Olympiad problems

2004 AMC 12/AHSME, 16

Tags: function , trigonometry , algebra , polynomial , complex number

A function $ f$ is defined by $ f(z) \equal{} i\bar z$, where $ i \equal{}\sqrt{\minus{}\!1}$ and $ \bar z$ is the complex conjugate of $ z$. How many values of $ z$ satisfy both $ |z| \equal{} 5$ and $ f (z) \equal{} z$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

1995 Cono Sur Olympiad, 2

Tags: ratio , function , geometry

The semicircle with centre $O$ and the diameter $AC$ is divided in two arcs $AB$ and $BC$ with ratio $1: 3$. $M$ is the midpoint of the radium $OC$. Let $T$ be the point of arc $BC$ such that the area of the cuadrylateral $OBTM$ is maximum. Find such area in fuction of the radium.

2012 China Team Selection Test, 3

Tags: function , group theory , abstract algebra , floor function , induction , algebra

$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.

2007 Balkan MO Shortlist, C2

Tags: function , inequalities , algebra , domain , combinatorics

Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find \[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]

2010 Poland - Second Round, 2

Tags: function , algebra

Find all monotonic functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[f(f(x) - y) + f(x+y) = 0,\] for every real $x, y$. (Note that monotonic means that function is not increasing or not decreasing)

Today's calculation of integrals, 893

Tags: calculus , integration , trigonometry , function , derivative , calculus computations , logarithm

Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

2008 Korea Junior Math Olympiad, 7

Tags: function , algebra , functional equation

Find all pairs of functions $f; g : R \to R$ such that for all reals $x.y \ne 0$ : $$f(x + y) = g \left(\frac{1}{x}+\frac{1}{y}\right) \cdot (xy)^{2008}$$

2010 Germany Team Selection Test, 1

Tags: function , algebra , modular arithmetic , divisibility , number theory

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

2003 Putnam, 1

Tags: algebra , polynomial , linear algebra , matrix , function , vector

Do there exist polynomials $a(x)$, $b(x)$, $c(y)$, $d(y)$ such that \[1 + xy + x^2y^2= a(x)c(y) + b(x)d(y)\] holds identically?

2004 National Olympiad First Round, 36

Tags: function

If the function $f$ satisfies the equation $f(x) + f\left ( \dfrac{1}{\sqrt[3]{1-x^3}}\right ) = x^3$ for every real $x \neq 1$, what is $f(-1)$? $ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ \dfrac 14 \qquad\textbf{(C)}\ \dfrac 12 \qquad\textbf{(D)}\ \dfrac 74 \qquad\textbf{(E)}\ \text{None of above} $

2011 Macedonia National Olympiad, 4

Tags: function , search , algebra

Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation \[ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . \]

2011 National Olympiad First Round, 35

Tags: inequalities , calculus , derivative , function

Which of these has the smallest maxima on positive real numbers? $\textbf{(A)}\ \frac{x^2}{1+x^{12}} \qquad\textbf{(B)}\ \frac{x^3}{1+x^{11}} \qquad\textbf{(C)}\ \frac{x^4}{1+x^{10}} \qquad\textbf{(D)}\ \frac{x^5}{1+x^{9}} \qquad\textbf{(E)}\ \frac{x^6}{1+x^{8}}$

1961 Putnam, A2

Tags: function , real analysis

For a real-valued function $f(x,y)$ of two positive real variables $x$ and $y$, define $f$ to be [i]linearly bounded[/i] if and only if there exists a positive number $K$ such that $|f(x,y)| < K(x+y)$ for all positive $x$ and $y.$ Find necessary and sufficient conditions on the real numbers $\alpha$ and $\beta$ such that $x^{\alpha}y^{\beta}$ is linearly bounded.

2008 District Olympiad, 2

Tags: function , domain , algebra

Consider the positive reals $ x$, $ y$ and $ z$. Prove that: a) $ \arctan(x) \plus{} \arctan(y) < \frac {\pi}{2}$ iff $ xy < 1$. b) $ \arctan(x) \plus{} \arctan(y) \plus{} \arctan(z) < \pi$ iff $ xyz < x \plus{} y \plus{} z$.

