Olympiad problems

1975 Miklós Schweitzer, 9

Let $ l_0,c,\alpha,g$ be positive constants, and let $ x(t)$ be the solution of the differential equation \[ ([l_0\plus{}ct^{\alpha}] ^2x')'\plus{}g[l_0\plus{}ct^{\alpha}] \sin x\equal{}0, \;t \geq 0,\ \;\minus{}\frac{\pi}{2} <x< \frac{\pi}{2},\] satisfying the initial conditions $ x(t_0)\equal{}x_0, \;x'(t_0)\equal{}0$. (This is the equation of the mathematical pendulum whose length changes according to the law $ l\equal{}l_0\plus{}ct^{\alpha}$.) Prove that $ x(t)$ is defined on the interval $ [t_0,\infty)$; furthermore, if $ \alpha >2$ then for every $ x_0 \not\equal{} 0$ there exists a $ t_0$ such that \[ \liminf_{t \rightarrow \infty} |x(t)| >0.\] [i]L. Hatvani[/i]

1977 Miklós Schweitzer, 6

Tags: function , logarithm , real analysis

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]

1985 Miklós Schweitzer, 10

Tags: set theory , real analysis

Show that any two intervals $A, B\subseteq \mathbb R$ of positive lengths can be countably disected into each other, that is, they can be written as countable unions $A=A_1\cup A_2\cup\ldots\,$ and $B=B_1\cup B_2\cup\ldots\,$ of pairwise disjoint sets, where $A_i$ and $B_i$ are congruent for every $i\in \mathbb N$ [Gy. Szabo]

2010 Contests, 3

Tags: limit , trigonometry , real analysis

Let $(x_n)_{n \in \mathbb{N}}$ be the sequence defined as $x_n = \sin(2 \pi n! e)$ for all $n \in \mathbb{N}$. Compute $\lim_{n \to \infty} x_n$.

1994 IMC, 2

Tags: real analysis

Let $f\colon \mathbb R ^2 \rightarrow \mathbb R$ be given by $f(x,y)=(x^2-y^2)e^{-x^2-y^2}$. a) Prove that $f$ attains its minimum and its maximum. b) Determine all points $(x,y)$ such that $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0$ and determine for which of them $f$ has global or local minimum or maximum.

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) ${a_1+a_2+(a_3)+(a_4)+(a_5+a_7)+(a_6+a_8)+(a_9+a_{11}+a_{13}+a_{15})+(a_{10}+a_{12}+a_{14}+a_{16})+(a_{17}+a_{19}+...}$

1961 Putnam, A2

Tags: function , real analysis

For a real-valued function $f(x,y)$ of two positive real variables $x$ and $y$, define $f$ to be [i]linearly bounded[/i] if and only if there exists a positive number $K$ such that $|f(x,y)| < K(x+y)$ for all positive $x$ and $y.$ Find necessary and sufficient conditions on the real numbers $\alpha$ and $\beta$ such that $x^{\alpha}y^{\beta}$ is linearly bounded.

2025 District Olympiad, P3

Tags: function , limit , real analysis

Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuous and bijective function, such that $$\lim_{x\rightarrow\infty}\frac{f^{-1}(f(x)/x)}{x}=1.$$ [list=a] [*] Show that $\lim_{x\rightarrow\infty}\frac{f(x)}{x}=\infty$ and $\lim_{x\rightarrow\infty}\frac{f^{-1}(ax)}{f^{-1}(x)}=1$ for any $a>0$. [*] Give an example of function which satisfies the hypothesis.

2022 CIIM, 5

Tags: vector , limit , limit sequence , norm , real analysis

Define in the plane the sequence of vectors $v_1, v_2, \ldots$ with initial values $v_1 = (1, 0)$, $v_2 = (-1/\sqrt{2}, 1/\sqrt{2})$ and satisfying the relationship $$v_n=\frac{v_{n-1}+v_{n-2}}{\lVert v_{n-1}+v_{n-2}\rVert},$$ for $n \geq 3$. Show that the sequence is convergent and determine its limit. [b]Note:[/b] The expression $\lVert v \rVert$ denotes the length of the vector $v$.

ICMC 6, 6

Tags: real analysis , convergence , floor function , sequence

Consider the sequence defined by $a_1 = 2022$ and $a_{n+1} = a_n + e^{-a_n}$ for $n \geq 1$. Prove that there exists a positive real number $r$ for which the sequence $$\{ra_1\}, \{ra_{10}\}, \{ra_{100}\}, . . . $$converges. [i]Note[/i]: $\{x \} = x - \lfloor x \rfloor$ denotes the part of $x$ after the decimal point. [i]Proposed by Ethan Tan[/i]

1964 Miklós Schweitzer, 9

Tags: function , topology , real analysis

Let $ E$ be the set of all real functions on $ I\equal{}[0,1]$. Prove that one cannot define a topology on $ E$ in which $ f_n\rightarrow f$ holds if and only if $ f_n$ converges to $ f$ almost everywhere.

