Olympiad problems

2012 China Second Round Olympiad, 9

Given a function $f(x)=a\sin x-\frac{1}{2}\cos 2x+a-\frac{3}{a}+\frac{1}{2}$, where $a\in\mathbb{R}, a\ne 0$. [b](1)[/b] If for any $x\in\mathbb{R}$, inequality $f(x)\le 0$ holds, find all possible value of $a$. [b](2)[/b] If $a\ge 2$, and there exists $x\in\mathbb{R}$, such that $f(x)\le 0$. Find all possible value of $a$.

PEN A Problems, 82

Tags: function , quadratic , vieta , divisibility theory , vieta jumping , inequalities

Which integers can be represented as \[\frac{(x+y+z)^{2}}{xyz}\] where $x$, $y$, and $z$ are positive integers?

2002 Iran Team Selection Test, 10

Tags: function , geometry , rectangle , linear algebra , vector , abstract algebra , matrix

Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.

2004 Korea Junior Math Olympiad, 5

Tags: functional equation , algebra , existence , function

Show that there exists no function $f:\mathbb {R}\rightarrow \mathbb {R}$ that satisfies $f(f(x))-x^2+x+3=0$ for arbitrary real variable $x$. (Same as KMO 2004 P1)

1991 AIME Problems, 4

Tags: function , logarithm , trigonometry

How many real numbers $x$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$?

2006 Bundeswettbewerb Mathematik, 2

Tags: function , induction , algebra

Find all functions $f: Q^{+}\rightarrow R$ such that $f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$ for all $x,y\in Q^{+}$

1990 Federal Competition For Advanced Students, P2, 4

Tags: algebra , function

For each nonzero integer $ n$ find all functions $ f: \mathbb{R} \minus{} \{\minus{}3,0 \} \rightarrow \mathbb{R}$ satisfying: $ f(x\plus{}3)\plus{}f \left( \minus{}\frac{9}{x} \right)\equal{}\frac{(1\minus{}n)(x^2\plus{}3x\minus{}9)}{9n(x\plus{}3)}\plus{}\frac{2}{n}$ for all $ x \not\equal{} 0,\minus{}3.$ Furthermore, for each fixed $ n$ find all integers $ x$ for which $ f(x)$ is an integer.

2016 German National Olympiad, 6

Tags: function , algebra , cyclic function , inequality , maximum

Let \[ f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3 \] be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$. Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.

2009 BMO TST, 1

Tags: algebra , function

Given the equation $x^4-x^3-1=0$ [b](a)[/b] Find the number of its real roots. [b](b)[/b] We denote by $S$ the sum of the real roots and by $P$ their product. Prove that $P< - \frac{11}{10}$ and $S> \frac {6}{11}$.

1995 IMO Shortlist, 5

Tags: function , algebra , functional equation

Let $ \mathbb{R}$ be the set of real numbers. Does there exist a function $ f: \mathbb{R} \mapsto \mathbb{R}$ which simultaneously satisfies the following three conditions? [b](a)[/b] There is a positive number $ M$ such that $ \forall x:$ $ \minus{} M \leq f(x) \leq M.$ [b](b)[/b] The value of $f(1)$ is $1$. [b](c)[/b] If $ x \neq 0,$ then \[ f \left(x \plus{} \frac {1}{x^2} \right) \equal{} f(x) \plus{} \left[ f \left(\frac {1}{x} \right) \right]^2 \]

1999 Turkey Team Selection Test, 3

Tags: function , algebra

Determine all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the set \[\left \{ \frac{f(x)}{x}: x \neq 0 \textnormal{ and } x \in \mathbb{R}\right \}\] is finite, and for all $x \in \mathbb{R}$ \[f(x-1-f(x)) = f(x) - x - 1\]

2019 Dutch IMO TST, 4

Tags: number theory , function , find all functions , prime number , functional equation

Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

2011 USAMO, 6

Tags: matrix , function , linear algebra , inequalities

Let $A$ be a set with $|A|=225$, meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1, \ldots, A_{11}$ of $A$ such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$ for $1\leq i<j\leq11$. Prove that $|A_1\cup A_2\cup\ldots\cup A_{11}|\geq 165$, and give an example for which equality holds.

2025 All-Russian Olympiad, 11.8

Tags: algebra , continuous function , function

Let $ f: \mathbb{R} \to \mathbb{R} $ be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of $ f $. It is known that the graph of $ f $ contains exactly $ N $ chords, one of which has length 2025. Find the minimum possible value of $ N $.

2016 Romania National Olympiad, 4

Tags: real analysis , function , find all functions , differentiability , darboux

Find all functions, $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ that have the properties that $ f^2 $ is differentiable and $ f=\left( f^2 \right)' . $

2014 Middle European Mathematical Olympiad, 1

Tags: function , functional equation , fe , real , algebra

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

1990 IberoAmerican, 1

Tags: induction , function

Let $f$ be a function defined for the non-negative integers, such that: a) $f(n)=0$ if $n=2^{j}-1$ for some $j \geq 0$. b) $f(n+1)=f(n)-1$ otherwise. i) Show that for every $n \geq 0$ there exists $k \geq 0$ such that $f(n)+n=2^{k}-1$. ii) Find $f(2^{1990})$.

2004 Turkey MO (2nd round), 4

Tags: function , algebra , induction

Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ satisfying the condition $f(n)-f(n+f(m))=m$ for all $m,n\in \mathbb{Z}$

1992 AIME Problems, 13

Tags: inequalities , function , calculus , quadratic , analytic geometry , geometry , ratio

Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?

2003 Alexandru Myller, 4

Tags: functional equation , real analysis , function , find all functions

Find the differentiable functions $ f:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ that verify $ f(0)=0 $ and $$ f'(x)=1/3\cdot f'\left( x/3 \right) +2/3\cdot f'\left( 2x/3 \right) , $$ for any nonnegative real number $ x. $

1997 Hungary-Israel Binational, 1

Tags: function , combinatorics

Determine the number of distinct sequences of letters of length 1997 which use each of the letters $A$, $B$, $C$ (and no others) an odd number of times.

2014 Miklós Schweitzer, 11

Tags: function , probability and stats , trigonometry

Let $U$ be a random variable that is uniformly distributed on the interval $[0,1]$, and let \[S_n= 2\sum_{k=1}^n \sin(2kU\pi).\] Show that, as $n\to \infty$, the limit distribution of $S_n$ is the Cauchy distribution with density function $f(x)=\frac1{\pi(1+x^2)}$.

2024 Indonesia TST, A

Tags: function , algebra

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

2012 Turkey Team Selection Test, 1

Tags: function , induction , combinatorics

Let $A=\{1,2,\ldots,2012\}, \: B=\{1,2,\ldots,19\}$ and $S$ be the set of all subsets of $A.$ Find the number of functions $f : S\to B$ satisfying $f(A_1\cap A_2)=\min\{f(A_1),f(A_2)\}$ for all $A_1, A_2 \in S.$

2010 Switzerland - Final Round, 6

Tags: algebra , function

Find all functions $ f: \mathbb{R}\mapsto\mathbb{R}$ such that for all $ x$, $ y$ $ \in\mathbb{R}$, \[ f(f(x))\plus{}f(f(y))\equal{}2y\plus{}f(x\minus{}y)\] holds.