Olympiad problems

2004 Putnam, B3

Determine all real numbers $a>0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region $R=\{(x,y): 0\le x\le a, 0\le y\le f(x)\}$ has perimeter $k$ units and area $k$ square units for some real number $k$.

2007 India IMO Training Camp, 3

Tags: function , probability , expected value , combinatorics

Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define \[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.\] Determine $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$ (Here $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$)

1969 IMO Longlists, 59

Tags: function , algebra , continuous function

$(SWE 2)$ For each $\lambda (0 < \lambda < 1$ and $\lambda = \frac{1}{n}$ for all $n = 1, 2, 3, \cdots)$, construct a continuous function $f$ such that there do not exist $x, y$ with $0 < \lambda < y = x + \lambda \le 1$ for which $f(x) = f(y).$

2003 District Olympiad, 4

Tags: function , algebra , logarithm

Let $\displaystyle a,b,c,d \in \mathbb R$ such that $\displaystyle a>c>d>b>1$ and $\displaystyle ab>cd$. Prove that $\displaystyle f : \left[ 0,\infty \right) \to \mathbb R$, defined through \[ \displaystyle f(x) = a^x+b^x-c^x-d^x, \, \forall x \geq 0 , \] is strictly increasing.

2018 IFYM, Sozopol, 6

Tags: functional equation , function , algebra

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$, such that $f(x+y) = f(y) f(x f(y))$ for every two real numbers $x$ and $y$.

2005 MOP Homework, 3

Tags: function , induction , algebra

Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that (a) $f(1)=1$ (b) $f(n+2)+(n^2+4n+3)f(n)=(2n+5)f(n+1)$ for all $n \in \mathbb{N}$. (c) $f(n)$ divides $f(m)$ if $m>n$.

1997 Moldova Team Selection Test, 9

Tags: function , algebra

Find all $t\in \mathbb Z$ such that: exists a function $f:\mathbb Z^+\to \mathbb Z$ such that: $f(1997)=1998$ $\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$

1985 IMO Longlists, 70

Tags: function , algebra

Let $C$ be a class of functions $f : \mathbb N \to \mathbb N$ that contains the functions $S(x) = x + 1$ and $E(x) = x - [\sqrt x]^2$ for every $x \in \mathbb N$. ($[x]$ is the integer part of $x$.) If $C$ has the property that for every $f, g \in C, f + g, fg, f \circ g \in C$, show that the function $\max(f(x) - g(x), 0)$ is in $C$, for all $f; g \in C$.

1954 AMC 12/AHSME, 17

Tags: function

The graph of the function $ f(x) \equal{} 2x^3 \minus{} 7$ goes: $ \textbf{(A)}\ \text{up to the right and down to the left} \\ \textbf{(B)}\ \text{down to the right and up to the left} \\ \textbf{(C)}\ \text{up to the right and up to the left} \\ \textbf{(D)}\ \text{down to the right and down to the left} \\ \textbf{(E)}\ \text{none of these ways.}$

1978 Austrian-Polish Competition, 1

Tags: function , functional equation , algebra

Determine all functions $f:(0;\infty)\to \mathbb{R}$ that satisfy $$f(x+y)=f(x^2+y^2)\quad \forall x,y\in (0;\infty)$$

1986 Kurschak Competition, 3

Tags: function , modular arithmetic , algebra , binomial theorem , combinatorics

A and B plays the following game: they choose randomly $k$ integers from $\{1,2,\dots,100\}$; if their sum is even, A wins, else B wins. For what values of $k$ does A and B have the same chance of winning?

2006 AMC 10, 7

Tags: function

Which of the following is equivalent to $ \displaystyle \sqrt {\frac {x}{1 \minus{} \frac {x \minus{} 1}{x}}}$ when $ x < 0$? $ \textbf{(A) } \minus{} x \qquad \textbf{(B) } x \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \sqrt {\frac x2} \qquad \textbf{(E) } x\sqrt { \minus{} 1}$

2019 Jozsef Wildt International Math Competition, W. 14

Tags: trigonometry , inequalities , function

If $a$, $b$, $c > 0$; $ab + bc + ca = 3$ then: $$4\left(\tan^{-1} 2\right)\left(\tan^{-1}\left(\sqrt[3]{abc}\right)\right) \leq \pi \tan^{-1}\left(1 + \sqrt[3]{abc}\right)$$

2006 China Team Selection Test, 2

Tags: function , polynomial , algebra

The function $f(n)$ satisfies $f(0)=0$, $f(n)=n-f \left( f(n-1) \right)$, $n=1,2,3 \cdots$. Find all polynomials $g(x)$ with real coefficient such that \[ f(n)= [ g(n) ], \qquad n=0,1,2 \cdots \] Where $[ g(n) ]$ denote the greatest integer that does not exceed $g(n)$.

