Olympiad problems

2013 Ukraine Team Selection Test, 9

Tags: function , algebra

Determine all functions $f:\Bbb{R}\to\Bbb{R}$ such that \[ f^2(x+y)=f^2(x)+2f(xy)+f^2(y), \] for all $x,y\in \Bbb{R}.$

2009 All-Russian Olympiad, 2

Tags: function , algebra

Consider the sequence of numbers $(a_n)$ ($n = 1, 2, \ldots$) defined as follows: $ a_1\in (1, 2)$, $ a_{k + 1} = a_k + \frac{k}{a_k}$ ($k = 1, 2, \ldots$). Prove that there exists at most one pair of distinct positive integers $(i, j)$ such that $a_i + a_j$ is an integer.

2015 Costa Rica - Final Round, F1

Tags: algebra , function

A function $f$ defined on integers such that $f (n) =n + 3$ if $n$ is odd $f (n) = \frac{n}{2}$ if $n$ is even If $k$ is an odd integer, determine the values for which $f (f (f (k))) = k$.

2006 Costa Rica - Final Round, 1

Tags: function

Let $f$ be a function that satisfies : \[ \displaystyle f(x)+2f\left(\frac{x+\frac{2001}2}{x-1}\right) = 4014-x. \] Find $f(2004)$.

2020 Turkey EGMO TST, 2

Tags: function , number theory

$p(m)$ is the number of distinct prime divisors of a positive integer $m>1$ and $f(m)$ is the $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$ th smallest prime divisor of $m$. Find all positive integers $n$ satisfying the equation: $$f(n^2+2) + f(n^2+5) = 2n-4$$

1976 IMO Longlists, 32

Tags: function , limit , random walk , martingale , combinatorics

We consider the infinite chessboard covering the whole plane. In every field of the chessboard there is a nonnegative real number. Every number is the arithmetic mean of the numbers in the four adjacent fields of the chessboard. Prove that the numbers occurring in the fields of the chessboard are all equal.

2007 Today's Calculation Of Integral, 181

Tags: calculus , integration , trigonometry , function , derivative , analytic geometry , logarithm

For real number $a,$ find the minimum value of $\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin 2x}{1+\sin^{2}x}-a\cos x\right| dx.$

2018 Thailand Mathematical Olympiad, 10

Tags: function , algebra

Let $a,b,c$ be non-zero real numbers.Prove that if function $f,g:\mathbb{R}\to\mathbb{R}$ satisfy $af(x+y)+bf(x-y)=cf(x)+g(y)$ for all real number $x,y$ that $y>2018$ then there exists a function $h:\mathbb{R}\to\mathbb{R}$ such that $f(x+y)+f(x-y)=2f(x)+h(y)$ for all real number $x,y$.

2009 Indonesia TST, 4

Tags: euler , function , number theory

Given positive integer $ n > 1$ and define \[ S \equal{} \{1,2,\dots,n\}. \] Suppose \[ T \equal{} \{t \in S: \gcd(t,n) \equal{} 1\}. \] Let $ A$ be arbitrary non-empty subset of $ A$ such thar for all $ x,y \in A$, we have $ (xy\mod n) \in A$. Prove that the number of elements of $ A$ divides $ \phi(n)$. ($ \phi(n)$ is Euler-Phi function)

2002 Federal Competition For Advanced Students, Part 2, 3

Tags: perimeter , trigonometry , function , geometry

Let $H$ be the orthocenter of an acute-angled triangle $ABC$. Show that the triangles $ABH,BCH$ and $CAH$ have the same perimeter if and only if the triangle $ABC$ is equilateral.

2004 Postal Coaching, 15

Tags: function , number theory

Show that for each integer $a$, there is a unique decomposition \[ a = \sum_{j=0}^{n} d_j 2^j , d_j \in (-1,0,1) \] such that no two consecutive $d_j$'s are nonzero. Show further that if $f$ is nondecreasing function from the set of all non-negative integers in to the set of all non-negative real numbers, and if $a = \sum_{j=0}^{n} c_j 2^j$ is any other decomposition of $a$ with $c_j \in (-1,0,1)$ , then \[ \sum_{j=0}^{n} |d_j| f(j) \leq \sum_{j=0}^{n} |c_j| f(j) \]

1978 Romania Team Selection Test, 8

Tags: function , domain , algebra

For any set $ A $ we say that two functions $ f,g:A\longrightarrow A $ are [i]similar,[/i] if there exists a bijection $ h:A\longrightarrow A $ such that $ f\circ h=h\circ g. $ [b]a)[/b] If $ A $ has three elements, construct a finite, arbitrary number functions, having as domain and codomain $ A, $ that are two by two similar, and every other function with the same domain and codomain as the ones determined is similar to, at least, one of them. [b]b)[/b] For $ A=\mathbb{R} , $ show that the functions $ \sin $ and $ -\sin $ are similar.