2019 Korea USCM, 6

Tags: real analysis , inequalities , integral inequality , function

A function $f:[0,\infty)\to[0,\infty)$ is integrable and $$\int_0^\infty f(x)^2 dx<\infty,\quad \int_0^\infty xf(x) dx <\infty$$ Prove the following inequality. $$\left(\int_0^\infty f(x) dx \right)^3 \leq 8\left(\int_0^\infty f(x)^2 dx \right) \left(\int_0^\infty xf(x) dx \right)$$

2023 CIIM, 4

Tags: limit superior , limit , sum of divisors , real analysis

For a positive integer $n$, $\sigma(n)$ denotes the sum of the positive divisors of $n$. Determine $$\limsup\limits_{n\rightarrow \infty} \frac{\sigma(n^{2023})}{(\sigma(n))^{2023}}$$ [b]Note:[/b] Given a sequence ($a_n$) of real numbers, we say that $\limsup\limits_{n\rightarrow \infty} a_n = +\infty$ if ($a_n$) is not upper bounded, and, otherwise, $\limsup\limits_{n\rightarrow \infty} a_n$ is the smallest constant $C$ such that, for every real $K > C$, there is a positive integer $N$ with $a_n < K$ for every $n > N$.

1964 Miklós Schweitzer, 7

Tags: invariant , function , real analysis

Find all linear homogeneous differential equations with continuous coefficients (on the whole real line) such that for any solution $ f(t)$ and any real number $ c,f(t\plus{}c)$ is also a solution.

2019 LIMIT Category B, Problem 1

Tags: real analysis

Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for $n\ge2$. Define $$p_n=\prod_{i=1}^n\left(1+\frac1{a_i}\right)$$Then $\lim_{n\to\infty}p_n$ is $\textbf{(A)}~1+e$ $\textbf{(B)}~e$ $\textbf{(C)}~1$ $\textbf{(D)}~\infty$

2001 District Olympiad, 4

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

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]

1986 Traian Lălescu, 1.3

Tags: algebra , polynom , supremum , polynomial function , real analysis , extremal

Let be four real numbers. Find the polynom of least degree such that two of these numbers are some locally extreme values, and the other two are the respective points of local extrema.

2015 District Olympiad, 4

Tags: real analysis , epsilon-delta , contest , sequence

Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of real numbers of the interval $ [1,\infty) . $ Suppose that the sequence $ \left( \left[ x_n^k\right]\right)_{n\ge 1} $ is convergent for all natural numbers $ k. $ Prove that $ \left( x_n\right)_{n\ge 1} $ is convergent. Here, $ [\beta ] $ means the greatest integer smaller than $ \beta . $

2006 Petru Moroșan-Trident, 2

Tags: real analysis , limit of sequence , cesaro-stolz , sequence

Study the convergence of the sequence $$ \left( \sum_{k=2}^{n+1} \sqrt[k]{n+1} -\sum_{k=2}^{n} \sqrt[k]{n} \right)_{n\ge 2} , $$ and calculate its limit. [i]Dan Negulescu[/i]

2024 IMAR Test, P1

Tags: algebra , real analysis

Fix integers $n\geq 2$ and $1\leq m\leq n-1$. Let $a_0, a_1, \dots, a_n$ be non-negative real numbers satisfying $a_0+a_1+\dots +a_n=1$. Prove that, if $\sum_{k=0}^n a_kx^k < x^m$ for some $0<x<1$, then $$\sum_{k=0}^{m-1}(m-k)a_k < \sum_{k=m+1}^n (k-m)a_k.$$

2024 Romania National Olympiad, 1

Tags: real analysis , calculus , integration , function

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x)+\sin(f(x)) \ge x,$ for all $x \in \mathbb{R}.$ Prove that $$\int\limits_0^{\pi} f(x) \mathrm{d}x \ge \frac{\pi^2}{2}-2.$$

2017 Miklós Schweitzer, 7

Tags: measure , measure theory , interval , real analysis

Characterize all increasing sequences $(s_n)$ of positive reals for which there exists a set $A\subset \mathbb{R}$ with positive measure such that $\lambda(A\cap I)<\frac{s_n}{n}$ holds for every interval $I$ with length $1/n$, where $\lambda$ denotes the Lebesgue measure.

1970 Miklós Schweitzer, 10

Tags: limit , real analysis

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]

2005 District Olympiad, 2

Tags: function , real analysis

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a continuous function such that for any $a,b\in \mathbb{R}$, with $a<b$ such that $f(a)=f(b)$, there exist some $c\in (a,b)$ such that $f(a)=f(b)=f(c)$. Prove that $f$ is monotonic over $\mathbb{R}$.

2007 Miklós Schweitzer, 5

Tags: real analysis , function

Let $D=\{ (x,y) \mid x>0, y\neq 0\}$ and let $u\in C^1(\overline {D})$ be a bounded function that is harmonic on $D$ and for which $u=0$ on the $y$-axis. Prove that $u$ is identically zero. (translated by Miklós Maróti)