Olympiad problems

1997 IMC, 1

Let $f\in C^3(\mathbb{R})$ nonnegative function with $f(0)=f'(0)=0, f''(0)>0$. Define $g(x)$ as follows: \[ \{ \begin{array}{ccc}g(x)= (\frac{\sqrt{f(x)}}{f'(x)})' &\text{for}& x\not=0 \\ g(x)=0 &\text{for}& x=0\end{array} \] (a) Show that $g$ is bounded in some neighbourhood of $0$. (b) Is the above true for $f\in C^2(\mathbb{R})$?

2008 Moldova National Olympiad, 12.2

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

Find the exact value of $ E\equal{}\displaystyle\int_0^{\frac\pi2}\cos^{1003}x\text{d}x\cdot\int_0^{\frac\pi2}\cos^{1004}x\text{d}x\cdot$.

1962 Miklós Schweitzer, 7

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

Prove that the function \[ f(\nu)= \int_1^{\frac{1}{\nu}} \frac{dx}{\sqrt{(x^2-1)(1-\nu^2x^2)}}\] (where the positive value of the square root is taken) is monotonically decreasing in the interval $ 0<\nu<1$. [P. Turan]

2014 Miklós Schweitzer, 4

Tags: real analysis , real analysis unsolved

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]

2014 Contests, 3

Tags: real analysis , real analysis unsolved

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.

2009 Romania National Olympiad, 1

Tags: limit , real analysis , real analysis unsolved

Let $(t_n)_n$ a convergent sequence of real numbers, $t_n\in (0,1),\ (\forall)n\in \mathbb{N}$ and $\lim_{n\to \infty} t_n\in (0,1)$. Define the sequences $(x_n)_n$ and $(y_n)_n$ by \[x_{n+1}=t_nx_n+(1-t_n)y_n,\ y_{n+1}=(1-t_n)x_n+t_n y_n,\ (\forall)n\in \mathbb{N}\] and $x_0,y_0$ are given real numbers. a) Prove that the sequences $(x_n)_n$ and $(y_n)_n$ are convergent and have the same limit. b) Prove that if $\lim_{n\to \infty} t_n\in \{0,1\}$, then the question is false.

2006 All-Russian Olympiad, 1

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

Prove that $\sin\sqrt{x}<\sqrt{\sin x}$ for every real $x$ such that $0<x<\frac{\pi}{2}$.

2014 ISI Entrance Examination, 3

Tags: real analysis , real analysis unsolved

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.

1999 Romania National Olympiad, 1

Tags: function , limit , inequalities , logarithms , integration , real analysis , real analysis unsolved

Find all continuous functions $ f: \mathbb{R}\to [1,\infty)$ for wich there exists $ a\in\mathbb{R}$ and a positive integer $ k$ such that \[ f(x)f(2x)\cdot...\cdot f(nx)\leq an^k\] for all real $ x$ and all positive integers $ n$. [i]author :Radu Gologan[/i]

2001 Romania National Olympiad, 4

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

Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$. a) Show that: \[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\] b) Show that: \[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]

2010 District Olympiad, 3

Tags: function , calculus , abstract algebra , real analysis , real analysis unsolved

Let $ a < c < b$ be three real numbers and let $ f: [a,b]\rightarrow \mathbb{R}$ be a continuos function in $ c$. If $ f$ has primitives on each of the intervals $ [a,c)$ and $ (c,b]$, then prove that it has primitives on the interval $ [a,b]$.

