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

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Found problems: 15925

2023 Romania National Olympiad, 2
Tags: algebra
Prove that: a) There are infinitely many pairs $(x,y)$ of real numbers from the interval $[0,\sqrt{3}]$ which satisfy the equation $x\sqrt{3-y^2}+y\sqrt{3-x^2}=3$. b) There do not exist any pairs $(x,y)$ of rational numbers from the interval $[0,\sqrt{3}]$ that satisfy the equation $x\sqrt{3-y^2}+y\sqrt{3-x^2}=3$.

1987 Traian Lălescu, 1.2
Tags: function , functional equation , algebra
Let $ A $ be a subset of $ \mathbb{R} $ and let be a function $ f:A\longrightarrow\mathbb{R} $ satisfying $$ f(x)-f(y)=(y-x)f(x)f(y),\quad\forall x,y\in A. $$ [b]a)[/b] Show that if $ A=\mathbb{R}, $ then $ f=0. $ [b]b)[/b] Find $ f, $ provided that $ A=\mathbb{R}\setminus\{1\} . $

1998 Brazil Team Selection Test, Problem 3
Tags: function , polynomial , algebra
Find all functions $f: \mathbb N \to \mathbb N$ for which \[ f(n) + f(n+1) = f(n+2)f(n+3)-1996\] holds for all positive integers $n$.

2024 Kyiv City MO Round 1, Problem 1
Tags: number theory , algebra , divisibility
Find all pairs of positive integers $(a, b)$ such that $4b - 1$ is divisible by $3a + 1$ and $3a - 1$ is divisible by $2b + 1$.

2012 Korea Junior Math Olympiad, 1
Tags: inequality , ineqstd , algebra
Prove the following inequality where positive reals $a$, $b$, $c$ satisfies $ab+bc+ca=1$. \[ \frac{a+b}{\sqrt{ab(1-ab)}} + \frac{b+c}{\sqrt{bc(1-bc)}} + \frac{c+a}{\sqrt{ca(1-ca)}} \le \frac{\sqrt{2}}{abc} \]

2007 Germany Team Selection Test, 1
Tags: algebra , sequence , periodic sequence , floor function , recurrence relation
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

2001 Singapore MO Open, 2
Tags: sum , algebra , inequalities
Let $n$ be a positive integer, and let $a_1,a_2,...,a_n$ be $n$ positive real numbers such that $a_1+a_2+...+a_n = 1$. Is it true that $\frac{a_1^4}{a_1^2+a_2^2}+\frac{a_2^4}{a_2^2+a_3^2}+\frac{a_3^4}{a_3^2+a_4^2}+...+\frac{a_{n-1}^4}{a_{n-1}^2+a_n^2}+\frac{a_n^4}{a_n^2+a_1^2}\ge \frac{1}{2n}$ ? Justify your answer.

2008 Czech and Slovak Olympiad III A, 1
Tags: algebra
Find all pairs of real numbers $(x,y)$ satisfying: \[x+y^2=y^3,\]\[y+x^2=x^3.\]

2012 India PRMO, 18
Tags: algebra , floor function , function
What is the sum of the squares of the roots of the equation $x^2 -7 \lfloor x\rfloor +5=0$ ?

2006 IMS, 5
Tags: function , limit , algebra
Suppose that $a_{1},a_{2},\dots,a_{k}\in\mathbb C$ that for each $1\leq i\leq k$ we know that $|a_{k}|=1$. Suppose that \[\lim_{n\to\infty}\sum_{i=1}^{k}a_{i}^{n}=c.\] Prove that $c=k$ and $a_{i}=1$ for each $i$.

2018 239 Open Mathematical Olympiad, 10-11.6
Tags: algebra
For which positive integers $n$, $m$ does there exist a polynomial of degree $n$, all coefficients of which are powers of $m$ with integer exponents, having $n$ rational roots, counting multiplicities? [i]Proposed by Fedor Petrov[/i]

2010 China Team Selection Test, 2
Tags: algebra , function , polynomial , complex number
Let $A=\{a_1,a_2,\cdots,a_{2010}\}$ and $B=\{b_1,b_2,\cdots,b_{2010}\}$ be two sets of complex numbers. Suppose \[\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k\] holds for every $k=1,2,\cdots, 2010$. Prove that $A=B$.

