This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 4776

2011 IMO Shortlist, 3
Tags: algebra , function , functional equation
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

1991 AMC 12/AHSME, 1
Tags: function
If for any three distinct numbers $a$, $b$ and $c$ we define \[\boxed{a,b,c} = \frac{c + a}{c - b},\] then $\boxed{1,-2,-3}=$ $ \textbf{(A)}\ -2\qquad\textbf{(B)}\ -\frac{2}{5}\qquad\textbf{(C)}\ -\frac{1}{4}\qquad\textbf{(D)}\ \frac{2}{5}\qquad\textbf{(E)}\ 2 $

1988 IMO, 3
Tags: binary representation , algebra , function , functional equation
A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$ for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$

2007 Today's Calculation Of Integral, 208
Tags: calculus computations , function , trigonometry , integration , calculus
Find the values of real numbers $a,\ b$ for which the function $f(x)=a|\cos x|+b|\sin x|$ has local minimum at $x=-\frac{\pi}{3}$ and satisfies $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\{f(x)\}^{2}dx=2$.

2018 Spain Mathematical Olympiad, 6
Tags: function , algebra
Find all functions such that $ f: \mathbb{R}^\plus{} \rightarrow \mathbb{R}^\plus{}$ and $ f(x\plus{}f(y))\equal{}yf(xy\plus{}1)$ for every $ x,y\in \mathbb{R}^\plus{}$.

2006 China Northern MO, 3
Tags: geometry , function , angle bisector , trigonometry
$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$. [i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.

2011 ISI B.Stat Entrance Exam, 3
Tags: function
Let $\mathbb{R}$ denote the set of real numbers. Suppose a function $f: \mathbb{R} \to \mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x\in \mathbb{R}$. Show that [b](i)[/b] $f$ is one-one, [b](ii)[/b] $f$ cannot be strictly decreasing, and [b](iii)[/b] if $f$ is strictly increasing, then $f(x)=x$ for all $x \in \mathbb{R}$.

2015 AMC 10, 23
Tags: quadratic , vieta , sfft , function , polynomial , algebra
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

2003 Canada National Olympiad, 2
Tags: euler , number theory , function , modular arithmetic
Find the last three digits of the number $2003^{{2002}^{2001}}$.

1991 China Team Selection Test, 2
Tags: number theory , function
Let $f$ be a function $f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},$ and satisfies the following conditions: (1) $f(0) = 0, f(1) = 1,$ (2) $f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.$ Prove that for any $m \in \mathbb{N}$, there exist a $d \in \mathbb{N}$ such that $m | f(f(n)) \Leftrightarrow d | n.$

2014 Online Math Open Problems, 21
Tags: function
Let $b = \tfrac 12 (-1 + 3\sqrt{5})$. Determine the number of rational numbers which can be written in the form \[ a_{2014}b^{2014} + a_{2013}b^{2013} + \dots + a_1b + a_0 \] where $a_0, a_1, \dots, a_{2014}$ are nonnegative integers less than $b$. [i]Proposed by Michael Kural and Evan Chen[/i]

2006 Bulgaria Team Selection Test, 3
Tags: function , floor function , inequalities , ceiling function , combinatorics
[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$? [i]Alexandar Ivanov[/i]

2013 Korea National Olympiad, 5
Tags: least common multiple , greatest common divisor , number theory , function
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying \[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \] for all positive integer $m,n$.

1992 Czech And Slovak Olympiad IIIA, 5
Tags: irrational , max , rational , algebra , function
The function $f : (0,1) \to R$ is defined by $f(x) = x$ if $x$ is irrational, $f(x) = \frac{p+1}{q}$ if $x =\frac{p}{q}$ , where $(p,q) = 1$. Find the maximum value of $f$ on the interval $(7/8,8/9)$.

