Olympiad problems

Gheorghe Țițeica 2025, P1

Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+y)=f(x+f(y))$ for all $x,y\in\mathbb{R}$.

2014-2015 SDML (Middle School), 12

Tags: function , real analysis

Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$. $\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$

2012 District Olympiad, 1

Tags: sequence , real analysis , limit

Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfying the equation $$ x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} . $$ Show that this sequence is convergent and find its limit.

2014 Cezar Ivănescu, 2

Tags: primitivability , antiderivative , darboux , derivability , function , real analysis

[b]a)[/b] Give an example of function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ that admits a primitive $ F:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ having the property that $ F^e $ is a primitive of $ f^e. $ [b]b)[/b] Prove that there is no derivable function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that has a primitive $ G:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ e^G $ is a primitive of $ e^g. $

2019 Korea USCM, 6

Tags: integral inequality , real analysis , function , inequalities

A function $f:[0,\infty)\to[0,\infty)$ is integrable and $$\int_0^\infty f(x)^2 dx<\infty,\quad \int_0^\infty xf(x) dx <\infty$$ Prove the following inequality. $$\left(\int_0^\infty f(x) dx \right)^3 \leq 8\left(\int_0^\infty f(x)^2 dx \right) \left(\int_0^\infty xf(x) dx \right)$$

2016 Miklós Schweitzer, 10

Tags: geometric probability , probability and stats , probability distribution , sphere , real analysis

Let $X$ and $Y$ be independent, identically distributed random points on the unit sphere in $\mathbb{R}^3$. For which distribution of $X$ will the expectation of the (Euclidean) distance of $X$ and $Y$ be maximal?

2021 IMO Shortlist, A5

Tags: inequalities , real analysis , algebra

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

2004 Nicolae Coculescu, 2

Tags: primitive , real analysis , function

Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that admits bounded primitives. Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\left\{ \begin{matrix} x, & \quad x\le 0 \\ f(1/x)\cdot\ln x ,& \quad x>0 \end{matrix}\right. $$ admits primitives. [i]Florian Dumitrel[/i]

1951 Miklós Schweitzer, 3

Tags: limit , real analysis

Consider the iterated sequence (1) $ x_0,x_1 \equal{} f(x_0),\dots,x_{n \plus{} 1} \equal{} f(x_n),\dots$, where $ f(x) \equal{} 4x \minus{} x^2$. Determine the points $ x_0$ of $ [0,1]$ for which (1) converges and find the limit of (1).

2008 IMS, 9

Tags: topology , real analysis

Let $ \gamma: [0,1]\rightarrow [0,1]\times [0,1]$ be a mapping such that for each $ s,t\in [0,1]$ \[ |\gamma(s) \minus{} \gamma(t)|\leq M|s \minus{} t|^\alpha \] in which $ \alpha,M$ are fixed numbers. Prove that if $ \gamma$ is surjective, then $ \alpha\leq\frac12$

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2

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

For real numbers $b>a>0$, let $f : [0,\ \infty)\rightarrow \mathbb{R}$ be a continuous function. Prove that : (i) $\lim_{\epsilon\rightarrow +0} \int_{a\epsilon}^{b\epsilon} \frac{f(x)}{x}dx=f(0)\ln \frac{b}{a}.$ (ii) If $\int_1^{\infty} \frac{f(x)}{x}dx$ converges, then $\int_0^{\infty} \frac{f(bx)-f(ax)}{x}dx=f(0)\ln \frac{a}{b}.$

2024 Brazil Undergrad MO, 2

Tags: real analysis , fixed point , polynomial , derivative , algebra

For each pair of integers $ j, k \geq 2 $, define the function $ f_{jk} : \mathbb{R} \to \mathbb{R} $ given by \[ f_{jk}(x) = 1 - (1 - x^j)^k. \] (a) Prove that for any integers $ j, k \geq 2 $, there exists a unique real number $ p_{jk} \in (0, 1) $ such that $ f_{jk}(p_{jk}) = p_{jk} $. Furthermore, defining $ \lambda_{jk} := f'_{jk}(p_{jk}) $, prove that $ \lambda_{jk} > 1 $. (b) Prove that $ p^j_{jk} = 1 - p_{kj} $ for any integers $ j, k \geq 2 $. (c) Prove that $ \lambda_{jk} = \lambda_{kj} $ for any integers $ j, k \geq 2 $.

2005 Romania National Olympiad, 2

Tags: real analysis , domain , trigonometry , function , limit , algebra

Let $f:[0,1)\to (0,1)$ a continous onto (surjective) function. a) Prove that, for all $a\in(0,1)$, the function $f_a:(a,1)\to (0,1)$, given by $f_a(x) = f(x)$, for all $x\in(a,1)$ is onto; b) Give an example of such a function.

