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.

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Found problems: 4776

2005 Harvard-MIT Mathematics Tournament, 2
Tags: function , calculus , derivative
How many real numbers $x$ are solutions to the following equation? \[ 2003^x + 2004^x = 2005^x \]

PEN K Problems, 5
Tags: functional equation , function , arithmetic sequence
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]

1955 AMC 12/AHSME, 24
Tags: function
The function $ 4x^2\minus{}12x\minus{}1$: $ \textbf{(A)}\ \text{always increases as }x\text{ increases}\\ \textbf{(B)}\ \text{always decreases as }x\text{ decreases to 1} \\ \textbf{(C)}\ \text{cannot equal 0} \\ \textbf{(D)}\ \text{has a maximum value when }x\text{ is negative} \\ \textbf{(E)}\ \text{has a minimum value of \minus{}10}$

2014 Contests, 3
Tags: quadratic , function , algebra , quadratic formula , inequalities
Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]

2013 Hanoi Open Mathematics Competitions, 2
Tags: function , minimum , algebra
The smallest value of the function $f(x) =|x| +\left|\frac{1 - 2013x}{2013 - x}\right|$ where $x \in [-1, 1] $ is: (A): $\frac{1}{2012}$, (B): $\frac{1}{2013}$, (C): $\frac{1}{2014}$, (D): $\frac{1}{2015}$, (E): None of the above.

1993 Swedish Mathematical Competition, 6
Tags: function , algebra
For real numbers $a$ and $b$ define $f(x) = \frac{1}{ax+b}$. For which $a$ and $b$ are there three distinct real numbers $x_1,x_2,x_3$ such that $f(x_1) = x_2$, $f(x_2) = x_3$ and $f(x_3) = x_1$?

1991 Romania Team Selection Test, 8
Tags: function , number theory , greatest common divisor , diophantine equation , algebra
Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$. If $a = 1$, show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$.

1997 National High School Mathematics League, 5
Tags: function , trigonometry
Let $f(x)=x^2-\pi x$, $\alpha=\arcsin\frac{1}{3},\beta=\arctan\frac{5}{4},\gamma=\arccos\left(-\frac{1}{3}\right),\delta=\text{arccot}\left(-\frac{5}{4}\right)$ $\text{(A)}f(\alpha)>f(\beta)>f(\delta)>f(\gamma)$ $\text{(B)}f(\alpha)>f(\delta)>f(\beta)>f(\gamma)$ $\text{(C)}f(\delta)>f(\alpha)>f(\beta)>f(\gamma)$ $\text{(D)}f(\delta)>f(\alpha)>f(\gamma)>f(\beta)$

2017 Philippine MO, 2
Tags: inequalities , function
Find all positive real numbers \((a,b,c) \leq 1\) which satisfy \[ \huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\]

2002 Italy TST, 3
Tags: function , algebra
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions: $(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$ $(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.

2014 ELMO Shortlist, 3
Tags: function , algebra , geometry , reflection
We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point). (a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$. (b) Find the largest possible size of a very set not contained in any line. (Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.) [i]Proposed by Sammy Luo[/i]

1991 China Team Selection Test, 2
Tags: function , number theory
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 $m | f(f(n)) \Leftrightarrow d | n.$

2005 AIME Problems, 13
Tags: polynomial , quadratic , vieta , function , algebra
Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17$. Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2$, find the product $n_1\cdot n_2$.

2007 Hungary-Israel Binational, 3
Tags: trigonometry , function , calculus , derivative , inequalities , geometry
Let $ AB$ be the diameter of a given circle with radius $ 1$ unit, and let $ P$ be a given point on $ AB$. A line through $ P$ meets the circle at points $ C$ and $ D$, so a convex quadrilateral $ ABCD$ is formed. Find the maximum possible area of the quadrilateral.