1974 Miklós Schweitzer, 7

Tags: algebra , polynomial , trigonometry , integration , real analysis , real analysis unsolved

Given a positive integer $ m$ and $ 0 < \delta <\pi$, construct a trigonometric polynomial $ f(x)\equal{}a_0\plus{} \sum_{n\equal{}1}^m (a_n \cos nx\plus{}b_n \sin nx)$ of degree $ m$ such that $ f(0)\equal{}1, \int_{ \delta \leq |x| \leq \pi} |f(x)|dx \leq c/m,$ and $ \max_{\minus{}\pi \leq x \leq \pi}|f'(x)| \leq c/{\delta}$, for some universal constant $ c$. [i]G. Halasz[/i]

1983 Miklós Schweitzer, 3

Tags: real analysis , real analysis unsolved

Let $ f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable, $ 2 \pi$-periodic even function. Prove that if \[ f''(x)\plus{}f(x)\equal{}\frac{1}{f(x\plus{} 3 \pi /2 )}\] holds for every $ x$, then $ f$ is $ \pi /2$-periodic. [i]Z. Szabo, J. Terjeki[/i]

2008 Moldova MO 11-12, 6

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

Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n\equal{}\frac1{\sqrt{n^2\plus{}8n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}16n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}24n\minus{}1}}\plus{}\ldots\plus{}\frac1{\sqrt{9n^2\minus{}1}}$.

2014 Contests, 2

Tags: topology , real analysis , real analysis unsolved

Let $(X,d)$ be a nonempty connected metric space such that the limit of every convergent sequence, is a term of that sequence. Prove that $X$ has exactly one element.

1998 Vietnam National Olympiad, 1

Tags: logarithms , integration , limit , real analysis , real analysis unsolved

Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$. Prove that the sequence $\{x_{n}\}$ converges and find its limit.

2008 Brazil Undergrad MO, 3

Tags: search , real analysis , real analysis unsolved , Brazilian Undergrad MO

Prove that there are real numbers $ a_1, a_2, ..$ such that: i) For all real numbers x, the serie $ f(x) \equal{} \sum_{n \equal{} 1}^\infty a_nx^n$ converge; ii) f is a bijection of R to R; iii) f'(x) >0; iv) f(Q) = A, where Q is the set of rational numbers and A is the set of algebraic numbers.

2008 District Olympiad, 4

Tags: function , real analysis , real analysis unsolved

Find the values of $a\in [0,\infty)$ for which there exist continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$, such that $f(f(x))=(x-a)^2,\ (\forall)x\in \mathbb{R}$.

2009 Olympic Revenge, 2

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

Prove that $\int_{0}^{\frac{\pi}{2}} arctg (1 - \sin^2x\cos^2x)dx = \frac{\pi^2}{4} - \pi arctg\sqrt{\frac{\sqrt{2}-1}{2}}$

2007 Romania National Olympiad, 4

Tags: function , calculus , derivative , symmetry , ratio , real analysis , real analysis unsolved

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$. a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$. b) Give an example of a non-constant function $f$ with property $(P)$. c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.

1968 Miklós Schweitzer, 9

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

Let $ f(x)$ be a real function such that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f(x)}{e^x}\equal{}1\] and $ |f''(x)|\leq c|f'(x)|$ for all sufficiently large $ x$. Prove that \[ \lim_{x \rightarrow \plus{}\infty} \frac{f'(x)}{e^x}\equal{}1.\] [i]P. Erdos[/i]

2010 Contests, 4

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

A real valued function $f$ is defined on the interval $(-1,2)$. A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$. Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$. Show that if $f'(1)>1$, then $f$ has a fixed point in the interval $(0,1)$.

2007 Moldova National Olympiad, 11.8

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

The continuous function and twice differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $2007^{2}\cdot f(x)+f''(x)=0$. Prove that there exist two such real numbers $k$ and $l$ such that $f(x)=l\cdot\sin(2007x)+k\cdot\cos(2007x)$.

1971 Miklós Schweitzer, 5

Tags: real analysis , real analysis unsolved

Let $ \lambda_1 \leq \lambda_2 \leq...$ be a positive sequence and let $ K$ be a constant such that \[ \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\] Prove that there exists a constant $ K'$ such that \[ \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\] [i]L. Leindler[/i]

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 4

Tags: parameterization , calculus , integration , logarithms , real analysis , real analysis unsolved

Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.