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

2009 USA Team Selection Test, 7
Tags: polynomial , system of equations , algebra , trigonometry
Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations \[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\] [i]Razvan Gelca.[/i]

III Soros Olympiad 1996 - 97 (Russia), 9.4
Tags: algebra , system of equations
Solve the system of equations $$\begin{cases} x^4-2x^3+x=y^2-y \\ y^4-2y^3+y=x^2-x \end{cases}$$

2009 JBMO Shortlist, 5
Tags: number theory , system of equations
Show that there are infinitely many positive integers $c$, such that the following equations both have solutions in positive integers: $(x^2 - c)(y^2 -c) = z^2 -c$ and $(x^2 + c)(y^2 - c) = z^2 - c$.

2014 Hanoi Open Mathematics Competitions, 1
Tags: system of equations , algebra
Let $a$ and $b$ satisfy the conditions $\begin{cases} a^3 - 6a^2 + 15a = 9 \\ b^3 - 3b^2 + 6b = -1 \end{cases}$ . The value of $(a - b)^{2014}$ is: (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

1949-56 Chisinau City MO, 40
Tags: logarithm , system of equations , algebra
Solve the system of equations: $$\begin{cases} \log_{2} x + \log_{4} y + \log_{4} z =2 \\ \log_{3} y + \log_{9} z + \log_{9} x =2 \\ \log_{4} z + \log_{16} x + \log_{16} y =2\end{cases}$$

2009 Mathcenter Contest, 4
Tags: algebra , system , system of equations
Find the values of the real numbers $x,y,z$ that correspond to the system of equations. $$8(x+\frac{1}{x}) =15(y+\frac{1}{y}) = 17(z+\frac{1}{z})$$ $$xy + yz + zx=1$$ [i](Heir of Ramanujan)[/i]

1997 Bosnia and Herzegovina Team Selection Test, 1
Tags: algebra , system of equations
Solve system of equation $$8(x^3+y^3+z^3)=73$$ $$2(x^2+y^2+z^2)=3(xy+yz+zx)$$ $$xyz=1$$ in set $\mathbb{R}^3$

2019 Brazil Undergrad MO, 3
Tags: calculus , algebra , system of equations
Let $a,b,c$ be constants and $a,b,c$ are positive real numbers. Prove that the equations $2x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2}$ $x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2}$ $x+y+2z=\sqrt{a^2+x^2}+\sqrt{a^2+y^2}$ have exactly one real solution $(x,y,z)$ with $x,y,z \geq 0$.

2012 Cuba MO, 1
Tags: algebra , system of equations
If $$\frac{x_1}{x_1+1} = \frac{x_2}{x_2+3} = \frac{x_3}{x_3+5} = ...= \frac{x_{1006}}{x_{1006}+2011}$$ and $x_1+x_2+...+x_{1006} = 503^2$, determine the value of $x_{1006}$.

2016 Dutch BxMO TST, 2
Tags: system of equations , algebra
Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations $\begin{cases} x^2 - y = (z - 1)^2\\ y^2 - z = (x - 1)^2 \\ z^2 - x = (y -1)^2 \end{cases}$.

2017 QEDMO 15th, 7
Tags: system , algebra , system of equations
Find all real solutions $x, y$ of the system of equations $$\begin{cases} x + \dfrac{3x-y}{x^2 + y^2} = 3 \\ \\ y-\dfrac{x + 3y}{x^2 + y^2} = 0 \end{cases}$$

2024 Centroamerican and Caribbean Math Olympiad, 5
Tags: algebra , system of equations
Let \(x\) and \(y\) be positive real numbers satisfying the following system of equations: \[ \begin{cases} \sqrt{x}\left(2 + \dfrac{5}{x+y}\right) = 3 \\\\ \sqrt{y}\left(2 - \dfrac{5}{x+y}\right) = 2 \end{cases} \] Find the maximum value of \(x + y\).

2019 Greece Junior Math Olympiad, 1
Tags: system of equations , algebra
Find all triplets of real numbers $(x,y,z)$ that are solutions to the system of equations $x^2+y^2+25z^2=6xz+8yz$ $ 3x^2+2y^2+z^2=240$

2014 Hanoi Open Mathematics Competitions, 1
Tags: system of equations , algebra
Let the numbers x and y satisfy the conditions $\begin{cases} x^2 + y^2 - xy = 2 \\ x^4 + y^4 + x^2y^2 = 8 \end{cases}$ The value of $P = x^8 + y^8 + x^{2014}y^{2014}$ is: (A): $46$, (B): $48$, (C): $50$, (D): $52$, (E) None of the above.