2006 Putnam, B5
Tags: function , integration , calculus , quadratic , inequalities , algebra
For each continuous function $f: [0,1]\to\mathbb{R},$ let $I(f)=\int_{0}^{1}x^{2}f(x)\,dx$ and $J(f)=\int_{0}^{1}x\left(f(x)\right)^{2}\,dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$

2005 Pan African, 3
Tags: function , induction
Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function such that: For all $a$ and $b$ in $\mathbb{Z} - \{0\}$, $f(ab) \geq f(a) + f(b)$. Show that for all $a \in \mathbb{Z} - \{0\}$ we have $f(a^n) = nf(a)$ for all $n \in \mathbb{N}$ if and only if $f(a^2) = 2f(a)$

1988 Nordic, 4
Tags: function , limit , minimum value , algebra
Let $m_n$ be the smallest value of the function ${{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}$ Show that $m_n \to \frac{1}{2}$, as $n \to \infty.$

1985 IMO Longlists, 56
Tags: function , geometry , rhombus , circumcircle
Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$

2010 IMO Shortlist, 5
Tags: function , algebra , functional equation
Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\] [i]Proposed by Thomas Huber, Switzerland[/i]

2005 Polish MO Finals, 1
Tags: inequalities , function , number theory
Find all triplets $(x,y,n)$ of positive integers which satisfy: \[ (x-y)^n=xy \]

2005 ISI B.Math Entrance Exam, 5
Tags: function , inequalities , analytic geometry , graphing lines , slope , calculus
Find the point in the closed unit disc $D=\{ (x,y) | x^2+y^2\le 1 \}$ at which the function $f(x,y)=x+y$ attains its maximum .

2020 Romanian Master of Mathematics Shortlist, N1
Tags: number theory , function
Determine all pairs of positive integers $(m, n)$ for which there exists a bijective function \[f : \mathbb{Z}_m \times \mathbb{Z}_n \to \mathbb{Z}_m \times \mathbb{Z}_n\]such that the vectors $f(\mathbf{v}) + \mathbf{v}$, as $\mathbf{v}$ runs through all of $\mathbb{Z}_m \times \mathbb{Z}_n$, are pairwise distinct. (For any integers $a$ and $b$, the vectors $[a, b], [a + m, b]$ and $[a, b + n]$ are treated as equal.) [i]Poland, Wojciech Nadara[/i]

2009 IberoAmerican Olympiad For University Students, 5
Tags: function , number theory
Let $\mathbb{N}$ and $\mathbb{N}^*$ be the sets containing the natural numbers/positive integers respectively. We define a binary relation on $\mathbb{N}$ by $a\acute{\in}b$ iff the $a$-th bit in the binary representation of $b$ is $1$. We define a binary relation on $\mathbb{N}^*$ by $a\tilde{\in}b$ iff $b$ is a multiple of the $a$-th prime number $p_a$. i) Prove that there is no bijection $f:\mathbb{N}\to \mathbb{N}^*$ such that $a\acute{\in}b\Leftrightarrow f(a)\tilde{\in}f(b)$. ii) Prove that there is a bijection $g:\mathbb{N}\to \mathbb{N}^*$ such that $(a\acute{\in}b \vee b\acute{\in}a)\Leftrightarrow (g(a)\tilde{\in}g(b) \vee g(b)\tilde{\in}g(a))$.

2003 Putnam, 4
Tags: quadratic , function , calculus , derivative , inequalities , absolute value
Suppose that $a, b, c, A, B, C$ are real numbers, $a \not= 0$ and $A \not= 0$, such that \[|ax^2+ bx + c| \le |Ax^2+ Bx + C|\] for all real numbers $x$. Show that \[|b^2- 4ac| \le |B^2- 4AC|\]

2008 Iran MO (3rd Round), 3
Tags: function , induction , combinatorics
Prove that for each $ n$: \[ \sum_{k\equal{}1}^n\binom{n\plus{}k\minus{}1}{2k\minus{}1}\equal{}F_{2n}\]