Olympiad problems

2014 Miklós Schweitzer, 8

Tags: polynomial , function , inequalities , functional analysis , real analysis , algebra

Let $n\ge 1$ be a fixed integer. Calculate the distance $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$ , where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x)= \sum_{k=n}^{\infty} c_k x^k$ defined on the closed interval $[0,1]$ , where $c_k \ge 0$ and $\sum_{k=n}^{\infty} c_k=1$.

Gheorghe Țițeica 2024, P1

Tags: real analysis , integral

Let $a>1$ and $b>1$ be rational numbers. Denote by $\mathcal{F}_{a,b}$ the set of functions $f:[0,\infty)\rightarrow\mathbb{R}$ such that $$f(ax)=bf(x), \text{ for all }x\geq 0.$$ a) Prove that the set $\mathcal{F}_{a,b}$ contains both Riemann integrable functions on any interval and functions that are not Riemann integrable on any interval. b) If $f\in\mathcal{F}_{a,b}$ is Riemann integrable on $[0,\infty)$ and $\int_{\frac{1}{a}}^{a}f(x)dx=1$, calculate $$\int_a^{a^2} f(x)dx\text{ and }\int_0^1 f(x)dx.$$ [i]Vasile Pop[/i]

2010 VTRMC, Problem 7

Tags: real analysis , sum , summation , sequence

Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$) and set $b_n=\frac1{na_n^2}$ for $n\ge1$. Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.

2024 Miklos Schweitzer, 2

Tags: measure theory , topology , real analysis

Does there exist a nowhere dense, nonempty compact set $C \subset [0,1]$ such that \[ \liminf_{h \to 0^+} \frac{\lambda(C \cap (x, x+h))}{h} > 0 \quad \text{or} \quad \liminf_{h \to 0^+} \frac{\lambda(C \cap (x-h, x))}{h} > 0 \] holds for every point $x \in C$, where $\lambda(A)$ denotes the Lebesgue measure of $A$?

2006 All-Russian Olympiad, 1

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

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

2011 Romania National Olympiad, 2

Tags: function , real analysis , integration , calculus

Let be a continuous function $ f:[0,1]\longrightarrow\left( 0,\infty \right) $ having the property that, for any natural number $ n\ge 2, $ there exist $ n-1 $ real numbers $ 0<t_1<t_2<\cdots <t_{n-1}<1, $ such that $$ \int_0^{t_1} f(t)dt=\int_{t_1}^{t_2} f(t)dt=\int_{t_2}^{t_3} f(t)dt=\cdots =\int_{t_{n-2}}^{t_{n-1}} f(t)dt=\int_{t_{n-1}}^{1} f(t)dt. $$ Calculate $ \lim_{n\to\infty } \frac{n}{\frac{1}{f(0)} +\sum_{i=1}^{n-1} \frac{1}{f\left( t_i \right)} +\frac{1}{f(1)}} . $

2021 Brazil Undergrad MO, Problem 4

Tags: limit , exponent , real analysis , mean

For every positive integeer $n>1$, let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$. Find $$lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}$$

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 $\sum^{\infty}_{k=1}\frac{1}{\sqrt{k}}\sqrt{\sum^{\infty}_{i=k}x_i^2} \le C\sum^{\infty}_{i=1}x_i$.

1997 IMC, 5

Tags: real analysis

For postive integer $n$ consider the hyperplane \[ R_0^n = {x=(x_1x_2...x_n)\in\mathbb{R}^n : \sum\limits^n_{i=1}x_i=0} \] and the lattice \[ Z_0^n = \{y \in R^n_0 : \ (\forall i: y_i \in \mathbb{N})\} \] Define the quasi-norm in $\mathbb{R}^n$ by $\|x\|_p= \sqrt[p]{\sum\limits^{n}_{i=1}|x_i|^p}$ if $0<p<\infty$ and $\|x\|_{\infty} = \max\limits_i |x_i|$. (a) If $x\in R^n_0$ so that $\max x_i - \min x_i \le 1$ then prove that $\forall p \in [1,\infty], \forall y \in Z^n_0$ we have $\|x\|_p\le\|x+y\|_p$ (b) Prove that for every $p\in ]0,1[$, there exist $n \in \mathbb{N}, x\in R^n_0, y\in Z^n_0$ with $\max x_i - \min x_i \le 1$ and $\|x\|_p>\|x+y\|_p$

