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: 91

2004 May Olympiad, 4
Tags: system of equations , diophantine , system , number theory
Find all the natural numbers $x, y, z$ that satisfy simultaneously $$\begin{cases} x y z=4104 \\ x+y+z=77 \end{cases}$$

2005 Austria Beginners' Competition, 3
Tags: algebra , floor function , system of equations , system
Determine all triples $(x,y,z)$ of real numbers that satisfy all of the following three equations: $$\begin{cases} \lfloor x \rfloor + \{y\} =z \\ \lfloor y \rfloor + \{z\} =x \\ \lfloor z \rfloor + \{x\} =y \end{cases}$$

1987 Spain Mathematical Olympiad, 4
Tags: algebra , system
If $a$ and $b$ are distinct real numbers, solve the systems (a) $\begin{cases} x+y = 1 \\ (ax+by)^2 \le a^2x+b^2y \end{cases}$ and (b) $\begin{cases} x+y = 1 \\ (ax+by)^4 \le a^4x+b^4y \end{cases}$

1926 Eotvos Mathematical Competition, 1
Tags: system of equations , number theory , diophantine , diophantine equation , system
Prove that, if $a$ and $b$ are given integers, the system of equatìons $$x + y + 2z + 2t = a$$ $$2x - 2y + z- t = b$$ has a solution in integers $x, y,z,t$.

1995 Swedish Mathematical Competition, 3
Tags: system of equations , system , algebra
Let $a,b,x,y$ be positive numbers with $a+b+x+y < 2$. Given that $$\begin{cases} a+b^2 = x+y^2 \\ a^2 +b = x^2 +y\end {cases} $$ show that $a = x$ and $b = y$

2006 Denmark MO - Mohr Contest, 2
Tags: algebra , system of equations , system
Determine all sets of real numbers $(x,y,z)$ which fulfills $$\begin{cases} x + y =2 \\ xy -z^2= 1\end{cases}$$

1964 Poland - Second Round, 4
Tags: algebra , system of equations , system
Find the real numbers $ x, y, z $ satisfying the system of equations $$(z - x)(x - y) = a $$ $$(x - y)(y - z) = b$$ $$(y - z)(z - x) = c$$ where $ a, b, c $ are given real numbers.

1983 Swedish Mathematical Competition, 3
Tags: diophantine , number theory , even , system of equations , system , diophantine equation
The systems of equations \[\left\{ \begin{array}{l} 2x_1 - x_2 = 1 \\ -x_1 + 2x_2 - x_3 = 1 \\ -x_2 + 2x_3 - x_4 = 1 \\ -x_3 + 3x_4 - x_5 =1 \\ \cdots\cdots\cdots\cdots\\ -x_{n-2} + 2x_{n-1} - x_n = 1 \\ -x_{n-1} + 2x_n = 1 \\ \end{array} \right. \] has a solution in positive integers $x_i$. Show that $n$ must be even.

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?

2011 Swedish Mathematical Competition, 3
Tags: system of equations , algebra , system
Find all positive real numbers $x, y, z$, such that $$x - \frac{1}{y^2} = y - \frac{1}{z^2}= z - \frac{1}{x^2}$$

1979 Dutch Mathematical Olympiad, 2
Tags: number theory , system of equations , system , diophantine
Solve in $N$: $$\begin{cases} a^3=b^3+c^3+12a \\ a^2=5(b+c) \end{cases}$$

2004 Peru MO (ONEM), 3
Tags: system of equations , algebra , trigonometry , system
Let $x,y,z$ be positive real numbers, less than $\pi$, such that: $$\cos x + \cos y + \cos z = 0$$ $$\cos 2x + \cos 2 y + \cos 2z = 0$$ $$\cos 3x + \cos 3y + \cos 3z = 0$$ Find all the values that $\sin x + \sin y + \sin z$ can take.

1968 German National Olympiad, 1
Tags: system of equations , system , algebra
Determine all ordered quadruples of real numbers $(x_1, x_2, x_3, x_4)$ for which the following system of equations exists, is fulfilled: $$x_1 + ax_2 + x_3 = b $$ $$x_2 + ax_3 + x_4 = b $$ $$x_3 + ax_4 + x_1 = b $$ $$x_4 + ax_1 + x_2 = b$$ Here $a$ and $b$ are real numbers (case distinction!).

1971 Czech and Slovak Olympiad III A, 1
Tags: inequalities , algebra , system
Let $a,b,c$ real numbers. Show that there are non-negative $x,y,z,xyz\neq0$ such that \begin{align*} cy-bz &\ge 0, \\ az-cx &\ge 0, \\ bx-ay &\ge 0. \end{align*}

2003 Junior Balkan Team Selection Tests - Moldova, 6
Tags: system of equations , system , algebra
The real numbers x and у satisfy the equations $$\begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\ \\ \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt2 \end{cases}$$ Find the numerical value of the ratio $y/x$.

1992 Swedish Mathematical Competition, 3
Tags: inequalities , system , algebra
Solve: $$\begin{cases} 2x_1 - 5x_2 + 3x_3 \ge 0 \\ 2x_2 - 5x_3 + 3x4 \ge 0 \\ ...\\ 2x_{23} - 5x_{24} + 3x_{25} \ge 0\\ 2x_{24} - 5x_{25} + 3x_1 \ge 0\\ 2x_{25} - 5x_1 + 3x_2 \ge 0 \end{cases}$$