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: 744

2014 Greece JBMO TST, 1
Tags: algebra , system of equations , solve system
Find all the pairs of real numbers $(x,y)$ that are solutions of the system: $(x^{2}+y^{2})^{2}-xy(x+y)^{2}=19 $ $| x - y | = 1$

1976 Czech and Slovak Olympiad III A, 4
Tags: algebra , system of equations , linear algebra , linear equation
Determine all solutions of the linear system of equations \begin{align*} &x_1& &-x_2& &-x_3& &-\cdots& &-x_n& &= 2a, \\ -&x_1& &+3x_2& &-x_3& &-\cdots& &-x_n& &= 4a, \\ -&x_1& &-x_2& &+7x_3& &-\cdots& &-x_n& &= 8a, \\ &&&&&&&&&&&\vdots \\ -&x_1& &-x_2& &-x_3& &-\cdots& &+\left(2^n-1\right)x_n& &= 2^na, \end{align*} with unknowns $x_1,\ldots,x_n$ and a real parameter $a.$

2017 Denmark MO - Mohr Contest, 1
Tags: algebra , system of equations , system
A system of equations $$\begin{cases} x^2 \,\, ? \,\, z^2 = -8 \\ y^2 \,\, ? \,\, z^2 = 7 \end{cases}$$ is written on a piece of paper, but unfortunately two of the symbols are a little blurred. However, it is known that the system has at least one solution, and that each of the two question marks stands for either $+$ or $-$. What are the two symbols?

2009 Greece JBMO TST, 4
Tags: algebra , system of equations , minimum , sum
Find positive real numbers $x,y,z$ that are solutions of the system $x+y+z=xy+yz+zx$ and $xyz=1$ , and have the smallest possible sum.

1986 AIME Problems, 4
Tags: algebra , system of equations
Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below. \[ \begin{array}{l} 2x_1+x_2+x_3+x_4+x_5=6 \\ x_1+2x_2+x_3+x_4+x_5=12 \\ x_1+x_2+2x_3+x_4+x_5=24 \\ x_1+x_2+x_3+2x_4+x_5=48 \\ x_1+x_2+x_3+x_4+2x_5=96 \\ \end{array} \]

1935 Moscow Mathematical Olympiad, 017
Tags: system of equations , algebra , parameter
Solve the system $\begin{cases} x^3 - y^3 = 26 \\ x^2y - xy^2 = 6 \end{cases}$ in $C$ [hide=other version]solved below Solve the system $\begin{cases} x^3 - y^3 = 2b \\ x^2y - xy^2 = b \end{cases}$[/hide]

2003 Czech And Slovak Olympiad III A, 1
Tags: system of equations , algebra
Solve the following system in the set of real numbers: $x^2 -xy+y^2 = 7$, $x^2y+xy^2 = -2$.

1983 Spain Mathematical Olympiad, 8
Tags: number theory , system of equations , algebra
In $1960$, the oldest of three brothers has an age that is the sum of the of his younger siblings. A few years later, the sum of the ages of two of brothers is double that of the other. A number of years have now passed since $1960$, which is equal to two thirds of the sum of the ages that the three brothers were at that year, and one of them has reached $21$ years. What is the age of each of the others two?

2004 India IMO Training Camp, 3
Tags: trigonometry , function , functional equation , system of equations , algebra
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.

2010 Indonesia TST, 1
Tags: algebra , system of equations
Find all triplets of real numbers $(x, y, z)$ that satisfies the system of equations $x^5 = 2y^3 + y - 2$ $y^5 = 2z^3 + z - 2$ $z^5 = 2x^3 + x - 2$

1969 Poland - Second Round, 1
Tags: algebra , system of equations
Prove that if the real numbers $ a, b, c, d $ satisfy the equations $$ \; a^2 + b^2 = 1,\; c^2 + d^2 = 1, \; ac + bd = -\frac{1}{2},$$ then $$a^2 + ac + c^2 = b^2 + bd + d^2.$$

