Olympiad problems

1996 IMO Shortlist, 8

Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that \[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0. \]

1981 AMC 12/AHSME, 18

Tags: trigonometry , function

The number of real solutions to the equation \[ \frac{x}{100} = \sin x \] is $\text{(A)} \ 61 \qquad \text{(B)} \ 62 \qquad \text{(C)} \ 63 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65$

1996 South africa National Olympiad, 2

Tags: function , trigonometry , inequalities , algebra

Find all real numbers for which $3^x+4^x=5^x$.

2021-IMOC qualification, A0

Tags: function

Consider the following function $ f(x)=\frac{1}{1+2^{1-2x}}$. Compute the value of $$f\left(\frac{1}{10}\right)+f\left(\frac{2}{10}\right)+...+f\left(\frac{9}{10}\right).$$

2009 AMC 12/AHSME, 24

Tags: function , inequalities , logarithm

The [i]tower function of twos[/i] is defined recursively as follows: $ T(1) \equal{} 2$ and $ T(n \plus{} 1) \equal{} 2^{T(n)}$ for $ n\ge1$. Let $ A \equal{} (T(2009))^{T(2009)}$ and $ B \equal{} (T(2009))^A$. What is the largest integer $ k$ such that \[ \underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}} \]is defined? $ \textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E)}\ 2013$

2015 Abels Math Contest (Norwegian MO) Final, 1b

Tags: function , functional equation , algebra

Find all functions $f : R \to R$ such that $x^2f(yf(x))= y^2f(x)f(f(x))$ for all real numbers $x$ and $y$.

1999 National Olympiad First Round, 28

Tags: function , trigonometry

Find the number of functions defined on positive real numbers such that $ f\left(1\right) \equal{} 1$ and for every $ x,y\in \Re$, $ f\left(x^{2} y^{2} \right) \equal{} f\left(x^{4} \plus{} y^{4} \right)$. $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many}$

2012 Pre-Preparation Course Examination, 1

Tags: function , real analysis

Suppose that $X$ and $Y$ are two metric spaces and $f:X \longrightarrow Y$ is a continious function. Also for every compact set $K \subseteq Y$, it's pre-image $f^{pre}(K)$ is a compact set in $X$. Prove that $f$ is a closed function, i.e for every close set $C\subseteq X$, it's image $f(C)$ is a closed subset of $Y$.

2004 Italy TST, 3

Tags: function , quadratic , algorithm , induction , floor function , algebra

Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$, \[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]

2007 Indonesia MO, 6

Tags: function , algebra

Find all triples $ (x,y,z)$ of real numbers which satisfy the simultaneous equations \[ x \equal{} y^3 \plus{} y \minus{} 8\] \[y \equal{} z^3 \plus{} z \minus{} 8\] \[ z \equal{} x^3 \plus{} x \minus{} 8.\]

2009 Today's Calculation Of Integral, 514

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

Prove the following inequalities: (1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$ (2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$

Oliforum Contest I 2008, 2

Tags: floor function , function , algebra

Let $ \{a_n\}_{n \in \mathbb{N}_0}$ be a sequence defined as follows: $ a_1=0$, $ a_n=a_{[\frac{n}{2}]}+(-1)^{n(n+1)/2}$, where $ [x]$ denotes the floor function. For every $ k \ge 0$, find the number $ n(k)$ of positive integers $ n$ such that $ 2^k \le n < 2^{k+1}$ and $ a_n=0$.

2009 Iran Team Selection Test, 3

Tags: calculus , function , three variable inequality , algebra , inequality

Suppose that $ a$,$ b$,$ c$ be three positive real numbers such that $ a\plus{}b\plus{}c\equal{}3$ . Prove that : $ \frac{1}{2\plus{}a^{2}\plus{}b^{2}}\plus{}\frac{1}{2\plus{}b^{2}\plus{}c^{2}}\plus{}\frac{1}{2\plus{}c^{2}\plus{}a^{2}} \leq \frac{3}{4}$

2007 IMS, 8

Tags: function , topology , real analysis

Let \[T=\{(tq,1-t) \in\mathbb R^{2}| t \in [0,1],q\in\mathbb Q\}\]Prove that each continuous function $f: T\longrightarrow T$ has a fixed point.

