Olympiad problems

1981 Miklós Schweitzer, 7

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]

1980 Miklós Schweitzer, 6

Tags: function , real analysis , real analysis unsolved

Let us call a continuous function $ f : [a,b] \rightarrow \mathbb{R}^2 \;\textit{reducible}$ if it has a double arc (that is, if there are $ a \leq \alpha < \beta \leq \gamma < \delta \leq b$ such that there exists a strictly monotone and continuous $ h : [\alpha,\beta] \rightarrow [\gamma,\delta]$ for which $ f(t)\equal{}f(h(t))$ is satisfied for every $ \alpha \leq t \leq \beta$); otherwise $ f$ is irreducible. Construct irreducible $ f : [a,b] \rightarrow \mathbb{R}^2$ and $ g : [c,d] \rightarrow \mathbb{R}^2$ such that $ f([a,b])\equal{}g([c,d])$ and (a) both $ f$ and $ g$ are rectifiable but their lengths are different; (b) $ f$ is rectifiable but $ g$ is not. [i]A. Csaszar[/i]

1964 Putnam, B6

Tags: Ross Mathematics Program , real analysis , real analysis unsolved

This is rather simple, but I liked it :). Show that a disk cannot be partitioned into two congruent subsets.

1998 IMC, 6

Tags: calculus , integration , inequalities , function , trigonometry , real analysis , real analysis unsolved

Let $f: [0,1]\rightarrow\mathbb{R}$ be a continuous function satisfying $xf(y)+yf(x)\le 1$ for every $x,y\in[0,1]$. (a) Show that $\int^1_0 f(x)dx \le \frac{\pi}4$. (b) Find such a funtion for which equality occurs.

2010 IMC, 1

Tags: trigonometry , real analysis , real analysis unsolved

[list] $(a)$ A sequence $x_1,x_2,\dots$ of real numbers satisfies \[x_{n+1}=x_n \cos x_n \textrm{ for all } n\geq 1.\] Does it follows that this sequence converges for all initial values $x_1?$ (5 points) $(b)$ A sequence $y_1,y_2,\dots$ of real numbers satisfies \[y_{n+1}=y_n \sin y_n \textrm{ for all } n\geq 1.\] Does it follows that this sequence converges for all initial values $y_1?$ (5 points)[/list]

2001 Romania National Olympiad, 3

Tags: calculus , integration , trigonometry , function , floor function , real analysis , real analysis unsolved

Let $f:[-1,1]\rightarrow\mathbb{R}$ be a continuous function. Show that: a) if $\int_0^1 f(\sin (x+\alpha ))\, dx=0$, for every $\alpha\in\mathbb{R}$, then $f(x)=0,\ \forall x\in [-1,1]$. b) if $\int_0^1 f(\sin (nx))\, dx=0$, for every $n\in\mathbb{Z}$, then $f(x)=0,\ \forall x\in [-1,1]$.

2002 Romania National Olympiad, 2

Tags: function , limit , real analysis , real analysis unsolved

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has limits at any point and has no local extrema. Show that: $a)$ $f$ is continuous; $b)$ $f$ is strictly monotone.

2012 Olympic Revenge, 1

Tags: inequalities , real analysis , real analysis unsolved

Let $a$ and $b$ real numbers. Let $f:[a,b] \rightarrow \mathbb{R}$ a continuous function. We say that f is "smp" if $[a,b]=[c_0,c_1]\cup[c_1,c_2]...\cup[c_{n-1},c_n]$ satisfying $c_0<c_1...<c_n$ and for each $i\in\{0,1,2...n-1\}$: $c_i<x<c_{i+1} \Rightarrow f(c_i)<f(x)<f(c_{i+1})$ or $c_i>x>c_{i+1} \Rightarrow f(c_i)>f(x)>f(c_{i+1})$ Prove that if $f:[a,b] \rightarrow \mathbb{R}$ is continuous such that for each $v\in\mathbb{R}$ there are only finitely many $x$ satisfying $f(x)=v$, then $f$ is "smp".

1952 Miklós Schweitzer, 10

Tags: trigonometry , function , integration , real analysis , real analysis unsolved

Let $ n$ be a positive integer. Prove that, for $ 0<x<\frac{\pi}{n\plus{}1}$, $ \sin{x}\minus{}\frac{\sin{2x}}{2}\plus{}\cdots\plus{}(\minus{}1)^{n\plus{}1}\frac{\sin{nx}}{n}\minus{}\frac{x}{2}$ is positive if $ n$ is odd and negative if $ n$ is even.

2014 District Olympiad, 2

Tags: function , calculus , derivative , integration , real analysis , real analysis unsolved

Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a differentiable function, with continuous derivative, and let \[ s_{n}=\sum_{k=1}^{n}f\left( \frac{k}{n}\right) \] Prove that the sequence $(s_{n+1}-s_{n})_{n\in{\mathbb{N}}^{\ast}}$ converges to $\int_{0}^{1}f(x)\mathrm{d}x$.

2008 Romania National Olympiad, 1

Tags: function , real analysis , real analysis unsolved

Let $ f : (0,\infty) \to \mathbb R$ be a continous function such that the sequences $ \{f(nx)\}_{n\geq 1}$ are nondecreasing for any real number $ x$. Prove that $ f$ is nondecreasing.

2012 Romania National Olympiad, 4

Tags: function , integration , calculus , inequalities , ratio , real analysis , real analysis unsolved

[color=darkred]Find all differentiable functions $f\colon [0,\infty)\to [0,\infty)$ for which $f(0)=0$ and $f^{\prime}(x^2)=f(x)$ for any $x\in [0,\infty)$ .[/color]

2006 District Olympiad, 1

Tags: inequalities , calculus , integration , function , real analysis , real analysis unsolved

Let $f_1,f_2,\ldots,f_n : [0,1]\to (0,\infty)$ be $n$ continuous functions, $n\geq 1$, and let $\sigma$ be a permutation of the set $\{1,2,\ldots, n\}$. Prove that \[ \prod^n_{i=1} \int^1_0 \frac{ f_i^2(x) }{ f_{\sigma(i)}(x) } dx \geq \prod^n_{i=1} \int^1_0 f_i(x) dx. \]

2011 Romania National Olympiad, 3

Tags: function , real analysis , real analysis unsolved

[color=darkred]Let $g:\mathbb{R}\to\mathbb{R}$ be a continuous and strictly decreasing function with $g(\mathbb{R})=(-\infty,0)$ . Prove that there are no continuous functions $f:\mathbb{R}\to\mathbb{R}$ with the property that there exists a natural number $k\ge 2$ so that : $\underbrace{f\circ f\circ\ldots\circ f}_{k\text{ times}}=g$ . [/color]

2000 IMC, 2

Tags: function , real analysis , real analysis unsolved

Let $f$ be continuous and nowhere monotone on $[0,1]$. Show that the set of points on which $f$ obtains a local minimum is dense.