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: 15925

2020 LIMIT Category 1, 3
Tags: limit , algebra
How many $2$ digit number $n=ab$ ($a$ and $b$ are digits) have the property that $$n=a+b+a\times b$$ (A)$20$ (B)$15$ (C)$9$ (D)$8$

2007 Canada National Olympiad, 4
Tags: algebra , polynomial , ratio , trigonometry , invariant , induction , combinatorics
For two real numbers $ a$, $ b$, with $ ab\neq 1$, define the $ \ast$ operation by \[ a\ast b=\frac{a+b-2ab}{1-ab}.\] Start with a list of $ n\geq 2$ real numbers whose entries $ x$ all satisfy $ 0<x<1$. Select any two numbers $ a$ and $ b$ in the list; remove them and put the number $ a\ast b$ at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains. $ a.$ Prove that this single number is the same regardless of the choice of pair at each stage. $ b.$ Suppose that the condition on the numbers $ x$ is weakened to $ 0<x\leq 1$. What happens if the list contains exactly one $ 1$?

2018 International Olympic Revenge, 4
Tags: algebra , functional equation
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R}$ such that \[ f(x)^2-f(y)^2=f(x+y)\cdot f(x-y), \] for all $x,y\in \mathbb{Q}$. [i]Proposed by Portugal.[/i]

1963 Czech and Slovak Olympiad III A, 4
Tags: real root , quadratic polynomial , root , quadratic , algebra
Consider two quadratic equations \begin{align*}x^2+ax+b&=0, \\ x^2+cx+d&=0,\end{align*} with real coefficients. Find necessary and sufficient conditions such that the first equation has (real) roots $x,x_1,$ the second $x,x_2$ and $x>0,x_1>x_2$.

2010 HMNT, 6
Tags: floor function , algebra
What is the sum of the positive solutions to $2x^2 -\lfloor x \rfloor = 5$, where $\lfloor x \rfloor$ is the largest integer less than or equal to $x$?

2023 VN Math Olympiad For High School Students, Problem 3
Tags: algebra , number theory , prime number
Given a polynomial with integer coefficents with degree $n>0:$$$P(x)=a_nx^n+...+a_1x+a_0.$$ Assume that there exists a prime number $p$ satisfying these conditions: [i]i)[/i] $p|a_i$ for all $0\le i<n,$ [i]ii)[/i] $p\nmid a_n,$ [i]iii)[/i] $p^2\nmid a_0.$ Prove that $P(x)$ is irreducible in $\mathbb{Z}[x].$

1995 Tuymaada Olympiad, 2
Tags: limit , limit of sequence , recurrence relation , maximum , algebra
Let $x_1=a, x_2=a^{x_1}, ..., x_n=a^{x_{n-1}}$ where $a>1$. What is the maximum value of $a$ for which lim exists $\lim_{n\to \infty} x_n$ and what is this limit?

1990 Swedish Mathematical Competition, 5
Tags: functional equation , functional , algebra
Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$.

2024 Korea Junior Math Olympiad (First Round), 14.
Tags: number theory , algebra , number
Find the number of positive integer $x$ that has $ {a}_{1},{a}_{2},\cdot \cdot \cdot {a}_{20} $ which follows the following ($x \ge 1000$) 1) $ {a}_{1}=2, {a}_{2}=1, {a}_{3}=x $ 2) for positive integer $n$, ($ 4 \le n \le 20 $), $ {a}_{n}={a}_{n-3}+\frac{(-2)^n}{{a}_{n-1}{a}_{n-2}} $

1963 Vietnam National Olympiad, 3
Tags: trigonometry , algebra
Solve the equation $ \sin^3x \cos 3x \plus{} \cos^3x \sin 3x \equal{} \frac{3}{8}$.

