Olympiad problems

2018 Rio de Janeiro Mathematical Olympiad, 3

Let $n$ be a positive integer. A function $f : \{1, 2, \dots, 2n\} \to \{1, 2, 3, 4, 5\}$ is [i]good[/i] if $f(j+2)$ and $f(j)$ have the same parity for every $j = 1, 2, \dots, 2n-2$. Prove that the number of good functions is a perfect square.

2010 Romania National Olympiad, 1

Tags: quadratic , vieta , function , algebra

Let $(a_n)_{n\ge0}$ be a sequence of positive real numbers such that \[\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\ \text{for any }n\ge 0.\] Prove that $(a_n)_{n\ge0}$ is a geometric sequence. [i]Lucian Dragomir[/i]

2009 VTRMC, Problem 3

Tags: calculus , integration , function

Define $f(x)=\int^x_0\int^x_0e^{u^2v^2}dudv$. Calculate $2f''(2)+f'(2)$.

1992 Iran MO (2nd round), 3

Tags: function , induction , number theory

Let $X \neq \varnothing$ be a finite set and let $f: X \to X$ be a function such that for every $x \in X$ and a fixed prime $p$ we have $f^p(x)=x.$ Let $Y=\{x \in X | f(x) \neq x\}.$ Prove that the number of the members of the set $Y$ is divisible by $p.$ [i]Note.[/i] ${f^p(x)=x = \underbrace{f(f(f(\cdots ((f}_{ p \text{ times}}(x) ) \cdots )))} .$

2012 District Olympiad, 2

Tags: function , equation , algebra

[b]a)[/b] Solve in $ \mathbb{R} $ the equation $ 2^x=x+1. $ [b]b)[/b] If a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has the property that $$ (f\circ f)(x)=2^x-1,\quad\forall x\in\mathbb{R} , $$ then $ f(0)+f(1)=1. $

Kvant 2020, M1069

Tags: combinatorics , function

Every day, some pairs of families living in a city may choose to exchange their apartments. A family may only participate in one exchange in a day. Prove that any complex exchange of apartments between several families can be carried out in two days. [i]Proposed by N. Konstantinov and A. Shnirelman[/i]

2006 IMC, 6

Tags: function , algebra , polynomial , trigonometry , real analysis

Find all sequences $a_{0}, a_{1},\ldots, a_{n}$ of real numbers such that $a_{n}\neq 0$, for which the following statement is true: If $f: \mathbb{R}\to\mathbb{R}$ is an $n$ times differentiable function and $x_{0}<x_{1}<\ldots <x_{n}$ are real numbers such that $f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0$ then there is $h\in (x_{0}, x_{n})$ for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]

2017 Miklós Schweitzer, 6

Tags: algebra , functional inequality , convex function , function , interval

Let $I$ and $J$ be intervals. Let $\varphi,\psi:I\to\mathbb{R}$ be strictly increasing continuous functions and let $\Phi,\Psi:J\to\mathbb{R}$ be continuous functions. Suppose that $\varphi(x)+\psi(x)=x$ and $\Phi(u)+\Psi(u)=u$ holds for all $x\in I$ and $u\in J$. Show that if $f:I\to J$ is a continuous solution of the functional inequality $$f\big(\varphi(x)+\psi(y)\big)\le \Phi\big(f(x)\big)+\Psi\big(f(y)\big)\qquad (x,y\in I),$$then $\Phi\circ f\circ \varphi^{-1}$ and $\Psi\circ f\circ \psi^{-1}$ are convex functions.

2010 Laurențiu Panaitopol, Tulcea, 1

Tags: function , primitivableness , antiderivative , differentiability , integral , indefinite integral , real analysis

Let be two real numbers $ a<b $ and a function $ f:[a,b]\longrightarrow\mathbb{R} $ having the property that if the sequence $ \left(f\left( x_n \right)\right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. [b]a)[/b] Prove that if $ f $ admits antiderivatives, then $ f $ is integrable. [b]b)[/b] Is the converse of [b]a)[/b] true? [i]Marcelina Popa[/i]

1981 IMO Shortlist, 16

Tags: function , limit , algebra , sequence , recurrence relation

A sequence of real numbers $u_1, u_2, u_3, \dots$ is determined by $u_1$ and the following recurrence relation for $n \geq 1$: \[4u_{n+1} = \sqrt[3]{ 64u_n + 15.}\] Describe, with proof, the behavior of $u_n$ as $n \to \infty.$

2010 Austria Beginners' Competition, 3

Tags: function

Let $x$ and $y$ be positive real numbers with $x + y =1 $. Prove that $$\frac{(3x-1)^2}{x}+ \frac{(3y-1)^2}{y} \ge1.$$ For which $x$ and $y$ equality holds? (K. Czakler, GRG 21, Vienna)

2008 German National Olympiad, 3

Tags: function , algebra

Find all functions $ f$ defined on non-negative real numbers having the following properties: (i) For all non-negative $ x$ it is $ f(x) \geq 0$. (ii) It is $ f\left(1\right)\equal{}\frac 12$. (iii) For all non-negative numbers $ x,y$ it is $ f\left( y \cdot f(x) \right) \cdot f(x) \equal{} f(x\plus{}y)$.

2016 CMIMC, 4

Tags: function , number theory , prime factorization

For some positive integer $n$, consider the usual prime factorization \[n = \displaystyle \prod_{i=1}^{k} p_{i}^{e_{i}}=p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k},\] where $k$ is the number of primes factors of $n$ and $p_{i}$ are the prime factors of $n$. Define $Q(n), R(n)$ by \[ Q(n) = \prod_{i=1}^{k} p_{i}^{p_{i}} \text{ and } R(n) = \prod_{i=1}^{k} e_{i}^{e_{i}}. \] For how many $1 \leq n \leq 70$ does $R(n)$ divide $Q(n)$?