2015 Indonesia MO, 4

Tags: function , algebra , functional equation

Let function pair $f,g : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies \[ f(g(x)y + f(x)) = (y+2015)f(x) \] for every $x,y \in \mathbb{R^+} $ a. Prove that $f(x) = 2015g(x)$ for every $x \in \mathbb{R^+}$ b. Give an example of function pair $(f,g)$ that satisfies the statement above and $f(x), g(x) \geq 1$ for every $x \in \mathbb{R^+}$

1983 AMC 12/AHSME, 8

Tags: function

Let $f(x) = \frac{x+1}{x-1}$. Then for $x^2 \neq 1$, $f(-x)$ is $ \textbf{(A)}\ \frac{1}{f(x)}\qquad\textbf{(B)}\ -f(x)\qquad\textbf{(C)}\ \frac{1}{f(-x)}\qquad\textbf{(D)}\ -f(-x)\qquad\textbf{(E)}\ f(x) $

2005 Junior Balkan Team Selection Tests - Moldova, 8

Tags: inequalities , trinomial , quadratic , quadratic polynomial , function , algebra

The families of second degree functions $f_m, g_m: R\to R, $ are considered , $f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1$, $g_m (x) = m^2x^2 + mx - 1$, where $m$ is a real nonzero parameter. Show that, for any function $h$ of the second degree with the property that $g_m (x) \le h (x) \le f_m (x)$ for any real $x$, there exists $\lambda \in [0, 1]$ which verifies the condition $h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)$, whatever real $x$ is.

1998 Harvard-MIT Mathematics Tournament, 4

Tags: calculus , integration , function , ratio , geometric series , logarithm

Let $f(x)=1+\dfrac{x}{2}+\dfrac{x^2}{4}+\dfrac{x^3}{8}+\cdots,$ for $-1\leq x \leq 1$. Find $\sqrt{e^{\int\limits_0^1 f(x)dx}}$.

2018 Bundeswettbewerb Mathematik, 2

Tags: real number , algebra , equation , floor function , function

Find all real numbers $x$ satisfying the equation \[\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.\]

2006 Federal Competition For Advanced Students, Part 2, 2

Tags: function , functional equation , algebra

Find all monotonous functions $ f: \mathbb{R} \to \mathbb{R}$ that satisfy the following functional equation: \[f(f(x)) \equal{} f( \minus{} f(x)) \equal{} f(x)^2.\]

1982 IMO Longlists, 22

Tags: function , topology , algebra , discrete intermediate value theorem

Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$

2016 Mathematical Talent Reward Programme, MCQ: P 14

Tags: floor function , greatest integer function , algebra , function

Let $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Find $x$ such that $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor \rfloor = 88$ [list=1] [*] $\pi$ [*] 3.14 [*] $\frac{22}{7}$ [*] All of these [/list]

2022 Iran MO (3rd Round), 2

Tags: function , function inequality , number theory , functional equation , algebra

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that for all $x,y\in\mathbb{N}$: $$0\le y+f(x)-f^{f(y)}(x)\le1$$ that here $$f^n(x)=\underbrace{f(f(\ldots(f}_{n}(x))\ldots)$$

2016 Vietnam National Olympiad, 1

Tags: algebra , functional equation , function

Find all $a\in\mathbb{R}$ such that there is function $f:\mathbb{R}\to\mathbb{R}$ i) $f(1)=2016$ ii) $f(x+y+f(y))=f(x)+ay\quad\forall x,y\in\mathbb{R}$

1987 IMO Shortlist, 10

Tags: geometry , 3d geometry , sphere , trigonometry , function

Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$? [i]Proposed by Iceland.[/i]

2004 Moldova Team Selection Test, 7

Tags: inequalities , geometry , circumcircle , inradius , function , trigonometry , geometric transformation

Let $ABC$ be a triangle, let $O$ be its circumcenter, and let $H$ be its orthocenter. Let $P$ be a point on the segment $OH$. Prove that $6r\leq PA+PB+PC\leq 3R$, where $r$ is the inradius and $R$ the circumradius of triangle $ABC$. [b]Moderator edit:[/b] This is true only if the point $P$ lies inside the triangle $ABC$. (Of course, this is always fulfilled if triangle $ABC$ is acute-angled, since in this case the segment $OH$ completely lies inside the triangle $ABC$; but if triangle $ABC$ is obtuse-angled, then the condition about $P$ lying inside the triangle $ABC$ is really necessary.)

2015 AMC 10, 11

Tags: rectangle , function , ratio , pythagorean theorem , geometry

The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$? $\textbf{(A) }\dfrac27\qquad\textbf{(B) }\dfrac37\qquad\textbf{(C) }\dfrac{12}{25}\qquad\textbf{(D) }\dfrac{16}{25}\qquad\textbf{(E) }\dfrac34$