Olympiad problems

2012 Romania National Olympiad, 1

[color=darkred]Let $f,g\colon [0,1]\to [0,1]$ be two functions such that $g$ is monotonic, surjective and $|f(x)-f(y)|\le |g(x)-g(y)|$ , for any $x,y\in [0,1]$ . [list] [b]a)[/b] Prove that $f$ is continuous and that there exists some $x_0\in [0,1]$ with $f(x_0)=g(x_0)$ . [b]b)[/b] Prove that the set $\{x\in [0,1]\, |\, f(x)=g(x)\}$ is a closed interval. [/list][/color]

1971 Miklós Schweitzer, 9

Tags: function , real analysis , real analysis unsolved

Given a positive, monotone function $ F(x)$ on $ (0, \infty)$ such that $ F(x)/x$ is monotone nondecreasing and $ F(x)/x^{1+d}$ is monotone nonincreasing for some positive $ d$, let $ \lambda_n >0$ and $ a_n \geq 0 , \;n \geq 1$. Prove that if \[ \sum_{n=1}^{\infty} \lambda_n F \left( a_n \sum _{k=1}^n \frac{\lambda_k}{\lambda_n} \right) < \infty,\] or \[ \sum_{n=1}^{\infty} \lambda_n F \left( \sum _{k=1}^n a_k \frac{\lambda_k}{\lambda_n} \right) < \infty,\] then $ \sum_{n=1}^ {\infty} a_n$ is convergent. [i]L. Leindler[/i]

2010 IMC, 1

Tags: integration , real analysis , real analysis unsolved

Let $0 < a < b$. Prove that $\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.

1973 Miklós Schweitzer, 5

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

Verify that for every $ x > 0$, \[ \frac{\Gamma'(x\plus{}1)}{\Gamma (x\plus{}1)} > \log x.\] [i]P. Medgyessy[/i]

2001 District Olympiad, 4

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

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]

2007 District Olympiad, 3

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

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 $A$.

2008 Romania National Olympiad, 1

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

Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)\equal{}0$ and for all nonnegative $ x$ we have \[ xf(x) \geq \int^x_0 f(t) dt ,\] then $ f$ admits primitives on $ [0,\infty)$.

2009 District Olympiad, 4

Tags: function , floor function , real analysis , real analysis unsolved

a) Prove that the function $F:\mathbb{R}\rightarrow \mathbb{R},\ F(x)=2\lfloor x\rfloor-\cos(3\pi\{x\})$ is continuous over $\mathbb{R}$ and for any $y\in \mathbb{R}$, the equation $F(x)=y$ has exactly three solutions. b) Let $k$ a positive even integer. Prove that there is no function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is continuous over $\mathbb{R}$ and that for any $y\in \text{Im}\ f$, the equation $f(x)=y$ has exactly $k$ solutions $(\text{Im}\ f=f(\mathbb{R}))$.

2003 IMC, 6

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

Let $ p=\sum\limits_{k=0}^n a_kX^k\in R[X] $ a polynomial such that all his roots lie in the half plane $ \{z\in C| Re(z)<0 \}. $ Prove that $ a_ka_{k+3}<a_{k+1}a_{k+2}, $ for every k=0,1,2...,n-3.

2009 Romania National Olympiad, 2

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

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a continuous function such that for any $x\in \mathbb{R}$, the limit $\lim_{h\to 0} \left|\frac{f(x+h)-f(x)}{h}\right|$ exists and it is finite. Prove that in any real point, $f$ is differentiable or it has finite one-side derivates, of the same modul, but different signs.

1997 Romania National Olympiad, 4

Tags: function , induction , real analysis , real analysis unsolved

Let two bijective and continuous functions$f,g: \mathbb{R}\to\mathbb{R}$ such that : $\left(f\circ g^{-1}\right)(x)+\left(g\circ f^{-1}\right)(x)=2x$ for any real $x$. Show that If we have a value $x_{0}\in\mathbb{R}$ such that $f(x_{0})=g(x_{0})$, then $f=g$.

2012 Pre-Preparation Course Examination, 6

Tags: limit , real analysis , real analysis unsolved

Suppose that $a_{ij}$ are real numbers in such a way that for each $i$, the series $\sum_{j=1}^{\infty}a_{ij}$ is absolutely convergent. In fact we have a series of absolutely convergent serieses. Also we know that for each bounded sequence $\{b_j\}_j$ we have $\lim_{i\to \infty} \sum_{j=1}^{\infty}a_{ij}b_j=0$. Prove that $\lim_{i\to \infty}\sum_{j=1}^{\infty}|a_{ij}|=0$.

1964 Miklós Schweitzer, 8

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

Let $ F$ be a closed set in the $ n$-dimensional Euclidean space. Construct a function that is $ 0$ on $ F$, positive outside $ F$ , and whose partial derivatives all exist.

