Olympiad problems

2012 CIIM, Problem 5

Tags: function

Let $D=\{0,1,\dots,9\}$. A direction function for $D$ is a function $f:D \times D \to \{0,1\}.$ A real $r\in [0,1]$ is compatible with $f$ if it can be written in the form $$r = \sum_{j=1}^{\infty} \frac{d_j}{10^j}$$ with $d_j \in D$ and $f(d_j,d_{j+1})=1$ for every positive integer $j$. Determine the least integer $k$ such that for any direction fuction $f$, if there are $k$ compatible reals with $f$ then there are infinite reals compatible with $f$.

2009 India Regional Mathematical Olympiad, 5

Tags: geometry , function , induction , circumcircle , conic , inequalities , ellipse

A convex polygon is such that the distance between any two vertices does not exceed $ 1$. $ (i)$ Prove that the distance between any two points on the boundary of the polygon does not exceed $ 1$. $ (ii)$ If $ X$ and $ Y$ are two distinct points inside the polygon, prove that there exists a point $ Z$ on the boundary of the polygon such that $ XZ \plus{} YZ\le1$.

2012 Kyoto University Entry Examination, 3

Tags: algebra , geometry , geometric transformation , function , rotation

When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$ Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$ 30 points

2019 Latvia Baltic Way TST, 2

Tags: functional equation , algebra , function

Let $\mathbb R$ be set of real numbers. Determine all functions $f:\mathbb R\to \mathbb R$ such that $$f(y^2 - f(x)) = yf(x)^2+f(x^2y+y)$$ holds for all real numbers $x; y$

2003 Romania National Olympiad, 3

Tags: function , calculus , integration

Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that $$ xf(x)\ge \int_0^x f(t)dt , $$ for all real numbers $ x. $ Prove that [b]a)[/b] the mapping $ x\mapsto \frac{1}{x}\int_0^x f(t) dt $ is nondecreasing on the restrictions $ \mathbb{R}_{<0 } $ and $ \mathbb{R}_{>0 } . $ [b]b)[/b] if $ \int_x^{x+1} f(t)dt=\int_{x-1}^x f(t)dt , $ for any real number $ x, $ then $ f $ is constant. [i]Mihai Piticari[/i]

2012 ELMO Problems, 3

Tags: function , algebra , polynomial

Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant. [i]Victor Wang.[/i]

2014 Contests, 1

Tags: algebra , function

Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]

1986 China Team Selection Test, 2

Tags: function , inequalities , perimeter , geometry , 3d geometry , tetrahedron

Given a tetrahedron $ABCD$, $E$, $F$, $G$, are on the respectively on the segments $AB$, $AC$ and $AD$. Prove that: i) area $EFG \leq$ max{area $ABC$,area $ABD$,area $ACD$,area $BCD$}. ii) The same as above replacing "area" for "perimeter".

2017 Bosnia and Herzegovina Team Selection Test, Problem 2

Tags: function , number theory , divisibility

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2008 Harvard-MIT Mathematics Tournament, 22

Tags: function , modular arithmetic

For a positive integer $ n$, let $ \theta(n)$ denote the number of integers $ 0 \leq x < 2010$ such that $ x^2 \minus{} n$ is divisible by $ 2010$. Determine the remainder when $ \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n)$ is divided by $ 2010$.

2001 Miklós Schweitzer, 1

Tags: function , set theory , real analysis

Let $f\colon 2^S\rightarrow \mathbb R$ be a function defined on the subsets of a finite set $S$. Prove that if $f(A)=F(S\backslash A)$ and $\max \{ f(A), f(B)\}\geq f(A\cup B)$ for all subsets $A, B$ of $S$, then $f$ assumes at most $|S|$ distinct values.

2016 Latvia National Olympiad, 5

Tags: function , analytic geometry , geometry

Consider the graphs of all the functions $y = x^2 + px + q$ having 3 different intersection points with the coordinate axes. For every such graph we pick these 3 intersection points and draw a circumcircle through them. Prove that all these circles have a common point!

2016 Turkey Team Selection Test, 5

Tags: number theory , function , functional equation , divisibility

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all $m,n \in \mathbb{N}$ holds $f(mn)=f(m)f(n)$ and $m+n \mid f(m)+f(n)$ .

2006 Germany Team Selection Test, 1

Tags: functional equation , algebra , function

We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers. Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property: \[f(x)f(y)\equal{}2f(x\plus{}yf(x))\] for all positive real numbers $x$ and $y$. [i]Proposed by Nikolai Nikolov, Bulgaria[/i]

2007 Bulgaria Team Selection Test, 2

Tags: inequalities , function , algebra

Find all $a\in\mathbb{R}$ for which there exists a non-constant function $f: (0,1]\rightarrow\mathbb{R}$ such that \[a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)\] for all $x,y\in(0,1].$

2015 Azerbaijan IMO TST, 2

Tags: function , algebra

Find all functions $f:[0,1] \to \mathbb{R}$ such that the inequality \[(x-y)^2\leq|f(x) -f(y)|\leq|x-y|\] is satisfied for all $x,y\in [0,1]$

PEN K Problems, 28

Tags: function , functional equation

Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n) \ge n+(-1)^{n}.\]

2014 Contests, 1

Tags: function , relatively prime , prime factorization

Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.

JOM 2015 Shortlist, C3

Tags: function , combinatorics

Let $ n\ge 2 $ be a positive integer and $ S= \{1,2,\cdots ,n\} $. Let two functions $ f:S \rightarrow \{1,-1\} $ and $ g:S \rightarrow S $ satisfy: i) $ f(x)f(y)=f(x+y) , \forall x,y \in S $ \\ ii) $ f(g(x))=f(x) , \forall x \in S $\\ iii) $f(x+n)=f(x) ,\forall x \in S$\\ iv) $ g $ is bijective.\\ Find the number of pair of such functions $ (f,g)$ for every $n$.

2005 AMC 12/AHSME, 20

Tags: function , induction

For each $ x$ in $ [0,1]$, define \[ f(x)=\begin{cases}2x, &\text { if } 0 \leq x \leq \frac {1}{2}; \\ 2 - 2x, &\text { if } \frac {1}{2} < x \leq 1. \end{cases} \]Let $ f^{[2]}(x) = f(f(x))$, and $ f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $ n \geq 2$. For how many values of $ x$ in $ [0,1]$ is $ f^{[2005]}(x) = \frac {1}{2}$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2005 \qquad \textbf{(C)}\ 4010 \qquad \textbf{(D)}\ 2005^2 \qquad \textbf{(E)}\ 2^{2005}$

2008 ITest, 6

Tags: rectangle , algebra , geometry , function , hmmt , similar triangles , ceiling function

Let $L$ be the length of the altitude to the hypotenuse of a right triangle with legs $5$ and $12$. Find the least integer greater than $L$.

2014 Miklós Schweitzer, 2

Tags: function , integration , combinatorics , inequalities

Let $ k\geq 1 $ and let $ I_{1},\dots, I_{k} $ be non-degenerate subintervals of the interval $ [0, 1] $. Prove that \[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2} \] where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$.

2014 Singapore MO Open, 2

Tags: algebra , function

Find all functions from the reals to the reals satisfying \[f(xf(y) + x) = xy + f(x)\]

2013 ELMO Shortlist, 7

Tags: induction , function , algebra , 3n-3 theorem , continuity

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

2014 China National Olympiad, 3

Tags: algebra , induction , function

Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying: i) $f(1)=f(2)=1$; ii)$f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$. For each integer $m\ge 2$, find the value of $f(2^m)$.