1979 Austrian-Polish Competition, 6
Tags: algebra , system of equations
A positive integer $n$ and a real number $a$ are given. Find all $n$-tuples $(x_1, ... ,x_n)$ of real numbers that satisfy the system of equations $$\sum_{i=1}^{n} x_i^k= a^k \,\,\,\, for \,\,\,\, k = 1,2, ... ,n$$

2011 Mathcenter Contest + Longlist, 5 sl6
Tags: system of equations , system , algebra
Given $x,y,z\in \mathbb{R^+}$. Find all sets of $x,y,z$ that correspond to $$x+y+z=x^2+y^2+z^2+18xyz=1$$ [i](Zhuge Liang)[/i]

2020 Chile National Olympiad, 4
Tags: diophantine equation , number theory , diophantine , system of equations
Determine all three integers $(x, y, z)$ that are solutions of the system $$x + y -z = 6$$ $$x^3 + y^3 -z^3 = 414$$

2020 MMATHS, 3
Tags: algebra , system of equations
Let $a, b$ be two real numbers such that $$\sqrt[3]{a}- \sqrt[3]{b} = 10, ,\,\,\,\,\,\, ab = \left( \frac{8 - a - b}{6}\right)^3$$ Find $a - b$.

2002 AMC 10, 25
Tags: algebra , system of equations
When $ 15$ is appended to a list of integers, the mean is increased by $ 2$. When $ 1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $ 1$. How many integers were in the original list? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

III Soros Olympiad 1996 - 97 (Russia), 11.8
Tags: algebra , trigonometry , system of equations
Solve the system of equations: $$ 2(3-2\cos y)^2+2(4-2\sin y)^2=2(3-x)^2+32=(x-2\cos y)^2+4\sin^2y$$

2009 Ukraine National Mathematical Olympiad, 1
Tags: system of equations , algebra
Find all possible real values of $a$ for which the system of equations \[\{\begin{array}{cc}x +y +z=0\\\text{ } \\ xy+yz+azx=0\end{array}\] has exactly one solution.

2015 Denmark MO - Mohr Contest, 4
Tags: algebra , system of equations , system
Determine all numbers $x, y$ and $z$ satisfying the system of equations $$\begin{cases} x^2 + yz = 1 \\ y^2 - xz = 0 \\ z^2 + xy = 1\end{cases}$$

1986 IMO Longlists, 52
Tags: trigonometry , system of equations , algebra
Solve the system of equations \[\tan x_1 +\cot x_1=3 \tan x_2,\]\[\tan x_2 +\cot x_2=3 \tan x_3,\]\[\vdots\]\[\tan x_n +\cot x_n=3 \tan x_1\]

2009 All-Russian Olympiad, 5
Tags: absolute value , algebra , system of equations
Let $ a$, $ b$, $ c$ be three real numbers satisfying that \[ \left\{\begin{array}{c c c} \left(a\plus{}b\right)\left(b\plus{}c\right)\left(c\plus{}a\right)&\equal{}&abc\\ \left(a^3\plus{}b^3\right)\left(b^3\plus{}c^3\right)\left(c^3\plus{}a^3\right)&\equal{}&a^3b^3c^3\end{array}\right.\] Prove that $ abc\equal{}0$.

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

2007 Puerto Rico Team Selection Test, 2
Tags: number theory , diophantine equation , diophantine , algebra , system of equations
Find the solutions of positive integers for the system $xy + x + y = 71$ and $x^2y + xy^2 = 880$.

2005 All-Russian Olympiad Regional Round, 8.7
Tags: number theory , diophantine , system of equations
Find all pairs $(x, y)$ of natural numbers such that $$x + y = a^n, x^2 + y^2 = a^m$$ for some natural $a, n, m$.

2019 Czech-Polish-Slovak Junior Match, 4
Tags: algebra , system of equations
Determine all possible values of the expression $xy+yz+zx$ with real numbers $x, y, z$ satisfying the conditions $x^2-yz = y^2-zx = z^2-xy = 2$.