Olympiad problems

2004 Unirea, 3

Hello, I've been trying to solve this for a while now, but no success! I mean, I have an expression for this but not a closed one. I derived something in terms of Tchebychev Polynomials : cos(nx) = P_n(cos(x)). Any help is appreciated. Compute the following primitive: \[ \int \frac{x\sin\left(2004 x\right)}{\tan x}\ dx\]

2017 Korea USCM, 3

Tags: real analysis , limit , recurrence relation

Sequence $\{a_n\}$ defined by recurrence relation $a_{n+1} = 1+\frac{n^2}{a_n}$. Given $a_1>1$, find the value of $\lim\limits_{n\to\infty} \frac{a_n}{n}$ with proof.

2019 Ramnicean Hope, 1

Tags: riemann sum , calculus , real analysis , definite integral , substitution

Calculate $ \lim_{n\to\infty }\sum_{t=1}^n\frac{1}{n+t+\sqrt{n^2+nt}} . $ [i]D.M. Bătinețu[/i] and [i]Neculai Stanciu[/i]

1998 IberoAmerican Olympiad For University Students, 6

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

Take the following differential equation: \[3(3+x^2)\frac{dx}{dt}=2(1+x^2)^2e^{-t^2}\] If $x(0)\leq 1$, prove that there exists $M>0$ such that $|x(t)|<M$ for all $t\geq 0$.

1968 Putnam, B6

Tags: topology , compact set , real analysis

Show that one cannot find compact sets $A_1, A_2, A_3, \ldots$ in $\mathbb{R}$ such that (1) All elements of $A_n$ are rational. (2) Any compact set $K\subset \mathbb{R}$ which only contains rational numbers is contained in some $A_{m}$.

2012 Pre-Preparation Course Examination, 1

Tags: real analysis , function

Suppose that $X$ and $Y$ are two metric spaces and $f:X \longrightarrow Y$ is a continious function. Also for every compact set $K \subseteq Y$, it's pre-image $f^{pre}(K)$ is a compact set in $X$. Prove that $f$ is a closed function, i.e for every close set $C\subseteq X$, it's image $f(C)$ is a closed subset of $Y$.

2000 IMC, 4

Tags: inequalities , real analysis

Let $(x_i)$ be a decreasing sequence of positive reals, then show that: (a) for every positive integer $n$ we have $\sqrt{\sum^n_{i=1}{x_i^2}} \leq \sum^n_{i=1}\frac{x_i}{\sqrt{i}}$. (b) there is a constant C for which we have $\sum^{\infty}_{k=1}\frac{1}{\sqrt{k}}\sqrt{\sum^{\infty}_{i=k}x_i^2} \le C\sum^{\infty}_{i=1}x_i$.

2006 Petru Moroșan-Trident, 3

Tags: real analysis , sequence

Let be a sequence $ \left( u_n \right)_{n\ge 1} $ given by the recurrence relation $ u_{n+1} =u_n+\sqrt{u_n^2-u_1^2} , $ and the constraints $ u_2\ge u_1>0. $ Calculate $ \lim_{n\to\infty }\frac{2^n}{u_n} . $ [i]Dan Negulescu[/i]

1988 Greece National Olympiad, 4

Tags: real analysis , limit , sequence

Let $a_1=5$ and $a_{n+1}= a^2_{n}-2$ for any $n=1,2,...$. a) Find $\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_1a_2 ...a_{n}}$ b) Find $\lim_{\nu \rightarrow \infty}\left(\frac{1}{a_1}+\frac{1}{a_1a_2}+...+\frac{1}{a_1a_2 ...a_{\nu}}\right)$

2001 Miklós Schweitzer, 1

Tags: function , set theory , real analysis

Let $f\colon 2^S\rightarrow \mathbb R$ be a function defined on the subsets of a finite set $S$. Prove that if $f(A)=F(S\backslash A)$ and $\max \{ f(A), f(B)\}\geq f(A\cup B)$ for all subsets $A, B$ of $S$, then $f$ assumes at most $|S|$ distinct values.

1969 Miklós Schweitzer, 4

Tags: inequalities , logarithm , real analysis

Show that the following inequality hold for all $ k \geq 1$, real numbers $ a_1,a_2,...,a_k$, and positive numbers $ x_1,x_2,...,x_k.$ \[ \ln \frac {\sum\limits_{i \equal{} 1}^kx_i}{\sum\limits_{i \equal{} 1}^kx_i^{1 \minus{} a_i}} \leq \frac {\sum\limits_{i \equal{} 1}^ka_ix_i \ln x_i}{\sum\limits_{i \equal{} 1}^kx_i} . \] [i]L. Losonczi[/i]

ICMC 7, 3

Tags: polynomial , real analysis , expected value

Let $N{}$ be a fixed positive integer, $S{}$ be the set $\{1, 2,\ldots , N\}$ and $\mathcal{F}$ be the set of functions $f:S\to S$ such that $f(i)\geqslant i$ for all $i\in S.$ For each $f\in\mathcal{F}$ let $P_f$ be the unique polynomial of degree less than $N{}$ satisfying $P_f(i) = f(i)$ for all $i\in S.$ If $f{}$ is chosen uniformly at random from $\mathcal{F}$ determine the expected value of $P_f'(0)$ where\[P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.\][i]Proposed by Ishan Nath[/i]

