Olympiad problems

2007 VJIMC, Problem 3

A function $f:[0,\infty)\to\mathbb R\setminus\{0\}$ is called [i]slowly changing[/i] if for any $t>1$ the limit $\lim_{x\to\infty}\frac{f(tx)}{f(x)}$ exists and is equal to $1$. Is it true that every slowly changing function has for sufficiently large $x$ a constant sign (i.e., is it true that for every slowly changing $f$ there exists an $N$ such that for every $x,y>N$ we have $f(x)f(y)>0$?)

2008 Moldova National Olympiad, 12.4

Tags: real analysis , limit

Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$. Find $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$.

2018 District Olympiad, 3

Tags: limit , real analysis , sequence , convergence

Let $(a_n)_{n\ge 1}$ be a sequence such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n + 2}$, for any $n\ge 1$. Show that the sequence $(x_n)_{n\ge 1}$ given by $x_n = \log_{a_n} a_{n + 1}$ for $n\ge 1$ is convergent and compute its limit.

2000 Miklós Schweitzer, 8

Tags: real analysis , function

Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a map such that the image of every compact set is compact, and the image of every connected set is connected. Prove that $f$ is continuous.

1981 Miklós Schweitzer, 10

Tags: real analysis , function , complex number , probability and stats , probability , integration

Let $ P$ be a probability distribution defined on the Borel sets of the real line. Suppose that $ P$ is symmetric with respect to the origin, absolutely continuous with respect to the Lebesgue measure, and its density function $ p$ is zero outside the interval $ [\minus{}1,1]$ and inside this interval it is between the positive numbers $ c$ and $ d$ ($ c < d$). Prove that there is no distribution whose convolution square equals $ P$. [i]T. F. Mori, G. J. Szekely[/i]

1999 IMC, 4

Tags: function , limit , real analysis , inequalities

Prove that there's no function $f: \mathbb{R}^+\rightarrow\mathbb{R}^+$ such that $f(x)^2\ge f(x+y)\left(f(x)+y\right)$ for all $x,y>0$.

2010 IMC, 1

Tags: real analysis , trigonometry

[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]

2005 Gheorghe Vranceanu, 3

Tags: limit , real analysis , calculus , euler s number

$ \lim_{n\to\infty }\left( \frac{1}{e}\sum_{i=0}^n \frac{1}{i!} \right)^{n!} $

1998 IMC, 5

Tags: algebra , induction , real analysis , polynomial

Let $P$ be a polynomial of degree $n$ with only real zeros and real coefficients. Prove that for every real $x$ we have $(n-1)(P'(x))^2\ge nP(x)P''(x)$. When does equality occur?

2011 Miklós Schweitzer, 6

Tags: real analysis , euclidean space , convex hull , combinatorial geometry

Let $C_1, ..., C_d$ be compact and connected sets in $R^d$, and suppose that each convex hull of $C_i$ contains the origin. Prove that for every i there is a $c_i \in C_i$ for which the origin is contained in the convex hull of the points $c_1, ..., c_d$.

2015 Romania National Olympiad, 1

Tags: real analysis , function

Find all differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the conditions: $ \text{(i)}\quad\forall x\in\mathbb{Z} \quad f'(x) =0 $ $ \text{(ii)}\quad\forall x\in\mathbb{R}\quad f'(x)=0\implies f(x)=0 $

1995 IMC, 10

Tags: real analysis , approximation , function

a) Prove that for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\lambda_{1},\dots,\lambda_{n}$ such that $$\max_{x\in [-1,1]}|x-\sum_{k=1}^{n}\lambda_{k}x^{2k+1}|<\epsilon.$$ b) Prove that for every odd continuous function $f$ on $[-1,1]$ and for every $\epsilon>0$ there is a positive integer $n$ and real numbers $\mu_{1},\dots,\mu_{n}$ such that $$\max_{x\in [-1,1]}|f(x)-\sum_{k=1}^{n}\mu_{k}x^{2k+1}|<\epsilon.$$

2012 District Olympiad, 1

Tags: real analysis , calculus , integration , limit

Let $a,b,c$ three positive distinct real numbers. Evaluate: \[\lim_{t\to \infty} \int_0^t \frac{1}{(x^2+a^2)(x^2+b^2)(x^2+c^2)}dx\]

