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: 744

2014 Contests, 1
Tags: algebra , solve system , system of equations
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 $ $| x - y | = 1$

1953 Moscow Mathematical Olympiad, 248
Tags: algebra , system of equations
a) Solve the system $\begin{cases} x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 = 1 \\ x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 = 2 \\ x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 = 3 \\ x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 = 4 \\ x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 = 5 \end{cases}$ b) Solve the system $\begin{cases} x_1 + 2x_2 + 2x_3 + 2x_4 + 2x_5 +...+ 2x_{100}= 1 \\ x_1 + 3x_2 + 4x_3 + 4x_4 + 4x_5 +...+ 4x_{100}= 2 \\ x_1 + 3x_2 + 5x_3 + 6x_4 + 6x_5 +...+ 6x_{100}= 3 \\ x_1 + 3x_2 + 5x_3 + 7x_4 + 8x_5 +...+ 8x_{100}= 4 \\ ... \\ x_1 + 3x_2 + 5x_3 + 7x_4 + 9x_5 +...+ 199x_{100}= 100 \end{cases}$

1946 Moscow Mathematical Olympiad, 109
Tags: algebra , system of equations
Solve the system of equations: $\begin{cases} x_1 + x_2 + x_3 = 6 \\ x_2 + x_3 + x_4 = 9 \\ x_3 + x_4 + x_5 = 3 \\ x_4 + x_5 + x_6 = -3 \\ x_5 + x_6 + x_7 = -9 \\ x_6 + x_7 + x_8 = -6 \\ x_7 + x_8 + x_1 = -2 \\ x_8 + x_1 + x_2 = 2 \end{cases}$

2011 Kosovo National Mathematical Olympiad, 3
Tags: algebra , function , trigonometry , inequalities , calculus , system of equations
Find maximal value of the function $f(x)=8-3\sin^2 (3x)+6 \sin (6x)$

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

2015 Harvard-MIT Mathematics Tournament, 10
Tags: algebra , system of equations
Find all ordered 4-tuples of integers $(a,b,c,d)$ (not necessarily distinct) satisfying the following system of equations: \begin{align*}a^2-b^2-c^2-d^2&=c-b-2\\2ab&=a-d-32\\2ac&=28-a-d\\2ad&=b+c+31.\end{align*}

2004 USAMTS Problems, 4
Tags: algebra , system of equations
Find, with proof, all integers $n$ such that there is a solution in nonnegative real numbers $(x,y,z)$ to the system of equations \[2x^2+3y^2+6z^2=n\text{ and }3x+4y+5z=23.\]

1991 IMO Shortlist, 16
Tags: number theory , arithmetic sequence , system of equations , eulers function , laurentiu panaitopol
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If \[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0, \] prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.

1985 Tournament Of Towns, (082) T3
Tags: system of equations , algebra
Find all real solutions of the system of equations $\begin{cases} (x + y) ^3 = z \\ (y + z) ^3 = x \\ ( z+ x) ^3 = y \end{cases} $ (Based on an idea by A . Aho , J. Hop croft , J. Ullman )

2005 AIME Problems, 3
Tags: number theory , algebra , ratio , geometric series , relatively prime , system of equations
An infinite geometric series has sum $2005$. A new series, obtained by squaring each term of the original series, has $10$ times the sum of the original series. The common ratio of the original series is $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m+n$.

2015 Dutch IMO TST, 2
Tags: diophantine equation , system of equations , diophantine , number theory
Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

2021 Dutch BxMO TST, 2
Tags: system of equations , system , algebra
Find all triplets $(x, y, z)$ of real numbers for which $$\begin{cases}x^2- yz = |y-z| +1 \\ y^2 - zx = |z-x| +1 \\ z^2 -xy = |x-y| + 1 \end{cases}$$

1966 IMO Longlists, 62
Tags: system of equations , algebra
Solve the system of equations \[ |a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1 \] \[ |a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1 \] \[ |a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1 \] \[ |a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1 \] where $a_1, a_2, a_3, a_4$ are four different real numbers.

2011 Mathcenter Contest + Longlist, 5 sl6
Tags: algebra , system of equations , system
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]

2013 BMT Spring, 1
Tags: system of equations , algebra
Find the value of $a$ satisfying \begin{align*} a+b&=3\\ b+c&=11\\ c+a&=61 \end{align*}

1999 Austrian-Polish Competition, 3
Tags: algebra , functional , functional equation , system of equations
Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$ $$\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}$$

1968 IMO, 3
Tags: polynomial , discriminant , algebra , system of equations
Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.

2020 Tuymaada Olympiad, 1
Tags: equation , system of equations , algebra , number theory
Does the system of equation \begin{align*} \begin{cases} x_1 + x_2 &= y_1 + y_2 + y_3 + y_4 \\ x_1^2 + x_2^2 &= y_1^2 + y_2^2 + y_3^2 + y_4^2 \\ x_1^3 + x_2^3 &= y_1^3 + y_2^3 + y_3^3 + y_4^3 \end{cases} \end{align*} admit a solution in integers such that the absolute value of each of these integers is greater than $2020$?

2022 IFYM, Sozopol, 1
Tags: system , system of equations , algebra
Find all triples of complex numbers $(x, y, z)$ for which $$(x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = x^2(y + z) + y^2(z + x ) + z^2(x + y) = 0$$

2000 Swedish Mathematical Competition, 6
Tags: system of equations , algebra , system
Solve \[\left\{ \begin{array}{l} y(x+y)^2 = 9 \\ y(x^3-y^3) = 7 \\ \end{array} \right. \]

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

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 $$a^2+ab=c$$ $$b^2+bc=a$$ $$c^2+ca=b$$

2015 Turkmenistan National Math Olympiad, 1
Tags: algebra , system of equations
Solve : $y(x+y)^2=9 $ ; $y(x^3-y^3)=7$

2024 Ecuador NMO (OMEC), 1
Tags: algebra , system of equations
Find all real solutions: $$\begin{cases}a^3=2024bc \\ b^3=2024cd \\ c^3=2024da \\ d^3=2024ab \end{cases}$$

2012 Romania National Olympiad, 1
Tags: system of equations , algebra
Determine the real numbers $a, b, c, d$ so that $$ab + c + d = 3, \,\, bc + d + a = 5, \,\, cd + a + b = 2 \,\,\,\, and \,\,\,\,da + b + c = 6$$