2008 iTest Tournament of Champions, 3

Tags: function , ratio

Let $\phi = \tfrac{1+\sqrt 5}2$ be the positive root of $x^2=x+1$. Define a function $f:\mathbb N\to\mathbb N$ by \begin{align*} f(0) &= 1\\ f(2x) &= \lfloor\phi f(x)\rfloor\\ f(2x+1) &= f(2x) + f(x). \end{align*} Find the remainder when $f(2007)$ is divided by $2008$.

2005 Mediterranean Mathematics Olympiad, 4

Tags: algebra , polynomial , function , geometric transformation , modular arithmetic , number theory , geometry

Let $A$ be the set of all polynomials $f(x)$ of order $3$ with integer coefficients and cubic coefficient $1$, so that for every $f(x)$ there exists a prime number $p$ which does not divide $2004$ and a number $q$ which is coprime to $p$ and $2004$, so that $f(p)=2004$ and $f(q)=0$. Prove that there exists a infinite subset $B\subset A$, so that the function graphs of the members of $B$ are identical except of translations

2025 District Olympiad, P3

Tags: function , limit , real analysis

Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuous and bijective function, such that $$\lim_{x\rightarrow\infty}\frac{f^{-1}(f(x)/x)}{x}=1.$$ [list=a] [*] Show that $\lim_{x\rightarrow\infty}\frac{f(x)}{x}=\infty$ and $\lim_{x\rightarrow\infty}\frac{f^{-1}(ax)}{f^{-1}(x)}=1$ for any $a>0$. [*] Give an example of function which satisfies the hypothesis.

1996 Estonia Team Selection Test, 3

Tags: function , algebra , functional equation

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$: $(i)$ $f(x)=-f(-x);$ $(ii)$ $f(x+1)=f(x)+1;$ $(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$

2013-2014 SDML (High School), 14

Tags: function

Let $X=\left\{1,2,3,4\right\}$. Consider a function $f:X\to X$. Let $f^1=f$ and $f^{k+1}=\left(f\circ f^k\right)$ for $k\geq1$. How many functions $f$ satisfy $f^{2014}\left(x\right)=x$ for all $x$ in $X$? $\text{(A) }9\qquad\text{(B) }10\qquad\text{(C) }12\qquad\text{(D) }15\qquad\text{(E) }18$

2010 Bundeswettbewerb Mathematik, 4

Tags: polynomial , function , algebra , functional equation , functional

In the following, let $N_0$ denotes the set of non-negative integers. Find all polynomials $P(x)$ that fulfill the following two properties: (1) All coefficients of $P(x)$ are from $N_0$. (2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$.

1991 AIME Problems, 7

Tags: quadratic , function , domain , ratio , algebra

Find $A^2$, where $A$ is the sum of the absolute values of all roots of the following equation: \begin{eqnarray*}x &=& \sqrt{19} + \frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{x}}}}}}}}}\end{eqnarray*}

2021 Romanian Master of Mathematics Shortlist, A1

Tags: algebra , functional equation , substitution , function

Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ f(xy+f(x)) + f(y) = xf(y) + f(x+y) \] for all real numbers $x$ and $y$.

1999 Irish Math Olympiad, 2

Tags: function , number theory , prime number , algebra

A function $ f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies: $ (a)$ $ f(ab)\equal{}f(a)f(b)$ whenever $ a$ and $ b$ are coprime; $ (b)$ $ f(p\plus{}q)\equal{}f(p)\plus{}f(q)$ for all prime numbers $ p$ and $ q$. Prove that $ f(2)\equal{}2,f(3)\equal{}3$ and $ f(1999)\equal{}1999.$

1975 Czech and Slovak Olympiad III A, 2

Tags: algebra , system of equations , floor function , inequalities , bounded , square , function

Show that the system of equations \begin{align*} \lfloor x\rfloor^2+\lfloor y\rfloor &=0, \\ 3x+y &=2, \end{align*} has infinitely many solutions and all these solutions satisfy bounds \begin{align*} 0<\ &x <4, \\ -9\le\ &y\le 1. \end{align*}

1977 IMO, 3

Tags: function , algebra , functional inequality , functional equation

Let $\mathbb{N}$ be the set of positive integers. Let $f$ be a function defined on $\mathbb{N}$, which satisfies the inequality $f(n + 1) > f(f(n))$ for all $n \in \mathbb{N}$. Prove that for any $n$ we have $f(n) = n.$