1970 Miklós Schweitzer, 9

Tags: real analysis , function

Construct a continuous function $ f(x)$, periodic with period $ 2 \pi$, such that the Fourier series of $ f(x)$ is divergent at $ x\equal{}0$, but the Fourier series of $ f^2(x)$ is uniformly convergent on $ [0,2 \pi].$ [i]P. Turan[/i]

2004 Alexandru Myller, 4

Tags: real analysis , darboux s property , darboux , monotone , continuity , function

Let be a real function that has the intermediate value property and is monotone on the irrationals. Show that it's continuous. [i]Mihai Piticari[/i]

2008 Grigore Moisil Intercounty, 4

Tags: real analysis , function

Let be a function $ f:(0,\infty )\longrightarrow\mathbb{R} . $ [b]a)[/b] Show that if $ f $ is differentiable and $ \lim_{x\to \infty } xf'(x)=1, $ then $ \lim_{x\to\infty } f(x)=\infty .$ [b]b)[/b] Prove that if $ f $ is twice differentiable and $ f''+5f'+6f $ has limit at plus infinity, then: $$ \lim_{x\to\infty } f(x)=\frac{1}{6}\lim_{x\to\infty } \left( f''(x)+5f'(x)+6f(x)\right) $$ [i]Dorel Duca[/i] and [i]Dorian Popa[/i]

2013 VTRMC, Problem 3

Tags: real analysis , sequence

Define a sequence $(a_n)$ for $n\ge1$ by $a_1=2$ and $a_{n+1}=a_n^{1+n^{-3/2}}$. Is $(a_n)$ convergent (i.e. $\lim_{n\to\infty}a_n<\infty$)?

2004 Vietnam National Olympiad, 2

Tags: real analysis , limit , inequalities , quadratic , function , algebra , polynomial

Let $x$, $y$, $z$ be positive reals satisfying $\left(x+y+z\right)^{3}=32xyz$ Find the minimum and the maximum of $P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}$

2018 Romania National Olympiad, 3

Tags: real analysis , inequalities , integral , function , calculus

Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$ $\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$ $\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$ [i]Nicolae Bourbacut[/i]

2010 IberoAmerican Olympiad For University Students, 2

Tags: limit , real analysis , trigonometry

Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.

1973 Miklós Schweitzer, 5

Tags: real analysis , limit , logarithm , integration

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

1996 IMC, 11

Tags: real analysis , limit

i) Prove that $$ \lim_{x\to \infty}\,\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}=\frac{1}{2}$$. ii) Prove that there is a positive constant $c$ such that for every $x\in [1,\infty)$ we have $$\left|\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}-\frac{1}{2} \right| \leq \frac{c}{x}$$

2009 IMS, 5

Tags: real analysis , calculus , function , integration

Suppose that $ f: \mathbb R^2\rightarrow \mathbb R$ is a non-negative and continuous function that $ \iint_{\mathbb R^2}f(x,y)dxdy\equal{}1$. Prove that there is a closed disc $ D$ with the least radius possible such that $ \iint_D f(x,y)dxdy\equal{}\frac12$.

2005 Alexandru Myller, 1

Tags: real analysis , integration , function

Let $f:[a,b]\to\mathbb R$ be a continous function with the property that there exists a constant $\lambda\in\mathbb R$ so that for every $x\in[a,b]$ there exists a $y\in[a,b]-\{x\}$ s.t. $\int_x^yf(x)dx=\lambda$. Prove that the function $f$ has at least two zeros in $(a,b)$. [i]Eugen Paltanea[/i]

2024 OMpD, 4

Tags: real analysis , function

Let $ n $ be a positive integer. Determine the largest possible value of $ k $ with the following property: there exists a bijective function $ \phi: [0, 1] \to [0, 1]^k $ and a constant $ C > 0 $ such that, for all $ x, y \in [0, 1] $, \[ \| \phi(x) - \phi(y) \| \leq C \| x - y \|^k. \] Note: $ \| \cdot \| $ denotes the Euclidean norm, that is, $ \| (a_1, \ldots, a_n) \| = \sqrt{a_1^2 + \cdots + a_n^2} $.