Olympiad problems

2021 Science ON all problems, 3

Define $E\subseteq \{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$ such that $E$ posseses the following properties:\\ $\textbf{(i)}$ If $\int_0^1 f(x)g(x) dx = 0$ for $f\in E$ with $\int_0^1f^2(t)dt \neq 0$, then $g\in E$; \\ $\textbf{(ii)}$ There exists $h\in E$ with $\int_0^1 h^2(t)dt\neq 0$.\\ Prove that $E=\{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$. \\ [i](Andrei Bâra)[/i]

2011 Morocco National Olympiad, 3

Tags: algebra , function

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y, \in \mathbb{R}$, \[xf(x+xy)=xf(x)+f(x^{2})\cdot f(y).\]

2004 Vietnam Team Selection Test, 2

Tags: search , integration , calculus , function , algebra

Find all real values of $\alpha$, for which there exists one and only one function $f: \mathbb{R} \mapsto \mathbb{R}$ and satisfying the equation \[ f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y \] for all $x, y \in \mathbb{R}$.

2014 USAMTS Problems, 3:

Tags: logarithm , floor function , inequalities , ceiling function , function

Let $a_1,a_2,a_3,...$ be a sequence of positive real numbers such that: (i) For all positive integers $m,n$, we have $a_{mn}=a_ma_n$ (ii) There exists a positive real number $B$ such that for all positive integers $m,n$ with $m<n$, we have $a_m < Ba_n$ Find all possible values of $\log_{2015}(a_{2015}) - \log_{2014}(a_{2014})$

1974 Miklós Schweitzer, 6

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

Let $ f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ ,$ where $ \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty$ for some $ \alpha > 2$. Let us assume that for some $ \beta > 1/{\alpha}$, we have $ f(x)\equal{}O(e^{\minus{}x^{\beta}})$ as $ x \rightarrow \infty$. Prove that $ a_n$ is identically $ 0$. [i]G. Halasz[/i]

1990 IMO Longlists, 66

Tags: algebra , limit , trigonometry , function

Find all the continuous bounded functions $f: \mathbb R \to \mathbb R$ such that \[(f(x))^2 -(f(y))^2 = f(x + y)f(x - y) \text{ for all } x, y \in \mathbb R.\]

2010 Kosovo National Mathematical Olympiad, 1

Tags: function , algebra

If the real function $f(x)=\cos x+\sum_{i=1}^{n}\cos(a_ix)$ is periodic, prove that $a_i,i\in\{1,2,...,n\}$, are rational numbers.

2006 MOP Homework, 7

Tags: function , inequalities

for real number $a,b,c$ in interval $ (0,1]$ prove that: $\frac{a}{bc+1}+\frac{b}{ac+1}+\frac{c}{ab+1} \leq 2$

2005 Today's Calculation Of Integral, 14

Tags: algebra , trigonometry , domain , logarithm , integration , calculus , function

Calculate the following indefinite integrals. [1] $\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx$ [2] $\int x\log_{10} x dx$ [3] $\int \frac{x}{\sqrt{2x-1}}dx$ [4] $\int (x^2+1)\ln x dx$ [5] $\int e^x\cos x dx$

2003 Bundeswettbewerb Mathematik, 2

Tags: algebra , system of equations , function

Find all triples $\left(x,\ y,\ z\right)$ of integers satisfying the following system of equations: $x^3-4x^2-16x+60=y$; $y^3-4y^2-16y+60=z$; $z^3-4z^2-16z+60=x$.

2007 South East Mathematical Olympiad, 3

Tags: number theory , function

Let $a_i=min\{ k+\dfrac{i}{k}|k \in N^*\}$, determine the value of $S_{n^2}=[a_1]+[a_2]+\cdots +[a_{n^2}]$, where $n\ge 2$ . ($[x]$ denotes the greatest integer not exceeding x)

PEN J Problems, 2

Tags: abstract algebra , real analysis , divisor function , group theory , function

Show that for all $n \in \mathbb{N}$, \[n = \sum^{}_{d \vert n}\phi(d).\]

1975 Putnam, B5

Tags: function , convergence

Define $f_{0}(x)=e^x$ and $f_{n+1}(x)=x f_{n}'(x)$. Show that $\sum_{n=0}^{\infty} \frac{f_{n}(1)}{n!}=e^e$.

