$. b) For each $n$, consider the $a_{i}$ above (that depend on $n$) and define $b_{n}=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}$. Show that the sequence $(b_{n})$ is convergent and compute it's limit.

2001 District Olympiad, 4

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

a)Prove that $\ln(1+x)\le x,\ (\forall)x\ge 0$. b)Let $a>0$. Prove that: \[\lim_{n\to \infty} n\int_0^1\frac{x^n}{a+x^n}dx=\ln \frac{a+1}{a}\] [i]***[/i]

2023 Brazil Undergrad MO, 6

Tags: real analysis , sequence , algebra

Determine all pairs $(c, d) \in \mathbb{R}^2$ of real constants such that there is a sequence $(a_n)_{n\geq1}$ of positive real numbers such that, for all $n \geq 1$, $$a_n \geq c \cdot a_{n+1} + d \cdot \sum_{1 \leq j < n} a_j .$$

2003 VJIMC, Problem 3

Tags: sequence , real analysis , limit

Let $\{a_n\}^\infty_{n=0}$ be the sequence of real numbers satisfying $a_0=0$, $a_1=1$ and $$a_{n+2}=a_{n+1}+\frac{a_n}{2^n}$$for every $n\ge0$. Prove that $$\lim_{n\to\infty}a_n=1+\sum_{n=1}^\infty\frac1{2^{\frac{n(n-1)}2}\displaystyle\prod_{k=1}^n(2^k-1)}.$$

1997 VJIMC, Problem 4-M

Tags: real analysis , limit , digit , number theory

Find all real numbers $a>0$ for which the series $$\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}$$is convergent; $f(n)$ denotes the number of $0$'s in the decimal expansion of $f$.

2010 Laurențiu Panaitopol, Tulcea, 2

Tags: real analysis , function , primitivableness , integration

Let be a real number $ c $ and a differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that $$ f(c)\neq \frac{1}{b-a}\int_a^b f(x)dx, $$ for any real numbers $ a\neq b. $ Prove that $ f'(c)=0. $ [i]Florin Rotaru[/i]

2009 Romania National Olympiad, 1

Tags: real analysis , limit

Let $(t_n)_n$ a convergent sequence of real numbers, $t_n\in (0,1),\ (\forall)n\in \mathbb{N}$ and $\lim_{n\to \infty} t_n\in (0,1)$. Define the sequences $(x_n)_n$ and $(y_n)_n$ by \[x_{n+1}=t_nx_n+(1-t_n)y_n,\ y_{n+1}=(1-t_n)x_n+t_n y_n,\ (\forall)n\in \mathbb{N}\] and $x_0,y_0$ are given real numbers. a) Prove that the sequences $(x_n)_n$ and $(y_n)_n$ are convergent and have the same limit. b) Prove that if $\lim_{n\to \infty} t_n\in \{0,1\}$, then the question is false.

1970 Miklós Schweitzer, 10

Tags: real analysis , limit

Prove that for every $ \vartheta$, $ 0<\vartheta<1$, there exist a sequence $ \lambda_n$ of positive integers and a series $ \sum_{n=1}^{\infty} a_n$ such that (i)$ \lambda_{n+1}-\lambda_n > (\lambda_n)^{\vartheta}$, (ii) $ \lim_{r\rightarrow 1^-} \sum_{n=1}^{\infty} a_nr^{\lambda_n}$ exists, (iii) $ \sum _{n=1}^{\infty} a_n$ is divergent. [i]P. Turan[/i]