2005 Brazil Undergrad MO, 5

Tags: real analysis , calculus , integration , function , induction , logarithm

Prove that \[ \sum_{n=1}^\infty {1\over n^n} = \int_0^1 x^{-x}\,dx. \]

2005 Grigore Moisil Urziceni, 2

Tags: limit , real analysis

[b]a)[/b] Prove that $ \lim_{x\to\infty } \sqrt{x}\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x}=1. $ [b]b)[/b] Show that $ \lim_{x\to\infty } \left( -\left\lfloor\sqrt{x}\right\rfloor +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) =\frac{-1}{2} $ [b]c)[/b] What about $ \lim_{x\to\infty } \left( -\sqrt{x} +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) ? $

2005 Harvard-MIT Mathematics Tournament, 4

Tags: calculus , function , derivative , real analysis

Let $ f : \mathbf {R} \to \mathbf {R} $ be a smooth function such that $ f'(x)^2 = f(x) f''(x) $ for all $x$. Suppose $f(0)=1$ and $f^{(4)} (0) = 9$. Find all possible values of $f'(0)$.

2005 Alexandru Myller, 3

Tags: real analysis , function , limit , topology

Let $f:[0,\infty)\to\mathbb R$ be a continuous function s.t. $\lim_{x\to\infty}\frac {f(x)}x=0$. Let $(x_n)_n$ be a sequence of positive real numbers s.t. $\left(\frac{x_n}n\right)_n$ is bounded. Prove that $\lim_{n\to\infty}\frac{f(x_n)}n=0$. [i]Dorin Andrica, Eugen Paltanea[/i]

Gheorghe Țițeica 2024, P1

Tags: real analysis , convergent sequence , continuous function

Find all continuous functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any sequences $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ such that the sequence $(a_n+b_n)_{n\geq 1}$ is convergent, the sequence $(f(a_n)+g(b_n))_{n\geq 1}$ is also convergent.

2012 Olympic Revenge, 1

Tags: inequalities , real analysis

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".

1974 Miklós Schweitzer, 5

Tags: function , integration , real analysis

Let $ \{f_n \}_{n=0}^{\infty}$ be a uniformly bounded sequence of real-valued measurable functions defined on $ [0,1]$ satisfying \[ \int_0^1 f_n^2=1.\] Further, let $ \{ c_n \}$ be a sequence of real numbers with \[ \sum_{n=0}^{\infty} c_n^2= +\infty.\] Prove that some re-arrangement of the series $ \sum_{n=0}^{\infty} c_nf_n$ is divergent on a set of positive measure. [i]J. Komlos[/i]

2003 Romania National Olympiad, 3

Tags: limit , function , continuity , real analysis

Let be two functions $ f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ having that properties that $ f $ is continuous, $ g $ is nondecreasing and unbounded, and for any sequence of rational numbers $ \left( x_n \right)_{n\ge 1} $ that diverges to $ \infty , $ we have $$ 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . $$ Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) . $ [i]Radu Gologan[/i]

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

Tags: parameterization , calculus , integration , logarithm , real analysis

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.

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.

1971 Miklós Schweitzer, 5

Tags: real analysis

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]

2014 Miklós Schweitzer, 8

Gheorghe Țițeica 2024, P1

2010 VTRMC, Problem 7

2024 Miklos Schweitzer, 2

2006 All-Russian Olympiad, 1

2011 Romania National Olympiad, 2

2021 Brazil Undergrad MO, Problem 4

2000 IMC, 4

1997 IMC, 5

2005 Brazil Undergrad MO, 5

2005 Grigore Moisil Urziceni, 2

2005 Harvard-MIT Mathematics Tournament, 4

2005 Alexandru Myller, 3

Gheorghe Țițeica 2024, P1

2012 Olympic Revenge, 1

1974 Miklós Schweitzer, 5

2003 Romania National Olympiad, 3

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

2014 ISI Entrance Examination, 3

1971 Miklós Schweitzer, 5

2003 Alexandru Myller, 4

2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 3

1996 IMC, 7

2008 ISI B.Stat Entrance Exam, 6

2006 Romania National Olympiad, 2