2009 AMC 12/AHSME, 25

Tags: algebra , domain , trigonometry , modular arithmetic , function

The first two terms of a sequence are $ a_1 \equal{} 1$ and $ a_2 \equal{} \frac {1}{\sqrt3}$. For $ n\ge1$, \[ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} a_{n \plus{} 1}}{1 \minus{} a_na_{n \plus{} 1}}. \]What is $ |a_{2009}|$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2 \minus{} \sqrt3\qquad \textbf{(C)}\ \frac {1}{\sqrt3}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ 2 \plus{} \sqrt3$

2006 Romania Team Selection Test, 4

Tags: function , inequalities , integration , derivative , calculus

Let $p$, $q$ be two integers, $q\geq p\geq 0$. Let $n \geq 2$ be an integer and $a_0=0, a_1 \geq 0, a_2, \ldots, a_{n-1},a_n = 1$ be real numbers such that \[ a_{k} \leq \frac{ a_{k-1} + a_{k+1} } 2 , \ \forall \ k=1,2,\ldots, n-1 . \] Prove that \[ (p+1) \sum_{k=1}^{n-1} a_k^p \geq (q+1) \sum_{k=1}^{n-1} a_k^q . \]

2023 USA IMOTST, 3

Tags: function , algebra

Let $\mathbb{N}$ denote the set of positive integers. Fix a function $f: \mathbb{N} \rightarrow \mathbb{N}$ and for any $m,n \in \mathbb{N}$ define $$\Delta(m,n)=\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(m)\ldots))-\underbrace{f(f(\ldots f}_{f(m)\text{ times}}(n)\ldots)).$$ Suppose $\Delta(m,n) \neq 0$ for any distinct $m,n \in \mathbb{N}$. Show that $\Delta$ is unbounded, meaning that for any constant $C$ there exists $m,n \in \mathbb{N}$ with $\left|\Delta(m,n)\right| > C$.

PEN K Problems, 11

Tags: algebra , functional equation , function

Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $m,n\in \mathbb{N}_{0}$: \[mf(n)+nf(m)=(m+n)f(m^{2}+n^{2}).\]

2004 Uzbekistan National Olympiad, 1

Tags: floor function , algebra , function

Solve the equation: $[\sqrt x+\sqrt{x+1}]+[\sqrt {4x+2}]=18$

2014 Contests, 2

Tags: algebra , function , limit , functional equation

Find all functions $f:R\rightarrow R$ such that \[ f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy)) \] for all $x,y\in R$.

2012 Indonesia TST, 1

Tags: function , floor function , algebra , induction

Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.

1985 Vietnam Team Selection Test, 3

Tags: algebra , function , functional equation

Suppose a function $ f: \mathbb R\to \mathbb R$ satisfies $ f(f(x)) \equal{} \minus{} x$ for all $ x\in \mathbb R$. Prove that $ f$ has infinitely many points of discontinuity.

2016 Mathematical Talent Reward Programme, MCQ: P 2

Tags: function , limit , summation , infinite sum

Let $f$ be a function satisfying $f(x+y+z)=f(x)+f(y)+f(z)$ for all integers $x$, $y$, $z$. Suppose $f(1)=1$, $f(2)=2$. Then $\lim \limits_{n\to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r)$ equals [list=1] [*] 4 [*] 6 [*] 12 [*] 24 [/list]

2011 IMAC Arhimede, 1

Tags: algebra , function

Find all functions $f: \mathbb{N} \rightarrow [0, +\infty)$ such that $f(1000)=10$ and $f(n+1)= \sum_{k=1}^n \frac{1}{f^2(k) + f(k)f(k+1) + f^2(k+1)}$ for all $n \in \mathbb{N}$. (Here, $f^2(i)$ means $(f(i))^2$.)

2009 AMC 12/AHSME, 23

Tags: rotation , geometry , algebra , analytic geometry , geometric transformation , quadratic , function

Functions $ f$ and $ g$ are quadratic, $ g(x) \equal{} \minus{} f(100 \minus{} x)$, and the graph of $ g$ contains the vertex of the graph of $ f$. The four $ x$-intercepts on the two graphs have $ x$-coordinates $ x_1$, $ x_2$, $ x_3$, and $ x_4$, in increasing order, and $ x_3 \minus{} x_2 \equal{} 150$. The value of $ x_4 \minus{} x_1$ is $ m \plus{} n\sqrt p$, where $ m$, $ n$, and $ p$ are positive integers, and $ p$ is not divisible by the square of any prime. What is $ m \plus{} n \plus{} p$? $ \textbf{(A)}\ 602\qquad \textbf{(B)}\ 652\qquad \textbf{(C)}\ 702\qquad \textbf{(D)}\ 752\qquad \textbf{(E)}\ 802$

2003 Brazil National Olympiad, 2

Tags: limit , algebra , function

Let $f(x)$ be a real-valued function defined on the positive reals such that (1) if $x < y$, then $f(x) < f(y)$, (2) $f\left(2xy\over x+y\right) \geq {f(x) + f(y)\over2}$ for all $x$. Show that $f(x) < 0$ for some value of $x$.