2010 Contests, 2
Tags: inequalities , function , algebra
Find all non-decreasing functions $f:\mathbb R^+\cup\{0\}\rightarrow\mathbb R^+\cup\{0\}$ such that for each $x,y\in \mathbb R^+\cup\{0\}$ \[f\left(\frac{x+f(x)}2+y\right)=2x-f(x)+f(f(y)).\]

2005 District Olympiad, 4
Tags: algebra
Let $\{a_k\}_{k\geq 1}$ be a sequence of non-negative integers, such that $a_k \geq a_{2k} + a_{2k+1}$, for all $k\geq 1$. a) Prove that for all positive integers $n\geq 1$ there exist $n$ consecutive terms equal with 0 in the sequence $\{a_k\}_k$; b) State an example of sequence with the property in the hypothesis which contains an infinite number of non-zero terms.

1992 Irish Math Olympiad, 2
Tags: algebra
How many ordered triples $(x,y,z)$ of real numbers satisfy the system of equations $$x^2+y^2+z^2=9,$$ $$x^4+y^4+z^4=33,$$ $$xyz=-4?$$

1995 Baltic Way, 1
Tags: number theory , algebra , system of equations
Find all triples $(x,y,z)$ of positive integers satisfying the system of equations \[\begin{cases} x^2=2(y+z)\\ x^6=y^6+z^6+31(y^2+z^2)\end{cases}\]

LMT Team Rounds 2021+, 1
Tags: algebra , geometry
Derek and Jacob have a cake in the shape a rectangle with dimensions $14$ inches by $9$ inches. They make a deal to split it: Derek takes home the portion of the cake that is less than one inch from the border, while Jacob takes home the remainder of the cake. Let $D : J$ be the ratio of the amount of cake Derek took to the amount of cake Jacob took, where $D$ and $J$ are relatively prime positive integers. Find $D + J$.

2014 Mediterranean Mathematics Olympiad, 1
Tags: induction , triangle inequality , n-variable inequality , inequalities , algebra
Let $a_1,\ldots,a_n$ and $b_1\ldots,b_n$ be $2n$ real numbers. Prove that there exists an integer $k$ with $1\le k\le n$ such that $ \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|.$ (Proposed by Gerhard Woeginger, Austria)

1974 IMO Longlists, 49
Tags: integration , polynomial , calculus , trigonometry , algebra
Determine an equation of third degree with integral coefficients having roots $\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}$ and $\sin \frac{-3 \pi}{14}.$

2010 Brazil National Olympiad, 1
Tags: algebra , function , vieta , functional equation
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.

2005 USAMO, 2
Tags: modular arithmetic , algebra , diophantine equation
Prove that the system \begin{align*} x^6+x^3+x^3y+y & = 147^{157} \\ x^3+x^3y+y^2+y+z^9 & = 157^{147} \end{align*} has no solutions in integers $x$, $y$, and $z$.

2019 Mediterranean Mathematics Olympiad, 2
Tags: algebra , sequence , inequalities
Let $m_1<m_2<\cdots<m_s$ be a sequence of $s\ge2$ positive integers, none of which can be written as the sum of (two or more) distinct other numbers in the sequence. For every integer $r$ with $1\le r<s$, prove that \[ r\cdot m_r+m_s ~\ge~ (r+1)(s-1). \] (Proposed by Gerhard Woeginger, Austria)

1896 Eotvos Mathematical Competition, 2
Tags: algebra
Prove that the equations $$x^2-3xy+2y^2+x-y=0 \text{ and } x^2-2xy+y^2-5x+7y=0$$ imply the equation $xy-12x+15y=0$.

2016 Azerbaijan BMO TST, 4
Tags: algebra
For all numbers $n\ge 1$ does there exist infinite positive numbers sequence $x_1,x_2,...,x_n$ such that $x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_n}$

2022 Iran MO (3rd Round), 6
Tags: inequalities , algebra , inequality
Prove that among any $9$ distinct real numbers, there exist $4$ distinct numbers $a,b,c,d$ such that $$(ac+bd)^2\ge\frac{9}{10}(a^2+b^2)(c^2+d^2)$$