2012 Pre-Preparation Course Examination, 5

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

The $2^{nd}$ order differentiable function $f:\mathbb R \longrightarrow \mathbb R$ is in such a way that for every $x\in \mathbb R$ we have $f''(x)+f(x)=0$. [b]a)[/b] Prove that if in addition, $f(0)=f'(0)=0$, then $f\equiv 0$. [b]b)[/b] Use the previous part to show that there exist $a,b\in \mathbb R$ such that $f(x)=a\sin x+b\cos x$.

1999 IMC, 6

Tags: function , real analysis , real analysis unsolved

(a) Let $p>1$ a real number. Find a real constant $c_p$ for which the following statement holds: If $f: [-1,1]\rightarrow\mathbb{R}$ is a continuously differentiable function with $f(1)>f(-1)$ and $|f'(y)|\le1 \forall y\in[-1,1]$, then $\exists x\in[-1,1]: f'(x)>0$ so that $\forall y\in[-1,1]: |f(y)-f(x)|\le c_p\sqrt[p]{f'(x)}|y-x|$. (b) What if $p=1$?

2005 Alexandru Myller, 4

Tags: ratio , real analysis , real analysis unsolved

Let $(a_n)_n$ be a sequence of positive irational numbers. a) Prove that for every $n\in\mathbb N^*$, the binomial development $(1+a_n)^n$ admits a unique maximum term and determine its rank $r_n\in\{1,2,\ldots,n+1\}$. b) We consider the sequences $x_n=a_n\sqrt n, n\in\mathbb N^*$ and $y_n=(1+a_n)^{r_n}, n\in\mathbb N^*$. Prove that $(x_n)_n$ is convergent if and only if the sequence $(y_n)_n$ is convergent. [i]Eugen Paltanea[/i]

2004 Romania National Olympiad, 3

Tags: function , real analysis , real analysis unsolved

Let $f : (a,b) \to \mathbb R$ be a function with the property that for all $x \in (a,b)$ there is a non-degenerated interval $[ a_x,b_x ]$ with $a < a_x \leq x \leq b_x < b$ such that $f$ is constant on $\left[ a_x,b_x \right]$. (a) Prove that $\textrm{Im} \, f$ is finite or numerable. (b) Find all continuous functions which have the property mentioned in the hypothesis.

1970 Miklós Schweitzer, 10

Tags: limit , real analysis , real analysis unsolved

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]

2006 IMC, 6

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

Find all sequences $a_{0}, a_{1},\ldots, a_{n}$ of real numbers such that $a_{n}\neq 0$, for which the following statement is true: If $f: \mathbb{R}\to\mathbb{R}$ is an $n$ times differentiable function and $x_{0}<x_{1}<\ldots <x_{n}$ are real numbers such that $f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0$ then there is $h\in (x_{0}, x_{n})$ for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]

1963 Miklós Schweitzer, 6

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

Show that if $ f(x)$ is a real-valued, continuous function on the half-line $ 0\leq x < \infty$, and \[ \int_0^{\infty} f^2(x)dx <\infty\] then the function \[ g(x)\equal{}f(x)\minus{}2e^{\minus{}x}\int_0^x e^tf(t)dt\] satisfies \[ \int _0^{\infty}g^2(x)dx\equal{}\int_0^{\infty}f^2(x)dx.\] [B. Szokefalvi-Nagy]

2010 Romania National Olympiad, 4

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

Let $a\in \mathbb{R}_+$ and define the sequence of real numbers $(x_n)_n$ by $x_1=a$ and $x_{n+1}=\left|x_n-\frac{1}{n}\right|,\ n\ge 1$. Prove that the sequence is convergent and find it's limit.

2005 Unirea, 4

Tags: real analysis , real analysis unsolved

Find all $a$ real number such that $x_n=n\{an! \}$ is convergeant Gabriel Dospinescu

2001 District Olympiad, 4

Tags: limit , induction , inequalities , real analysis , real analysis unsolved

Prove that: a) the sequence $a_n=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+n},\ n\ge 1$ is monotonic. b) there is a sequence $(a_n)_{n\ge 1}\in \{0,1\}$ such that: \[\lim_{n\to \infty} \left(\frac{a_1}{n+1}+\frac{a_2}{n+2}+\ldots +\frac{a_n}{n+n}\right)=\frac{1}{2}\] [i]Radu Gologan[/i]

2008 Moldova National Olympiad, 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}}$.

1997 IMC, 4

Tags: real analysis , real analysis unsolved

Let $\alpha$ be a real number, $1<\alpha<2$. (a) Show that $\alpha$ can uniquely be represented as the infinte product \[ \alpha = \left(1+\dfrac1{n_1}\right)\left(1+\dfrac1{n_2}\right)\cdots \] with $n_i$ positive integers satisfying $n_i^2\le n_{i+1}$. (b) Show that $\alpha\in\mathbb{Q}$ iff from some $k$ onwards we have $n_{k+1}=n_k^2$.