1999 Switzerland Team Selection Test, 9
Tags: algebra , polynomial , number theory
Suppose that $P(x)$ is a polynomial with degree $10$ and integer coefficients. Prove that, there is an infinite arithmetic progression (open to bothside) not contain value of $P(k)$ with $k\in\mathbb{Z}$

1983 AMC 12/AHSME, 6
Tags: polynomial , algebra
When \[x^5, \quad x+\frac{1}{x}\quad \text{and}\quad 1+\frac{2}{x} + \frac{3}{x^2}\] are multiplied, the product is a polynomial of degree $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 $

2014 239 Open Mathematical Olympiad, 6
Tags: algebra , inequality
Given posetive real numbers $a_1,a_2,\dots,a_n$ such that $a_1^2+2a_2^3+\dots+na_n^{n+1} <1.$ Prove that $2a_1+3a_2^2+\dots+(n+1)a_{n}^n <3.$

2014 JBMO Shortlist, 4
Tags: algebra , inequalities
With the conditions $a,b,c\in\mathbb{R^+}$ and $a+b+c=1$, prove that \[\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\geq\frac{69}{4}\]

2017 International Olympic Revenge, 2
Tags: algebra
A polynomial is [i]good[/i] if it has integer coefficients, it is monic, all its roots are distinct, and there exists a disk with radius $0.99$ on the complex plane that contains all the roots. Prove that there is no [i]good[/i] polynomial for a sufficient large degree. [i]Proposed by Rodrigo Sanches Angelo (rsa365), Brazil.[/i]

2011 Princeton University Math Competition, A2
Tags: algebra
Define the sequence of real numbers $\{x_n\}_{n \geq 1}$, where $x_1$ is any real number and \[x_n = 1 - x_1x_2\ldots x_{n-1} \text{ for all } n > 1.\] Show that $x_{2011} > \frac{2011}{2012}$.

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]

1998 Czech and Slovak Match, 4
Tags: natural number , functional equation in n , algebra
Find all functions $f : N\rightarrow N - \{1\}$ satisfying $f (n)+ f (n+1)= f (n+2) +f (n+3) -168$ for all $n \in N$ .

2023 Thailand Mathematical Olympiad, 10
Tags: combinatorics , number theory , algebra
To celebrate the 20th Thailand Mathematical Olympiad (TMO), Ratchasima Witthayalai School put up flags around the Thao Suranari Monument so that [list=i] [*] Each flag is painted in exactly one color, and at least $2$ distinct colors are used. [*] The number of flags are odd. [*] Every flags are on a regular polygon such that each vertex has one flag. [*] Every flags with the same color are on a regular polygon. [/list] Prove that there are at least $3$ colors with the same amount of flags.

2019 India Regional Mathematical Olympiad, 3
Tags: algebra , inequality , system of equations
Find all triples of non-negative real numbers $(a,b,c)$ which satisfy the following set of equations $$a^2+ab=c$$ $$b^2+bc=a$$ $$c^2+ca=b$$

2017 Romania National Olympiad, 2
Tags: function , algebra
A function $ f:\mathbb{Q}_{>0}\longrightarrow\mathbb{Q} $ has the following property: $$ f(xy)=f(x)+f(y),\quad x,y\in\mathbb{Q}_{>0} $$ [b]a)[/b] Demonstrate that there are no injective functions with this property. [b]b)[/b] Do exist surjective functions having this property?

1992 French Mathematical Olympiad, Problem 4
Tags: sequence , recurrence relation , algebra
Given $u_0,u_1$ with $0<u_0,u_1<1$, define the sequence $(u_n)$ recurrently by the formula $$u_{n+2}=\frac12\left(\sqrt{u_{n+1}}+\sqrt{u_n}\right).$$(a) Prove that the sequence $u_n$ is convergent and find its limit. (b) Prove that, starting from some index $n_0$, the sequence $u_n$ is monotonous.

2002 Czech and Slovak Olympiad III A, 1
Tags: algebra , function , domain , number theory
Solve the system \[(4x)_5+7y=14 \\ (2y)_5 -(3x)_7=74\] in the domain of integers, where $(n)_k$ stands for the multiple of the number $k$ closest to the number $n$.

2007 Hungary-Israel Binational, 3
Tags: polynomial , algebra
Let $ t \ge 3$ be a given real number and assume that the polynomial $ f(x)$ satisfies $|f(k)\minus{}t^k|<1$, for $ k\equal{}0,1,2,\ldots ,n$. Prove that the degree of $f(x)$ is at least $n$.

2001 Estonia Team Selection Test, 3
Tags: functional , functional equation , algebra
Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.