2010 Today's Calculation Of Integral, 565

Tags: calculus , integration , function , derivative , symmetry , calculus computations

Prove that $ f(x)\equal{}\int_0^1 e^{\minus{}|t\minus{}x|}t(1\minus{}t)dt$ has maximal value at $ x\equal{}\frac 12$.

VMEO II 2005, 7

Tags: algebra , functional equation , polynomial , function

Find all function $f:[0,\infty )\to\mathbb{R}$ such that $f$ is monotonic and \[ [f(x)+f(y)]^2=f(x^2-y^2)+f(2xy) \] for all $x\geq y\geq 0$

2012 Iran Team Selection Test, 2

Tags: binomial coefficient , function , induction , ceiling function , algebra

The function $f:\mathbb R^{\ge 0} \longrightarrow \mathbb R^{\ge 0}$ satisfies the following properties for all $a,b\in \mathbb R^{\ge 0}$: [b]a)[/b] $f(a)=0 \Leftrightarrow a=0$ [b]b)[/b] $f(ab)=f(a)f(b)$ [b]c)[/b] $f(a+b)\le 2 \max \{f(a),f(b)\}$. Prove that for all $a,b\in \mathbb R^{\ge 0}$ we have $f(a+b)\le f(a)+f(b)$. [i]Proposed by Masoud Shafaei[/i]

2003 AMC 12-AHSME, 6

Tags: function

Define $ x \heartsuit y$ to be $ |x\minus{}y|$ for all real numbers $ x$ and $ y$. Which of the following statements is [b]not[/b] true? $\textbf{(A)}\ x \heartsuit y \equal{} y \heartsuit x \text{ for all } x \text{ and } y$ $\textbf{(B)}\ 2(x \heartsuit y) \equal{} (2x) \heartsuit (2y) \text{ for all } x \text{ and } y$ $\textbf{(C)}\ x \heartsuit 0 \equal{} x \text{ for all } x$ $\textbf{(D)}\ x \heartsuit x \equal{} 0 \text{ for all } x$ $\textbf{(E)}\ x \heartsuit y > 0 \text{ if } x \ne y$

1989 IberoAmerican, 2

Tags: function , algebra

Let the function $f$ be defined on the set $\mathbb{N}$ such that $\text{(i)}\ \ \quad f(1)=1$ $\text{(ii)}\ \quad f(2n+1)=f(2n)+1$ $\text{(iii)}\quad f(2n)=3f(n)$ Determine the set of values taken $f$.

1998 Switzerland Team Selection Test, 1

Tags: functional , periodic , function , algebra

A function $f : R -\{0\} \to R$ has the following properties: (i) $f(x)- f(y) = f(x)f\left(\frac{1}{y}\right)- f(y)f\left(\frac{1}{x}\right)$ for all $x,y \ne 0$, (ii) $f$ takes the value $\frac12$ at least once. Determine $f(-1)$. Prove that $f$ is a periodic function

2010 VJIMC, Problem 4

Tags: function , geometry , rectangle

Let $f:[0,1]\to\mathbb R$ be a function satisfying $$|f(x)-f(y)|\le|x-y|$$for every $x,y\in[0,1]$. Show that for every $\varepsilon>0$ there exists a countable family of rectangles $(R_i)$ of dimensions $a_i\times b_i$, $a_i\le b_i$ in the plane such that $$\{(x,f(x)):x\in[0,1]\}\subset\bigcup_iR_i\text{ and }\sum_ia_i<\varepsilon.$$(The edges of the rectangles are not necessarily parallel to the coordinate axes.)

2004 Alexandru Myller, 4

Tags: continuity , function , darboux s property , darboux , real analysis , monotone

Let be a real function that has the intermediate value property and is monotone on the irrationals. Show that it's continuous. [i]Mihai Piticari[/i]

2013 Albania Team Selection Test, 1

Tags: ratio , function , number theory

Find the 3-digit number whose ratio with the sum of its digits it's minimal.

2000 Belarus Team Selection Test, 5.3

Tags: function , linear algebra , combinatorics , ramsey theory

Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.

2012 ISI Entrance Examination, 2

Tags: function , limit , calculus , derivative , calculus computations

Consider the following function \[g(x)=(\alpha+|x|)^{2}e^{(5-|x|)^{2}}\] [b]i)[/b] Find all the values of $\alpha$ for which $g(x)$ is continuous for all $x\in\mathbb{R}$ [b]ii)[/b]Find all the values of $\alpha$ for which $g(x)$ is differentiable for all $x\in\mathbb{R}$.