Olympiad problems

2015 Romania National Olympiad, 3

Let be two nonnegative real numbers $ a,b $ with $ b>a, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ of real numbers such that the sequence $ \left( \frac{x_1+x_2+\cdots +x_n}{n^a} \right)_{n\ge 1} $ is bounded. Show that the sequence $ \left( x_1+\frac{x_2}{2^b} +\frac{x_3}{3^b} +\cdots +\frac{x_n}{n^b} \right)_{n\ge 1} $ is convergent.

1996 Romania National Olympiad, 4

Tags: real analysis , integral , monotone functions , limit , function , calculus

Let $f:[0,1) \to \mathbb{R}$ be a monotonic function. Prove that the limits [center]$\lim_{x \nearrow 1} \int_0^x f(t) \mathrm{d}t$ and $\lim_{n \to \infty} \frac{1}{n} \left[ f(0) + f \left(\frac{1}{n}\right) + \ldots + f \left( \frac{n-1}{n} \right) \right]$[/center] exist and are equal.

2014 Paenza, 6

Tags: real analysis , integration , function

(a) Show that if $f:[-1,1]\to \mathbb{R}$ is a convex and $C^2$ function such that $f(1),f(-1)\geq 0$, then: \[\min_{x\in[-1,1]} \{f(x)\} \geq - \int_{-1}^1 f''\] (b) Let $B\subset \mathbb{R}^2$ the closed ball with center $0$ and radius $1$. Show that if $f: B \to \mathbb{R}$ is a convex and $C^2$ function and $f\geq 0$ in $\partial B$, then: \[f(0)\geq -\frac{1}{\sqrt{\pi}} \left( \int_{B} (f_{xx}f_{yy}-f_{xy}^2) \right)^{1/2}\]

2000 District Olympiad (Hunedoara), 3

Tags: sequence , limit , real analysis , calculus

Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies $$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$ Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $

2021 Science ON grade XII, 3

Tags: integrability , riemann integrability , function , real analysis

Define $E\subseteq \{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$ such that $E$ posseses the following properties:\\ $\textbf{(i)}$ If $\int_0^1 f(x)g(x) dx = 0$ for $f\in E$ with $\int_0^1f^2(t)dt \neq 0$, then $g\in E$; \\ $\textbf{(ii)}$ There exists $h\in E$ with $\int_0^1 h^2(t)dt\neq 0$.\\ Prove that $E=\{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$. \\ [i](Andrei Bâra)[/i]

2012 Miklós Schweitzer, 1

Tags: recursive , computable analysis , discrete mathematics , real analysis

Is there any real number $\alpha$ for which there exist two functions $f,g: \mathbb{N} \to \mathbb{N}$ such that $$\alpha=\lim_{n \to \infty} \frac{f(n)}{g(n)},$$ but the function which associates to $n$ the $n$-th decimal digit of $\alpha$ is not recursive?

2008 District Olympiad, 4

Tags: real analysis , function

Find the values of $a\in [0,\infty)$ for which there exist continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$, such that $f(f(x))=(x-a)^2,\ (\forall)x\in \mathbb{R}$.

1997 Romania National Olympiad, 2

Tags: real analysis , calculus , integration , function

Prove that: $\int_{-1}^1f^2(x)dx\ge \frac 1 2 (\int_{-1}^1f(x)dx)^2 +\frac 3 2(\int_{-1}^1xf(x)dx)^2$ Please give a proof without using even and odd functions. (the oficial proof uses those and seems to be un-natural) :D

2008 Moldova MO 11-12, 2

Tags: real analysis , integration , trigonometry

Find the exact value of $ E\equal{}\displaystyle\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x\cdot$.

1997 IMC, 2

Tags: real analysis

Let $a_n$ be a sequence of reals. Suppose $\sum a_n$ converges. Do these sums converge aswell? (a) $a_1+a_2+(a_4+a_3)+(a_8+...+a_5)+(a_{16}+...+a_9)+...$ (b) ${a_1+a_2+(a_3)+(a_4)+(a_5+a_7)+(a_6+a_8)+(a_9+a_{11}+a_{13}+a_{15})+(a_{10}+a_{12}+a_{14}+a_{16})+(a_{17}+a_{19}+...}$

1995 Miklós Schweitzer, 9

Tags: real analysis

A serpentine is a sequence of points $P_1 , ..., P_m$ in a plane, not necessarily all different, such that the distance between $P_i$ and $P_{i+1}$ is at least 1, and the segments $P_i P_{i +1}$ are alternately horizontal and vertical. Construct a compact set in which there is a sequence of serpentines with arbitrary long lengths but there is no closed serpentine ($P_m = P_i$ for some i < m).