1968 IMO, 5

Tags: function , algebra , periodic function , functional equation

Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$. Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.

2009 IMS, 5

Tags: calculus , integration , function , real analysis

Suppose that $ f: \mathbb R^2\rightarrow \mathbb R$ is a non-negative and continuous function that $ \iint_{\mathbb R^2}f(x,y)dxdy\equal{}1$. Prove that there is a closed disc $ D$ with the least radius possible such that $ \iint_D f(x,y)dxdy\equal{}\frac12$.

2002 District Olympiad, 3

Tags: matrix , function , linear algebra

a)Find a matrix $A\in \mathcal{M}_3(\mathbb{C})$ such that $A^2\neq O_3$ and $A^3=O_3$. b)Let $n,p\in\{2,3\}$. Prove that if there is bijective function $f:\mathcal{M}_n(\mathbb{C})\rightarrow \mathcal{M}_p(\mathbb{C})$ such that $f(XY)=f(X)f(Y),\ \forall X,Y\in \mathcal{M}_n(\mathbb{C})$, then $n=p$. [i]Ion Savu[/i]

2005 Italy TST, 3

Tags: function , number theory

The function $\psi : \mathbb{N}\rightarrow\mathbb{N}$ is defined by $\psi (n)=\sum_{k=1}^n\gcd (k,n)$. $(a)$ Prove that $\psi (mn)=\psi (m)\psi (n)$ for every two coprime $m,n \in \mathbb{N}$. $(b)$ Prove that for each $a\in\mathbb{N}$ the equation $\psi (x)=ax$ has a solution.

2009 Indonesia TST, 3

Tags: function , algebra

Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(x \plus{} y)(f(x) \minus{} y) \equal{} xf(x) \minus{} yf(y) \] for all $ x,y \in \mathbb{R}$.

1996 IMO Shortlist, 7

Tags: function , symmetry , combinatorics , partition

let $ V$ be a finitive set and $ g$ and $ f$ be two injective surjective functions from $ V$to$ V$.let $ T$ and $ S$ be two sets such that they are defined as following" $ S \equal{} \{w \in V: f(f(w)) \equal{} g(g(w))\}$ $ T \equal{} \{w \in V: f(g(w)) \equal{} g(f(w))\}$ we know that $ S \cup T \equal{} V$, prove: for each $ w \in V : f(w) \in S$ if and only if $ g(w) \in S$

2001 Romania Team Selection Test, 2

Tags: function , pigeonhole principle , number theory , prime number , combinatorics , algebra

a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function. b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.

2004 Vietnam Team Selection Test, 3

Tags: geometric transformation , rotation , homothety , function , ratio , geometry

In the plane, there are two circles $\Gamma_1, \Gamma_2$ intersecting each other at two points $A$ and $B$. Tangents of $\Gamma_1$ at $A$ and $B$ meet each other at $K$. Let us consider an arbitrary point $M$ (which is different of $A$ and $B$) on $\Gamma_1$. The line $MA$ meets $\Gamma_2$ again at $P$. The line $MK$ meets $\Gamma_1$ again at $C$. The line $CA$ meets $\Gamma_2 $ again at $Q$. Show that the midpoint of $PQ$ lies on the line $MC$ and the line $PQ$ passes through a fixed point when $M$ moves on $\Gamma_1$. [color=red][Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .][/color]

2002 Irish Math Olympiad, 3

Tags: function , functional equation , algebra

Find all functions $ f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that: $ f(x\plus{}f(y))\equal{}y\plus{}f(x)$ for all $ x,y \in \mathbb{Q}$.

2014 Online Math Open Problems, 26

Tags: circumcircle , analytic geometry , incenter , trigonometry , function , geometry

Let $ABC$ be a triangle with $AB=26$, $AC=28$, $BC=30$. Let $X$, $Y$, $Z$ be the midpoints of arcs $BC$, $CA$, $AB$ (not containing the opposite vertices) respectively on the circumcircle of $ABC$. Let $P$ be the midpoint of arc $BC$ containing point $A$. Suppose lines $BP$ and $XZ$ meet at $M$ , while lines $CP$ and $XY$ meet at $N$. Find the square of the distance from $X$ to $MN$. [i]Proposed by Michael Kural[/i]

2005 Greece National Olympiad, 3

Tags: function , algebra

We know that $k$ is a positive integer and the equation \[ x^3+y^3-2y(x^2-xy+y^2)=k^2(x-y) \quad (1) \] has one solution $(x_0,y_0)$ with $x_0,y_0\in \mathbb{Z}-\{0\}$ and $x_0\neq y_0$. Prove that i) the equation (1) has a finite number of solutions $(x,y)$ with $x,y\in \mathbb{Z}$ and $x\neq y$; ii) it is possible to find $11$ addition different solutions $(X,Y)$ of the equation (1) with $X,Y\in \mathbb{Z}-\{0\}$ and $X\neq Y$ where $X,Y$ are functions of $x_0,y_0$.

1992 India National Olympiad, 10

Tags: function

Determine all functions $f : \mathbb{R} - [0,1] \to \mathbb{R}$ such that \[ f(x) + f \left( \dfrac{1}{1-x} \right) = \dfrac{2(1-2x)}{x(1-x)} . \]