Olympiad problems

1996 German National Olympiad, 4

Find all pairs of real numbers $(x,y)$ which satisfy the system $$\begin{cases} x-y = 7 \\ \sqrt[3]{x^2}+\sqrt[3]{xy}+\sqrt[3]{y^2} = 7\end{cases}$$

2022 Chile National Olympiad, 1

Tags: algebra , system of equations

Find all real numbers $x, y, z$ that satisfy the following system $$\sqrt{x^3 - y} = z - 1$$ $$\sqrt{y^3 - z} = x - 1$$ $$\sqrt{z^3 - x} = y - 1$$

2002 Greece JBMO TST, 1

Tags: algebra , system of equations

Real numbers $x,y,a$ are such that $x+y=x^2+y^2=x^3+y^3=a$. Find all the possible values of $a$.

2010 Stanford Mathematics Tournament, 9

Tags: geometry , incenter , algebra , circumcircle , system of equations , trigonometry

For an acute triangle $ABC$ and a point $X$ satisfying $\angle{ABX}+\angle{ACX}=\angle{CBX}+\angle{BCX}$. Find the minimum length of $AX$ if $AB=13$, $BC=14$, and $CA=15$.

2023 Israel National Olympiad, P4

Tags: parameterization , algebra , system of equations

For each positive integer $n$, find all triples $a,b,c$ of real numbers for which \[\begin{cases}a=b^n+c^n\\ b=c^n+a^n\\ c=a^n+b^n\end{cases}\]

How many ordered triples $(x,y,z)$ of integers satisfy the system of equations below? \[ \begin{array}{l} x^2-3xy+2yz-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array} \] $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{a finite number greater than 2} \qquad \text{(E)} \ \text{infinately many}$

III Soros Olympiad 1996 - 97 (Russia), 10.5

Tags: system of equations , algebra

Solve the system of equations $$\begin{cases} \dfrac{x+y}{1+xy}=\dfrac{1-2y}{2-y} \\ \dfrac{x-y}{1-xy}=\dfrac{1-3x}{3-x} \end{cases}$$

2018 Polish Junior MO Second Round, 1

Tags: system of equations , pythagorean triple , algebra

Do positive reals $a, b, c, x$ such that $a^2+ b^2 = c^2$ and $(a + x)^2+ (b +x)^2 = (c + x)^2$ exist?

2016 Korea Summer Program Practice Test, 1

Tags: system of equations , algebra

Find all real numbers $x_1, \dots, x_{2016}$ that satisfy the following equation for each $1 \le i \le 2016$. (Here $x_{2017} = x_1$.) \[ x_i^2 + x_i - 1 = x_{i+1} \]

2003 Bundeswettbewerb Mathematik, 2

Tags: algebra , system of equations , function

Find all triples $\left(x,\ y,\ z\right)$ of integers satisfying the following system of equations: $x^3-4x^2-16x+60=y$; $y^3-4y^2-16y+60=z$; $z^3-4z^2-16z+60=x$.

2007 Cuba MO, 1

Tags: algebra , system of equations , system

Find all the real numbers $x, y$ such that $x^3 - y^3 = 7(x - y)$ and $x^3 + y^3 = 5(x + y).$

1972 Swedish Mathematical Competition, 1

Tags: diophantine , system of equations , system , number theory

Find the largest real number $a$ such that \[\left\{ \begin{array}{l} x - 4y = 1 \\ ax + 3y = 1\\ \end{array} \right. \] has an integer solution.

2019 Stars of Mathematics, 1

Tags: algebra , system of equations

Let $m$ be a positive integer and $n=m^2+1$. Determine all real numbers $x_1,x_2,\dotsc ,x_n$ satisfying $$x_i=1+\frac{2mx_i^2}{x_1^2+x_2^2+\cdots +x_n^2}\quad \text{for all }i=1,2,\dotsc ,n.$$

2016 CMIMC, 3

Tags: algebra , system of equations , yuh.

Suppose $x$ and $y$ are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then $x^2+y^2$ can be written in the form $\sqrt[m]{n}$, where $m$ and $n$ are positive integers and $m$ is as small as possible. Find $m+n$.

2022-23 IOQM India, 16

Tags: algebra , system of equations

Let $a,b,c$ be reals satisfying\\ $\hspace{2cm} 3ab+2=6b, \hspace{0.5cm} 3bc+2=5c, \hspace{0.5cm} 3ca+2=4a.$\\ \\ Let $\mathbb{Q}$ denote the set of all rational numbers. Given that the product $abc$ can take two values $\frac{r}{s}\in \mathbb{Q}$ and $\frac{t}{u}\in \mathbb{Q}$ , in lowest form, find $r+s+t+u$.

2006 Lithuania National Olympiad, 1

Tags: algebra , system of equations

Solve the system of equations: $\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$

2010 Contests, 2

Tags: system of equations , algebra

Find all real $x,y,z$ such that $\frac{x-2y}{y}+\frac{2y-4}{x}+\frac{4}{xy}=0$ and $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$.

2025 Kosovo National Mathematical Olympiad`, P2

Tags: system of equations , algebra

Find all real numbers $a$ and $b$ that satisfy the system of equations: $$\begin{cases} a &= \frac{2}{a+b} \\ \\ b &= \frac{2}{3a-b} \\ \end{cases}$$

2023 Brazil National Olympiad, 3

Tags: diophantine equation , system of equations , number theory

Let $n$ be a positive integer. Show that there are integers $x_1, x_2, \ldots , x_n$, [i]not all equal[/i], satisfying $$\begin{cases} x_1^2+x_2+x_3+\ldots+x_n=0 \\ x_1+x_2^2+x_3+\ldots+x_n=0 \\ x_1+x_2+x_3^2+\ldots+x_n=0 \\ \vdots \\ x_1+x_2+x_3+\ldots+x_n^2=0 \end{cases}$$ if, and only if, $2n-1$ is not prime.

1996 German National Olympiad, 4

2022 Chile National Olympiad, 1

2002 Greece JBMO TST, 1

2010 Stanford Mathematics Tournament, 9

2023 Israel National Olympiad, P4

1980 AMC 12/AHSME, 29

III Soros Olympiad 1996 - 97 (Russia), 10.5

2018 Polish Junior MO Second Round, 1

2016 Korea Summer Program Practice Test, 1

2003 Bundeswettbewerb Mathematik, 2

2007 Cuba MO, 1

1972 Swedish Mathematical Competition, 1

2019 Stars of Mathematics, 1

2016 CMIMC, 3

2022-23 IOQM India, 16

2006 Lithuania National Olympiad, 1

2010 Contests, 2

2025 Kosovo National Mathematical Olympiad`, P2

2023 Brazil National Olympiad, 3

2000 Moldova National Olympiad, Problem 2

1998 Romania National Olympiad, 1

1905 Eotvos Mathematical Competition, 1

2005 Denmark MO - Mohr Contest, 2

2004 Germany Team Selection Test, 1

2009 Ukraine National Mathematical Olympiad, 1