Olympiad problems

2013 China Team Selection Test, 1

Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]

1999 Romania National Olympiad, 3

Tags: real analysis , mean value theorem

Let $a,b \in \mathbb{R},$ $a<b$ and $f,g:[a,b] \to \mathbb{R}$ two differentiable functions with increasing derivatives and $f'(a)>0,$ $g'(a)>0.$ Prove that there exists $c \in [a,b]$ such that $$\frac{f(b)-f(a)}{b-a} \cdot \frac{g(b)-g(a)}{b-a}=f'(c)g'(c).$$

2005 Harvard-MIT Mathematics Tournament, 4

Tags: function , real analysis , derivative , calculus

Let $ f : \mathbf {R} \to \mathbf {R} $ be a smooth function such that $ f'(x)^2 = f(x) f''(x) $ for all $x$. Suppose $f(0)=1$ and $f^{(4)} (0) = 9$. Find all possible values of $f'(0)$.

2018 Putnam, A5

Tags: taylor series , real analysis

Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1) = 1$, and $f(x) \ge 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$.

2018 Romania National Olympiad, 2

Tags: function , real analysis , integration , calculus

Let $\mathcal{F}$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R}$$ For $f \in \mathcal{F},$ let $$I(f)=\int_0^ef(x) dx$$ Determine $\min_{f \in \mathcal{F}}I(f).$ [i]Liviu Vlaicu[/i]

1977 Miklós Schweitzer, 6

Tags: real analysis , logarithm , function

Let $ f$ be a real function defined on the positive half-axis for which $ f(xy)\equal{}xf(y)\plus{}yf(x)$ and $ f(x\plus{}1) \leq f(x)$ hold for every positive $ x$ and $ y$. Show that if $ f(1/2)\equal{}1/2$, then \[ f(x)\plus{}f(1\minus{}x) \geq \minus{}x \log_2 x \minus{}(1\minus{}x) \log_2 (1\minus{}x)\] for every $ x\in (0,1)$. [i]Z. Daroczy, Gy. Maksa[/i]

2017 Korea USCM, 7

Tags: trigonometry , real analysis , series , trigonometric series , inequalities

Prove the following inequality holds if $\{a_n\}$ is a deceasing sequence of positive reals, and $0<\theta<\frac{\pi}{2}$. $$\left|\sum_{n=1}^{2017} a_n \cos n\theta \right| \leq \frac{\pi a_1}{\theta}$$

1959 Miklós Schweitzer, 2

Tags: real analysis , topology , geometry

[b]2.[/b] Omit the vertices of a closed rectangle; the configuration obtained in such a way will be called a reduced rectangle. Prove tha the set-union of any system of reduced rectangles with parallel sides is equal to the union of countably many elements of the system. [b](St. 3)[/b]

2000 VJIMC, Problem 1

Tags: real analysis , set , set theory , infinity

Is there a countable set $Y$ and an uncountable family $\mathcal F$ of its subsets such that for every two distinct $A,B\in\mathcal F$, their intersection $A\cap B$ is finite?

2003 IMC, 5

Tags: real analysis , function

Let $g:[0,1]\rightarrow \mathbb{R}$ be a continuous function and let $f_{n}:[0,1]\rightarrow \mathbb{R}$ be a sequence of functions defined by $f_{0}(x)=g(x)$ and $$f_{n+1}(x)=\frac{1}{x}\int_{0}^{x}f_{n}(t)dt.$$ Determine $\lim_{n\to \infty}f_{n}(x)$ for every $x\in (0,1]$.

2007 District Olympiad, 3

Tags: floor function , limit , real analysis

Let $a,b\in \mathbb{R}$. Evaluate: \[\lim_{n\to \infty}\left(\sqrt{a^2n^2+bn}-an\right)\] Consider the sequence $(x_n)_{n\ge 1}$, defined by $x_n=\sqrt{n}-\lfloor \sqrt{n}\rfloor$. Denote by $A$ the set of the points $x\in \mathbb{R}$, for which there is a subsequence of $(x_n)_{n\ge 1}$ tending to $x$. a) Prove that $\mathbb{Q}\cap [0,1]\subset A$. b) Find $A$.

2021 Miklós Schweitzer, 6

Tags: real analysis , fourier analysis

Let $f$ and $g$ be $2 \pi$-periodic integrable functions such that in some neighborhood of $0$, $g(x) = f(ax)$ with some $a \neq 0$. Prove that the Fourier series of $f$ and $g$ are simultaneously convergent or divergent at $0$.