2008 Alexandru Myller, 4

Tags: real analysis , function

Let be a function $ f:\mathbb{R}\rightarrow\mathbb{R} $ satisfying the following properties: $ \text{(i)} $ is continuous on the rational numbers. $ \text{(ii)} f(x)<f\left( x+\frac{1}{n}\right) , $ for any real $ x $ and natural $ n. $ Prove that $ f $ is increasing. [i]Gabriel Mârşanu, Mihai Piticari[/i]

2016 District Olympiad, 3

Tags: function , continuity , real analysis , algebra , functional equation

Find the continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property: $$ f\left( x+\frac{1}{n}\right) \le f(x) +\frac{1}{n},\quad\forall n\in\mathbb{Z}^* ,\quad\forall x\in\mathbb{R} . $$

ICMC 6, 5

Tags: continuous function , real analysis , function

Let $[0, 1]$ be the set $\{x \in \mathbb{R} : 0 \leq x \leq 1\}$. Does there exist a continuous function $g : [0, 1] \to [0, 1]$ such that no line intersects the graph of $g$ infinitely many times, but for any positive integer $n$ there is a line intersecting $g$ more than $n$ times? [i]Proposed by Ethan Tan[/i]

2015 VJIMC, 2

Tags: number theory , real analysis , modular arithmetic

[b]Problem 2[/b] Determine all pairs $(n, m)$ of positive integers satisfying the equation $$5^n = 6m^2 + 1\ . $$

1961 Miklós Schweitzer, 6

Tags: real analysis

[b]6.[/b] Consider a sequence $\{ a_n \}_{n=1}^{\infty}$ such that, for any convergent subsequence $\{ a_{n_k} \}$ of $\{a_n\}$, the sequence $\{ a_{n_k +1} \}$ also is convergent and has the same limit as $\{ a_{n_k}\}$. Prove that the sequence $\{ a_n \}$ is either convergent of has infinitely many accumulation points the set of which is dense in itself. Give an example for the second case. (A sequence $ x_n \to \infty $ or $-\infty$ is considered to be convergente, too) [b](S. 13)[/b]

2012 Romania National Olympiad, 2

Tags: real analysis , domain , reflection , algebra , geometric transformation , function , geometry

[color=darkred]Find all functions $f:\mathbb{R}\to\mathbb{R}$ with the following property: for any open bounded interval $I$, the set $f(I)$ is an open interval having the same length with $I$ .[/color]

1996 Miklós Schweitzer, 6

Tags: real analysis

Let $\{a_n\}$ be a bounded real sequence. (a) Prove that if X is a positive-measure subset of $\mathbb R$, then for almost all $x\in X$, there exist a subsequence $\{y_n\}$ of X such that $$\sum_{n=1}^\infty (n(y_n-x)-a_n)=1$$ (b) construct an unbounded sequence $\{a_n\}$ for which the above equation is also true.

2006 Iran MO (3rd Round), 6

Tags: trigonometry , dilation , geometric transformation , vector , real analysis , ratio , geometry

Assume that $C$ is a convex subset of $\mathbb R^{d}$. Suppose that $C_{1},C_{2},\dots,C_{n}$ are translations of $C$ that $C_{i}\cap C\neq\emptyset$ but $C_{i}\cap C_{j}=\emptyset$. Prove that \[n\leq 3^{d}-1\] Prove that $3^{d}-1$ is the best bound. P.S. In the exam problem was given for $n=3$.

1974 Miklós Schweitzer, 8

Tags: real analysis , advanced fields , function

Prove that there exists a topological space $ T$ containing the real line as a subset, such that the Lebesgue-measurable functions, and only those, extend continuously over $ T$. Show that the real line cannot be an everywhere-dense subset of such a space $ T$. [i]A. Csaszar[/i]

2020 Miklós Schweitzer, 11

Tags: vector , real analysis , function , inequalities

Given a real number $p>1$, a continuous function $h\colon [0,\infty)\to [0,\infty)$, and a smooth vector field $Y\colon \mathbb{R}^n \to \mathbb{R}^n$ with $\mathrm{div}~Y=0$, prove the following inequality \[\int_{\mathbb{R}^n}h(|x|)|x|^{p}\leq \int_{\mathbb{R}^{n}}h(|x|)|x+Y(x)|^{p}.\]

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]

2019 VJIMC, 2

Tags: functional equation , real analysis

Find all twice differentiable functions $f : \mathbb{R} \to \mathbb{R}$ such that $$f''(x) \cos(f(x))\geq(f'(x))^2 \sin(f(x)) $$ for every $x\in \mathbb{R}$. [i]Proposed by Orif Ibrogimov (Czech Technical University of Prague), Karim Rakhimov (University of Pisa)[/i]