2002 District Olympiad, 3

Tags: real analysis , integration , logarithm , calculus

[b]a)[/b] Calculate $ \lim_{n\to\infty} \int_0^{\alpha } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx , $ for all $ \alpha\in (0,1) . $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \int_0^{1 } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx . $

1997 Romania National Olympiad, 4

Tags: function , calculus , integration , real analysis

Suppose that $(f_n)_{n\in N}$ be the sequence from all functions $f_n:[0,1]\rightarrow \mathbb{R^+}$ s.t. $f_0$ be the continuous function and $\forall x\in [0,1] , \forall n\in \mathbb {N} , f_{n+1}(x)=\int_0^x \frac {1}{1+f_n (t)}dt$. Prove that for every $x\in [0,1]$ the sequence of $(f_n(x))_{n\in N}$ be the convergent sequence and calculate the limitation.

1995 Miklós Schweitzer, 10

Tags: geometry , real analysis

Let $X =\{ X_1 , X_2 , ...\}$ be a countable set of points in space. Show that there is a positive sequence $\{a_k\}$ such that for any point $Z\not\in X$ the distance between the point Z and the set $\{X_1,X_2 , ...,X_k\}$ is at least $a_k$ for infinitely many k.

1949 Miklós Schweitzer, 4

Tags: real analysis

Let $ A$ and $ B$ be two disjoint sets in the interval $ (0,1)$ . Denoting by $ \mu$ the Lebesgue measure on the real line, let $ \mu(A)>0$ and $ \mu(B)>0$ . Let further $ n$ be a positive integer and $ \lambda \equal{}\frac1n$ . Show that there exists a subinterval $ (c,d)$ of $ (0,1)$ for which $ \mu(A\cap (c,d))\equal{}\lambda \mu(A)$ and $ \mu(B\cap (c,d))\equal{}\lambda \mu(B)$ . Show further that this is not true if $ \lambda$ is not of the form $ \frac1n$.

2006 Moldova MO 11-12, 2

Tags: function , limit , real analysis

Function $f: [a,b]\to\mathbb{R}$, $0<a<b$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Prove that there exists $c\in(a,b)$ such that \[ f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}. \]

2009 IberoAmerican Olympiad For University Students, 4

Tags: function , real analysis

Given two positive integers $m,n$, we say that a function $f : [0,m] \to \mathbb{R}$ is $(m,n)$-[i]slippery[/i] if it has the following properties: i) $f$ is continuous; ii) $f(0) = 0$, $f(m) = n$; iii) If $t_1, t_2\in [0,m]$ with $t_1 < t_2$ are such that $t_2-t_1\in \mathbb{Z}$ and $f(t_2)-f(t_1)\in\mathbb{Z}$, then $t_2-t_1 \in \{0,m\}$. Find all the possible values for $m, n$ such that there is a function $f$ that is $(m,n)$-slippery.

2020 Miklós Schweitzer, 5

Tags: real analysis

Prove that for a nowhere dense, compact set $K\subset \mathbb{R}^2$ the following are equivalent: (i) $K=\bigcup_{i=1}^{\infty}K_n$ where $K_n$ is a compact set with connected complement for all $n$. (ii) $K$ does not have a nonempty closed subset $S\subseteq K$ such that any neighborhood of any point in $S$ contains a connected component of $\mathbb{R}^2 \setminus S$.

2001 Romania National Olympiad, 4

Tags: function , real analysis , limit

The continuous function $f:[0,1]\rightarrow\mathbb{R}$ has the property: \[\lim_{x\rightarrow\infty}\ n\left(f\left(x+\frac{1}{n}\right)-f(x)\right)=0 \] for every $x\in [0,1)$. Show that: a) For every $\epsilon >0$ and $\lambda\in (0,1)$, we have: \[ \sup\ \{x\in[0,\lambda )\mid |f(x)-f(0)|\le \epsilon x \}=\lambda \] b) $f$ is a constant function.

1959 Miklós Schweitzer, 1

Tags: number theory , real analysis , prime number

[b]1.[/b] Let $p_n$ be the $n$th prime number. Prove that $\sum_{n=2}^{\infty} \frac{1}{np_n-(n-1)p_{n-1}}= \infty$ [b](N.17)[/b]

2022 Germany Team Selection Test, 1

Tags: algebra , inequalities , real analysis

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 $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

2014 ISI Entrance Examination, 3

Tags: real analysis

Consider $f(x)=x^4+ax^3+bx^2+cx+d\; (a,b,c,d\in\mathbb{R})$. It is known that $f$ intersects X-axis in at least $3$ (distinct) points. Show either $f$ has $4$ $\mathbf{distinct}$ real roots or it has $3$ $\mathbf{distinct}$ real roots and one of them is a point of local maxima or minima.

1962 Putnam, B3

Tags: real analysis , bounded , convex set , geometry

Let $S$ be a convex region in the euclidean plane containing the origin. Assume that every ray from the origin has at least one point outside $S$. Prove that $S$ is bounded.

2014 IMC, 3

Tags: polynomial , real analysis , algebra

Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial $$\pm a_nx^n\pm a_{n-1}x^{n-1} \pm \dots \pm a_1x \pm a_0$$ has $n$ distinct real roots. (Proposed by Stephan Neupert, TUM, München)