1998 Iran MO (3rd Round), 1

Tags: function , number theory

Find all functions $f: \mathbb N \to \mathbb N$ such that for all positive integers $m,n$, [b](i)[/b] $mf(f(m))=\left( f(m) \right)^2$, [b](ii)[/b] If $\gcd(m,n)=d$, then $f(mn) \cdot f(d)=d \cdot f(m) \cdot f(n)$, [b](iii)[/b] $f(m)=m$ if and only if $m=1$.

2003 IMO Shortlist, 6

Tags: inequalities , function , calculus

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$. Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right). \] [hide="comment"] [i]Edited by Orl.[/i] [/hide] [i]Proposed by Reid Barton, USA[/i]

2007 Romania Team Selection Test, 4

Tags: function , geometric transformation , reflection , geometry

Let $\mathcal O_{1}$ and $\mathcal O_{2}$ two exterior circles. Let $A$, $B$, $C$ be points on $\mathcal O_{1}$ and $D$, $E$, $F$ points on $\mathcal O_{1}$ such that $AD$ and $BE$ are the common exterior tangents to these two circles and $CF$ is one of the interior tangents to these two circles, and such that $C$, $F$ are in the interior of the quadrilateral $ABED$. If $CO_{1}\cap AB=\{M\}$ and $FO_{2}\cap DE=\{N\}$ then prove that $MN$ passes through the middle of $CF$.

2005 MOP Homework, 1

Tags: algebra , polynomial , parameterization , modular arithmetic , function , number theory

Let $n$ be a natural number and $f_1$, $f_2$, ..., $f_n$ be polynomials with integers coeffcients. Show that there exists a polynomial $g(x)$ which can be factored (with at least two terms of degree at least $1$) over the integers such that $f_i(x)+g(x)$ cannot be factored (with at least two terms of degree at least $1$) over the integers for every $i$.

1997 Korea - Final Round, 3

Tags: function , algebra

Find all pairs of functions $ f, g: \mathbb R \to \mathbb R$ such that [list] (i) if $ x < y$, then $ f(x) < f(y)$; (ii) $ f(xy) \equal{} g(y)f(x) \plus{} f(y)$ for all $ x, y \in \mathbb R$. [/list]

1987 Traian Lălescu, 1.1

Tags: algebra , function

Let $ a\in\mathbb{R}. $ Prove the following proposition: $$ \left( x,y\in\mathbb{R}\implies x^4+y^4+axy+2\ge 0 \right)\iff |a|\le 4. $$

2004 Miklós Schweitzer, 6

Tags: real analysis , function , topology

Is is true that if the perfect set $F\subseteq [0,1]$ is of zero Lebesgue measure then those functions in $C^1[0,1]$ which are one-to-one on $F$ form a dense subset of $C^1[0,1]$? (We use the metric $$d(f,g)=\sup_{x\in[0,1]} |f(x)-g(x)| + \sup_{x\in[0,1]} |f'(x)-g'(x)|$$ to define the topology in the space $C^1[0,1]$ of continuously differentiable real functions on $[0,1]$.)

2009 Today's Calculation Of Integral, 489

Tags: calculus , integration , limit , function , calculus computations

Find the following limit. $ \lim_{n\to\infty} \int_{\minus{}1}^1 |x|\left(1\plus{}x\plus{}\frac{x^2}{2}\plus{}\frac{x^3}{3}\plus{}\cdots \plus{}\frac{x^{2n}}{2n}\right)\ dx$.

Today's calculation of integrals, 855

Tags: calculus , integration , function , limit , calculus computations

Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$. For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$

2006 District Olympiad, 4

Tags: inequalities , integration , function , real analysis

Let $\mathcal F = \{ f: [0,1] \to [0,\infty) \mid f$ continuous $\}$ and $n$ an integer, $n\geq 2$. Find the smallest real constant $c$ such that for any $f\in \mathcal F$ the following inequality takes place \[ \int^1_0 f \left( \sqrt [n] x \right) dx \leq c \int^1_0 f(x) dx. \]

2008 iTest Tournament of Champions, 3

Tags: function , modular arithmetic , euler

For how many integers $1\leq n\leq 9999$ is there a solution to the congruence \[\phi(n)\equiv 2\,\,\,\pmod{12},\] where $\phi(n)$ is the Euler phi-function?