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) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}\geq \frac{2}{3}$

2020 Jozsef Wildt International Math Competition, W20

Tags: real analysis , sequence , asymptotics

Let $p\in(0,1)$ and $a>0$ be real numbers. Determine the asymptotic behavior of the sequence $\{a_n\}_{n=1}^\infty$ defined recursively by $$a_1=a,a_{n+1}=\frac{a_n}{1+a_n^p},n\in\mathbb N$$ [i]Proposed by Arkady Alt[/i]

1979 Miklós Schweitzer, 8

Tags: integration , function , real analysis

Let $ K_n(n=1,2,\ldots)$ be periodical continuous functions of period $ 2 \pi$, and write \[ k_n(f;x)= \int_0^{2\pi}f(t)K_n(x-t)dt .\] Prove that the following statements are equivalent: (i) $ \int_0^{2\pi}|k_n(f;x)-f(x)|dx \rightarrow 0 \;(n \rightarrow \infty)$ for all $ f \in \mathcal{L}_1[0,2 \pi]$. (ii) $ k_n(f;0) \rightarrow f(0)$ for all continuous, $ 2 \pi$-periodic functions $ f$. [i]V. Totik[/i]

2014 Contests, 4

Tags: real analysis

For a positive integer $n$, define $f(n)$ to be the number of sequences $(a_1,a_2,\dots,a_k)$ such that $a_1a_2\cdots a_k=n$ where $a_i\geq 2$ and $k\ge 0$ is arbitrary. Also we define $f(1)=1$. Now let $\alpha>1$ be the unique real number satisfying $\zeta(\alpha)=2$, i.e $ \sum_{n=1}^{\infty}\frac{1}{n^\alpha}=2 $ Prove that [list] (a) \[ \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\alpha) \] (b) There is no real number $\beta<\alpha$ such that \[ \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\beta) \] [/list]

2013 Taiwan TST Round 1, 2

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

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.

2022 Germany Team Selection Test, 1

Tags: algebra , real analysis , inequalities

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

1993 Vietnam National Olympiad, 1

Tags: calculus , real analysis , algebra , trigonometry

$f : [-\sqrt{1995},\sqrt{1995}] \to\mathbb{R}$ is defined by $f(x) = x(1993+\sqrt{1995-x^{2}})$. Find its maximum and minimum values.

2016 Romania National Olympiad, 4

Tags: real analysis , function , find all functions , differentiability , darboux

Find all functions, $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ that have the properties that $ f^2 $ is differentiable and $ f=\left( f^2 \right)' . $

2003 Alexandru Myller, 4

Tags: functional equation , real analysis , function , find all functions

Find the differentiable functions $ f:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ that verify $ f(0)=0 $ and $$ f'(x)=1/3\cdot f'\left( x/3 \right) +2/3\cdot f'\left( 2x/3 \right) , $$ for any nonnegative real number $ x. $

1970 Miklós Schweitzer, 8

Tags: polynomial , algebra , real analysis

Let $ \pi_n(x)$ be a polynomial of degree not exceeding $ n$ with real coefficients such that \[ |\pi_n(x)| \leq \sqrt{1\minus{}x^2} \;\textrm{for}\ \;\minus{}1\leq x \leq 1 \ .\] Then \[ |\pi'_n(x)| \leq 2(n\minus{}1).\] [i]P. Turan[/i]

1969 Miklós Schweitzer, 9

Tags: real analysis

In $ n$-dimensional Euclidean space, the union of any set of closed balls (of positive radii) is measurable in the sense of Lebesgue. [i]A. Csaszar[/i]

2016 Brazil Undergrad MO, 6

Tags: real analysis

Let it $C,D > 0$. We call a function $f:\mathbb{R} \rightarrow \mathbb{R}$ [i]pretty[/i] if $f$ is a $C^2$-class, $|x^3f(x)| \leq C$ and $|xf''(x)| \leq D$. [list='i'] [*] Show that if $f$ is pretty, then, given $\epsilon \geq 0$, there is a $x_0 \geq 0$ such that for every $x$ with $|x| \geq x_0$, we have $|x^2f'(x)| < \sqrt{2CD}+\epsilon$. [*] Show that if $0 < E < \sqrt{2CD}$ then there is a pretty function $f$ such that for every $x_0 \geq 0$ there is a $x > x_0$ such that $|x^2f'(x)| > E$. [/list]

1981 Miklós Schweitzer, 7

Tags: real analysis , induction , abstract algebra

Let $ U$ be a real normed space such that, for an finite-dimensional, real normed space $ X,U$ contains a subspace isometrically isomorphic to $ X$. Prove that every (not necessarily closed) subspace $ V$ of $ U$ of finite codimension has the same property. (We call $ V$ of finite codimension if there exists a finite-dimensional subspace $ N$ of $ U$ such that $ V\plus{}N\equal{}U$.) [i]A. Bosznay[/i]