1963 IMO Shortlist, 4
Tags: algebra , parameterization , system of equations
Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system \[ x_5+x_2=yx_1 \] \[ x_1+x_3=yx_2 \] \[ x_2+x_4=yx_3 \] \[ x_3+x_5=yx_4 \] \[ x_4+x_1=yx_5 \] where $y$ is a parameter.

1996 Austrian-Polish Competition, 6
Tags: system of equations , algebra
Given natural numbers $n > k > 1$, find all real solutions $x_1,..., x_n$ of the system $$x_i^3(x_i^2 + x_{i+1}^2+... +x_{i+k-1}^2) = x_{i-1}^2$$ for 1 $\le i \le n$. Here $x_{n+i} = x_i$ for all$ i$.

2021 Latvia Baltic Way TST, P3
Tags: algebra , system of equations
Find all triplets of real numbers $(x,y,z)$ such that the following equations are satisfied simultaneously: \begin{align*} x^3+y=z^2 \\ y^3+z=x^2 \\ z^3+x =y^2 \end{align*}

2010 AMC 12/AHSME, 10
Tags: algebra , arithmetic sequence , system of equations
The first four terms of an arithmetic sequence are $ p,9,3p\minus{}q,$ and $ 3p\plus{}q$. What is the $ 2010^{\text{th}}$ term of the sequence? $ \textbf{(A)}\ 8041\qquad \textbf{(B)}\ 8043\qquad \textbf{(C)}\ 8045\qquad \textbf{(D)}\ 8047\qquad \textbf{(E)}\ 8049$

2015 Swedish Mathematical Competition, 4
Tags: logarithm , system , system of equations , algebra
Solve the system of equations $$ \left\{\begin{array}{l} x \log x+y \log y+z \log x=0\\ \\ \dfrac{\log x}{x}+\dfrac{\log y}{y}+\dfrac{\log z}{z}=0 \end{array} \right. $$

2007 Federal Competition For Advanced Students, Part 2, 2
Tags: system of equations , algebra
Find all tuples $ (x_1,x_2,x_3,x_4,x_5,x_6)$ of non-negative integers, such that the following system of equations holds: $ x_1x_2(1\minus{}x_3)\equal{}x_4x_5 \\ x_2x_3(1\minus{}x_4)\equal{}x_5x_6 \\ x_3x_4(1\minus{}x_5)\equal{}x_6x_1 \\ x_4x_5(1\minus{}x_6)\equal{}x_1x_2 \\ x_5x_6(1\minus{}x_1)\equal{}x_2x_3 \\ x_6x_1(1\minus{}x_2)\equal{}x_3x_4$

2012 Purple Comet Problems, 7
Tags: system of equations , algebra
Two convex polygons have a total of 33 sides and 243 diagonals. Find the number of diagonals in the polygon with the greater number of sides.

2011-2012 SDML (High School), 13
Tags: system of equations , algebra
The number of solutions, in real numbers $a$, $b$, and $c$, to the system of equations $$a+bc=1,$$$$b+ac=1,$$$$c+ab=1,$$ is $\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) more than }5\text{, but finitely many}\qquad\text{(E) infinitely many}$

2022 Chile Junior Math Olympiad, 1
Tags: algebra , system of equations
Find all real numbers $x, y, z$ that satisfy the following system $$\sqrt{x^3 - y} = z - 1$$ $$\sqrt{y^3 - z} = x - 1$$ $$\sqrt{z^3 - x} = y - 1$$

1977 Swedish Mathematical Competition, 3
Tags: diophantine , system , system of equations , number theory
Show that the only integral solution to \[\left\{ \begin{array}{l} xy + yz + zx = 3n^2 - 1\\ x + y + z = 3n \\ \end{array} \right. \] with $x \geq y \geq z$ is $x=n+1$, $y=n$, $z=n-1$.

2010 Dutch IMO TST, 5
Tags: algebra , system of equations
Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying $3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.