Olympiad problems

2014 Miklós Schweitzer, 7

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and let $g : \mathbb{R} \to \mathbb{R}$ be arbitrary. Suppose that the Minkowski sum of the graph of $f$ and the graph of $g$ (i.e., the set $\{( x+y; f(x)+g(y) ) \mid x, y \in \mathbb{R}\}$) has Lebesgue measure zero. Does it follow then that the function $f$ is of the form $f(x) = ax + b$ with suitable constants $a, b \in \mathbb{R}$ ?

1995 IMC, 3

Tags: differentiable function , real analysis

Let $f$ be twice continuously differentiable on $(0,\infty)$ such that $\lim_{x \to 0^{+}}f'(x)=-\infty$ and $\lim_{x \to 0^{+}}f''(x)=\infty$. Show that $$\lim_{x\to 0^{+}}\frac{f(x)}{f'(x)}=0.$$

2012 Centers of Excellency of Suceava, 3

Tags: function , real analysis

Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ that has a root, and for which the line $ y=0 $ in the Cartesian plane is an horizontal asymptote. Show that $ f $ is bounded and touches its boundaries. [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

2021 Science ON all problems, 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]

2006 Mathematics for Its Sake, 3

Tags: limit of sequence , arithmetic sequence , arithmetic series , study convergence , real analysis , sequence

Let be two positive real numbers $ a,b, $ and an infinite arithmetic sequence of natural numbers $ \left( x_n \right)_{n\ge 1} . $ Study the convergence of the sequences $$ \left( \frac{1}{x_n}\sum_{i=1}^n\sqrt[x_i]{b} \right)_{n\ge 1}\text{ and } \left( \left(\sum_{i=1}^n \sqrt[x_i]{a}/\sqrt[x_i]{b} \right)^\frac{x_n}{\ln x_n} \right)_{n\ge 1} , $$ and calculate their limits. [i]Dumitru Acu[/i]

1960 Miklós Schweitzer, 6

Tags: real analysis

[b]6.[/b] Let $\{ n_k \}_{k=1}^{\infty}$ be a stricly increasing sequence of positive integers such that $\lim_{k \to \infty} n_k^{\frac {1}{2^k}}= \infty$ Show that the sum of the series $\sum_{k=1}^{\infty} \frac {1}{n_k} $ is an irrational number. [b](N. 19)[/b]

1975 Miklós Schweitzer, 7

Tags: function , calculus , derivative , analytic geometry , graphing lines , slope , real analysis

Let $ a<a'<b<b'$ be real numbers and let the real function $ f$ be continuous on the interval $ [a,b']$ and differentiable in its interior. Prove that there exist $ c \in (a,b), c'\in (a',b')$ such that \[ f(b)\minus{}f(a)\equal{}f'(c)(b\minus{}a),\] \[ f(b')\minus{}f(a')\equal{}f'(c')(b'\minus{}a'),\] and $ c<c'$. [i]B. Szokefalvi Nagy[/i]

2012 Centers of Excellency of Suceava, 3

Tags: factorial , series , definite integral , real analysis , integral sequence

Consider the sequence $ \left( I_n \right)_{n\ge 1} , $ where $ I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, $ for any natural number $ n. $ [b]a)[/b] Find a relation between any two consecutive terms of $ I_n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } nI_n. $ [i]c)[/i] Show that $ \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx. $ [i]Cătălin Țigăeru[/i]

1994 IMC, 3

Tags: real analysis , function , calculus , derivative

Let $f$ be a real-valued function with $n+1$ derivatives at each point of $\mathbb R$. Show that for each pair of real numbers $a$, $b$, $a<b$, such that $$\ln\left( \frac{f(b)+f'(b)+\cdots + f^{(n)} (b)}{f(a)+f'(a)+\cdots + f^{(n)}(a)}\right)=b-a$$ there is a number $c$ in the open interval $(a,b)$ for which $$f^{(n+1)}(c)=f(c)$$

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]

2008 Moldova MO 11-12, 8

Tags: integration , trigonometry , logarithm , real analysis

Evaluate $ \displaystyle I \equal{} \int_0^{\frac\pi4}\left(\sin^62x \plus{} \cos^62x\right)\cdot \ln(1 \plus{} \tan x)\text{d}x$.

1994 Miklós Schweitzer, 5

Tags: real analysis , measure theory

Let H be a $G_{\delta}$ subset of $\mathbb R$ whose closure has a positive Lebesgue measure. Prove that the set $H + H + H + H = \{ x + y + z + u : x , y , z , u \in H \}$ contains an interval.