2006 ISI B.Math Entrance Exam, 4
Tags: calculus computations , derivative , induction , function , calculus
Let $f:\mathbb{R} \to \mathbb{R}$ be a function that is a function that is differentiable $n+1$ times for some positive integer $n$ . The $i^{th}$ derivative of $f$ is denoted by $f^{(i)}$ . Suppose- $f(1)=f(0)=f^{(1)}(0)=...=f^{(n)}(0)=0$. Prove that $f^{(n+1)}(x)=0$ for some $x \in (0,1)$

2007 China Team Selection Test, 3
Tags: arithmetic sequence , invariant , algebra , calculus , function , polynomial
Consider a $ 7\times 7$ numbers table $ a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.

1981 National High School Mathematics League, 3
Tags: function
Let $\alpha$ be a real number and $\alpha\neq\frac{k\pi}{2} , k\in\mathbb{Z}$, $$T=\frac{\sin\alpha+\tan\alpha}{\cos\alpha+\cot\alpha}$$. $\text{(A)}$$T$ is negative. $\text{(B)}$$T$ is nonnegative. $\text{(C)}$$T$ is positive. $\text{(D)}$$T$ can be either positive or negative.

2002 Romania National Olympiad, 4
Tags: function , algebra
Let $I\subseteq \mathbb{R}$ be an interval and $f:I\rightarrow\mathbb{R}$ a function such that: \[|f(x)-f(y)|\le |x-y|,\quad\text{for all}\ x,y\in I. \] Show that $f$ is monotonic on $I$ if and only if, for any $x,y\in I$, either $f(x)\le f\left(\frac{x+y}{2}\right)\le f(y)$ or $f(y)\le f\left(\frac{x+y}{2}\right)\le f(x)$.

2018 Macedonia National Olympiad, Problem 3
Tags: function , algebra
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that:$$f(\max \left\{ x, y \right\} + \min \left\{ f(x), f(y) \right\}) = x+y $$ for all real $x,y \in \mathbb{R}$ [i]Proposed by Nikola Velov[/i]

2012 Putnam, 3
Tags: topology , algebra , limit , function , functional equation
Let $f:[-1,1]\to\mathbb{R}$ be a continuous function such that (i) $f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)$ for every $x$ in $[-1,1],$ (ii) $ f(0)=1,$ and (iii) $\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}$ exists and is finite. Prove that $f$ is unique, and express $f(x)$ in closed form.

2005 Romania National Olympiad, 2
Tags: real analysis , domain , trigonometry , function , limit , algebra
Let $f:[0,1)\to (0,1)$ a continous onto (surjective) function. a) Prove that, for all $a\in(0,1)$, the function $f_a:(a,1)\to (0,1)$, given by $f_a(x) = f(x)$, for all $x\in(a,1)$ is onto; b) Give an example of such a function.

2022 Romania National Olympiad, P1
Tags: function , integral , calculus
Let $\mathcal{F}$ be the set of functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(2x)=f(x)$ for all $x\in\mathbb{R}.$ [list=a] [*]Determine all functions $f\in\mathcal{F}$ which admit antiderivatives on $\mathbb{R}.$ [*]Give an example of a non-constant function $f\in\mathcal{F}$ which is integrable on any interval $[a,b]\subset\mathbb{R}$ and satisfies \[\int_a^bf(x) \ dx=0\]for all real numbers $a$ and $b.$ [/list][i]Mihai Piticari and Sorin Rădulescu[/i]

2001 IMC, 5
Tags: function , functional equation
Prove that there is no function $f: \mathbb{R} \rightarrow \mathbb{R}$ with $f(0) >0$, and such that \[f(x+y) \geq f(x) +yf(f(x)) \text{ for all } x,y \in \mathbb{R}. \]

1990 IMO Longlists, 80
Tags: algebra , induction , function
Function $f(x, y): \mathbb N \times \mathbb N \to \mathbb Q$ satisfies the conditions: (i) $f(1, 1) =1$, (ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \in \mathbb N$, and (iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \in \mathbb N$. Find $f(1990, 31).$

2012 IMO Shortlist, A5
Tags: function , algebra , functional equation
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions \[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\] and $f(-1) \neq 0$.