2010 Laurențiu Panaitopol, Tulcea, 1

Tags: differentiability , primitivableness , antiderivative , integral , indefinite integral , real analysis , function

Let be two real numbers $ a<b $ and a function $ f:[a,b]\longrightarrow\mathbb{R} $ having the property that if the sequence $ \left(f\left( x_n \right)\right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. [b]a)[/b] Prove that if $ f $ admits antiderivatives, then $ f $ is integrable. [b]b)[/b] Is the converse of [b]a)[/b] true? [i]Marcelina Popa[/i]

1986 Traian Lălescu, 1.4

Tags: darboux , function , real analysis

Let $ f:(0,1)\longrightarrow \mathbb{R} $ be a bounded function having the property of Darboux. Then: [b]a)[/b] There exists $ g:[0,1)\longrightarrow\mathbb{R} $ with Darboux’s property such that $ g\bigg|_{(0,1)} =f\bigg|_{(0,1)} . $ [b]b)[/b] The function above is uniquely determined if and only if $ f $ has limit at $ 0. $

1967 Miklós Schweitzer, 6

Tags: vector , functional analysis , real analysis

Let $ A$ be a family of proper closed subspaces of the Hilbert space $ H\equal{}l^2$ totally ordered with respect to inclusion (that is , if $ L_1,L_2 \in A$, then either $ L_1\subset L_2$ or $ L_2\subset L_1$). Prove that there exists a vector $ x \in H$ not contaied in any of the subspaces $ L$ belonging to $ A$. [i]B. Szokefalvi Nagy[/i]

2002 IMC, 2

Tags: function , real analysis

Does there exist a continuously differentiable function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for every $x \in \mathbb{R}$ we have $f(x) > 0$ and $f'(x) = f(f(x))$?

2019 Romania National Olympiad, 2

Tags: real analysis , calculus , function

Let $f:[0, \infty) \to \mathbb{R}$ a continuous function, constant on $\mathbb{Z}_{\geq 0}.$ For any $0 \leq a < b < c < d$ which satisfy $f(a)=f(c)$ and $f(b)=f(d)$ we also have $f \left( \frac{a+b}{2} \right) = f \left( \frac{c+d}{2} \right).$ Prove that $f$ is constant.

1980 IMO, 4

Tags: real analysis , limit

Given a real number $x>1$, prove that there exists a real number $y >0$ such that \[\lim_{n \to \infty} \underbrace{\sqrt{y+\sqrt {y + \cdots+\sqrt y}}}_{n \text{ roots}}=x.\]

2010 Contests, 3

Tags: real analysis , limit , function

Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$. Compute $\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.

2014 ISI Entrance Examination, 6

Tags: real analysis , rotation , trigonometry

Define $\mathcal{A}=\{(x,y)|x=u+v,y=v, u^2+v^2\le 1\}$. Find the length of the longest segment that is contained in $\mathcal{A}$.

1993 Miklós Schweitzer, 5

Tags: ordered set , measure theory , real analysis

Does the set of real numbers have a well-order $\prec$ such that the intersection of the subset $\{(x,y) : x\prec y\}$ of the plane with every line is Lebesgue measurable on the line?

1971 Miklós Schweitzer, 3

Tags: real analysis

Let $ 0<a_k<1$ for $ k=1,2,... .$ Give a necessary and sufficient condition for the existence, for every $ 0<x<1$, of a permutation $ \pi_x$ of the positive integers such that \[ x= \sum_{k=1}^{\infty} \frac{a_{\pi_x}(k)}{2^k}.\] [i]P. Erdos[/i]

2010 Today's Calculation Of Integral, 547

Tags: logarithm , real analysis , derivative , integration , function , calculus computations , calculus

Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$.

2004 VTRMC, Problem 7

Tags: real analysis , limit

Let $\{a_n\}$ be a sequence of positive real numbers such that $\lim_{n\to\infty}a_n=0$. Prove that $\sum^\infty_{n=1}\left|1-\frac{a_{n+1}}{a_n}\right|$ is divergent.

1979 Miklós Schweitzer, 10

Tags: real analysis , function

Prove that if $ a_i(i=1,2,3,4)$ are positive constants, $ a_2-a_4 > 2$, and $ a_1a_3-a_2 > 2$, then the solution $ (x(t),y(t))$ of the system of differential equations \[ \.{x}=a_1-a_2x+a_3xy,\] \[ \.{y}=a_4x-y-a_3xy \;\;\;(x,y \in \mathbb{R}) \] with the initial conditions $ x(0)=0, y(0) \geq a_1$ is such that the function $ x(t)$ has exactly one strict local maximum on the interval $ [0, \infty)$. [i]L. Pinter, L. Hatvani[/i]

2011 N.N. Mihăileanu Individual, 3

Tags: function , real analysis , continuity

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function having the property that $$ f(f(x))=f(x)-\frac{1}{4}x +1, $$ for all real numbers $ x. $ [b]a)[/b] Prove that $ f $ is increasing. [b]b)[/b] Show that the equation $ f(x)=ax $ has at least a real solution in $ x, $ for any real number $ a\ge 1. $ [b]c)[/b] Calculate $ \lim_{x\to\infty } \frac{f(x)}{x} $ supposing that it exists, it's finite, and that $ \lim_{x\to\infty } f(f(x))=\infty . $