2013 Gheorghe Vranceanu, 1

Tags: function , real analysis , can be done wo calculus

Find both extrema of the function $ x\to\frac{\sin x-3}{\cos x +2} .$

2012 Grigore Moisil Intercounty, 3

Tags: riemann sum , definite integral , claculus , real analysis , limit

$ \lim_{n\to\infty } \frac{1}{n}\sum_{i,j=1}^n \frac{i+j}{i^2+j^2} $

2010 Contests, 2

Tags: quadratic , real analysis , complex number , algebra , polynomial , sum of roots , number theory

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

2012 IMO Shortlist, A4

Tags: algebra , polynomial , number theory , rational roots , real analysis

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

2016 Korea USCM, 4

Tags: inequalities , real analysis , mean value theorem , function

Suppose a continuous function $f:[-\frac{\pi}{4},\frac{\pi}{4}]\to[-1,1]$ and differentiable on $(-\frac{\pi}{4},\frac{\pi}{4})$. Then, there exists a point $x_0\in (-\frac{\pi}{4},\frac{\pi}{4})$ such that $$|f'(x_0)|\leq 1+f(x_0)^2$$

2007 Today's Calculation Of Integral, 195

Tags: calculus , integration , function , real analysis , algebra , partial fractions , calculus computations

Find continuous functions $x(t),\ y(t)$ such that $\ \ \ \ \ \ \ \ \ x(t)=1+\int_{0}^{t}e^{-2(t-s)}x(s)ds$ $\ \ \ \ \ \ \ \ \ y(t)=\int_{0}^{t}e^{-2(t-s)}\{2x(s)+3y(s)\}ds$

2005 Brazil Undergrad MO, 2

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

Let $f$ and $g$ be two continuous, distinct functions from $[0,1] \rightarrow (0,+\infty)$ such that $\int_{0}^{1}f(x)dx = \int_{0}^{1}g(x)dx$ Let $y_n=\int_{0}^{1}{\frac{f^{n+1}(x)}{g^{n}(x)}dx}$, for $n\geq 0$, natural. Prove that $(y_n)$ is an increasing and divergent sequence.

1996 IMC, 4

Tags: real analysis , sequence

Let $a_{1}=1$, $a_{n}=\frac{1}{n} \sum_{k=1}^{n-1}a_{k}a_{n-k}$ for $n\geq 2$. Show that i) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}<2^{-\frac{1}{2}}$; ii) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}\geq \frac{2}{3}$

2002 IMC, 9

Tags: real analysis , summation

For each $n\geq 1$ let $$a_{n}=\sum_{k=0}^{\infty}\frac{k^{n}}{k!}, \;\; b_{n}=\sum_{k=0}^{\infty}(-1)^{k}\frac{k^{n}}{k!}.$$ Show that $a_{n}\cdot b_{n}$ is an integer.

2007 Grigore Moisil Intercounty, 3

Tags: function , real analysis , composition

Let be two functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ g $ has infinite limit at $ \infty . $ [b]a)[/b] Prove that if $ g $ continuous then $ \lim_{x\to\infty } f(x) =\lim_{x\to\infty } f(g(x)) $ [b]b)[/b] Provide an example of what $ f,g $ could be if $ f $ has no limit at $ \infty $ and $ \lim_{x\to\infty } f(g(x)) =0. $

2007 Miklós Schweitzer, 1

Tags: real analysis

Prove that there exist subfields of $\mathbb R$ that are a) non-measurable and b) of measure zero and continuum cardinality. (translated by Miklós Maróti)

2014 District Olympiad, 1

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

For each positive integer $n$ we consider the function $f_{n}:[0,n]\rightarrow{\mathbb{R}}$ defined by $f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} $, where $\left\lfloor x\right\rfloor $ denotes the floor of the real number $x$. Prove that $f_{n}$ is a Riemann Integrable function and find $\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\int_{0}^{n}f_{n}(x)\mathrm{d}x.$

1972 Miklós Schweitzer, 5

Tags: function , topology , real analysis

We say that the real-valued function $ f(x)$ defined on the interval $ (0,1)$ is approximately continuous on $ (0,1)$ if for any $ x_0 \in (0,1)$ and $ \varepsilon >0$ the point $ x_0$ is a point of interior density $ 1$ of the set \[ H\equal{} \{x : \;|f(x)\minus{}f(x_0)|< \varepsilon \ \}.\] Let $ F \subset (0,1)$ be a countable closed set, and $ g(x)$ a real-valued function defined on $ F$. Prove the existence of an approximately continuous function $ f(x)$ defined on $ (0,1)$ such that \[ f(x)\equal{}g(x) \;\textrm{for all}\ \;x \in F\ .\] [i]M. Laczkovich, Gy. Petruska[/i]