Olympiad problems

1993 Denmark MO - Mohr Contest, 3

Determine all real solutions $x,y$ to the system of equations $$\begin{cases} x^2 + y^2 = 1 \\ x^6 + y^6 = \dfrac{7}{16} \end{cases}$$

2010 Saudi Arabia Pre-TST, 2.1

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

Find all triples $(x,y,z)$ of positive integers such that $$\begin{cases} x + y +z = 2010 \\x^2 + y^2 + z^2 - xy - yz - zx =3 \end{cases}$$

2016 Hanoi Open Mathematics Competitions, 9

Tags: inequalities , system , maximum , algebra

Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\ | x - 2y + 3z| \le 6 \\ | x - 2y - 3z| \le 6 \\ | x + 2y + 3z| \le 6 \end{cases}$ Determine the greatest value of $M = |x| + |y| + |z|$.

2015 Denmark MO - Mohr Contest, 4

Tags: system of equations , system , algebra

Determine all numbers $x, y$ and $z$ satisfying the system of equations $$\begin{cases} x^2 + yz = 1 \\ y^2 - xz = 0 \\ z^2 + xy = 1\end{cases}$$

2021 Dutch BxMO TST, 2

Tags: system of equations , system , algebra

Find all triplets $(x, y, z)$ of real numbers for which $$\begin{cases}x^2- yz = |y-z| +1 \\ y^2 - zx = |z-x| +1 \\ z^2 -xy = |x-y| + 1 \end{cases}$$

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]

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 \[\left\{ \begin{array}{l} y(x+y)^2 = 9 \\ y(x^3-y^3) = 7 \\ \end{array} \right. \]

1977 Swedish Mathematical Competition, 6

Tags: algebra , system , inequalities

Show that there are positive reals $a$, $b$, $c$ such that \[\left\{ \begin{array}{l} a^2 + b^2 + c^2 > 2 \\ a^3 + b^3 + c^3 <2 \\ a^4 + b^4 + c^4 > 2 \\ \end{array} \right. \]

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.

2005 Denmark MO - Mohr Contest, 2

Tags: algebra , parameterization , system of equations , system

Determine, for any positive real number $a$, the number of solutions $(x,y)$ to the system of equations $$\begin{cases} |x|+|y|= 1 \\ x^2 + y^2 = a \end{cases}$$ where $x$ and $y$ are real numbers.

2009 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: algebra , system , inequalities , minimum

Find minimal value of $a \in \mathbb{R}$ such that system $$\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}=a-1$$ $$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}=a+1$$ has solution in set of real numbers

2014 IMAC Arhimede, 5

Tags: number theory , numerical systems , system , binomial , product , modulo , prime number

Let $p$ be a prime number. The natural numbers $m$ and $n$ are written in the system with the base $p$ as $n = a_0 + a_1p +...+ a_kp^k$ and $m = b_0 + b_1p +..+ b_kp^k$. Prove that $${n \choose m} \equiv \prod_{i=0}^{k}{a_i \choose b_i} (mod p)$$

2008 Mathcenter Contest, 1

Tags: system of equations , system , algebra

Given $x,y,z\in \mathbb{R} ^+$ , that are the solutions to the system of equations : $$x^2+xy+y^2=57$$ $$y^2+yz+z^2=84$$ $$z^2+zx+x^2=111$$ What is the value of $xy+3yz+5zx$? [i](maphybich)[/i]

2019 Durer Math Competition Finals, 13

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

Let $k > 1$ be a positive integer and $n \ge 2019$ be an odd positive integer. The non-zero rational numbers $x_1, x_2,..., x_n$ are not all equal, and satisfy the following chain of equalities: $$x_1 +\frac{k}{x_2}= x_2 +\frac{k}{x_3}= x_3 +\frac{k}{x_4}= ... = x_{n-1} +\frac{k}{x_n}= x_n +\frac{k}{x_1}.$$ What is the smallest possible value of $k$?

2012 Argentina National Olympiad, 1

Tags: inequalities , system , system of equations , algebra

Determine if there are triplets ($x,y,z)$ of real numbers such that $$\begin{cases} x+y+z=7 \\ xy+yz+zx=11\end{cases}$$ If the answer is affirmative, find the minimum and maximum values of $z$ in such a triplet.

1983 Swedish Mathematical Competition, 6

Tags: algebra , system of equations , system

Show that the only real solution to \[\left\{ \begin{array}{l} x(x+y)^2 = 9 \\ x(y^3 - x^3) = 7 \\ \end{array} \right. \] is $x = 1$, $y = 2$.

1981 Swedish Mathematical Competition, 2

Tags: system of equations , system , algebra

Does \[\left\{ \begin{array}{l} x^y = z \\ y^z = x \\ z^x = y \\ \end{array} \right. \] have any solutions in positive reals apart from $x = y = z= 1$?

1976 Swedish Mathematical Competition, 2

Tags: algebra , system , system of equations

For which real $a$ are there distinct reals $x$, $y$ such that $$\begin{cases} x = a - y^2 \\ y = a - x^2 \,\,\, ? \end {cases}$$

1950 Poland - Second Round, 1

Tags: algebra , system , system of equations

Solve the system of equations $$\begin{cases} x^2+x+y=8\\ y^2+2xy+z=168\\ z^2+2yz+2xz=12480 \end{cases}$$

2010 Saudi Arabia Pre-TST, 4.1

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

Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$

2011 Mathcenter Contest + Longlist, 7

Tags: algebra , system , system of equations

Given $k_1,k_2,...,k_n\in R^+$, find all the naturals $n$ such that $$k_1+k_2+...+k_n=2n-3$$ $$\frac{1}{k_1}+\frac{1}{k_2}+...+\frac{1}{k_n}=3$$ [i](Zhuge Liang)[/i]

1966 German National Olympiad, 4

Tags: algebra , system of equations , system

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: $$x_1x_2 + x_1x_3 + x_2x_3 + x_4 = 2$$ $$x_1x_2 + x_1x_4 + x_2x_4 + x_3 = 2$$ $$x_1x_3 + x_1x_4 + x_3x_4 + x_2 = 2$$ $$x_2x_3 + x_2x_4 + x_3x_4 + x_1 = 2$$

1973 Spain Mathematical Olympiad, 2

Tags: inequalities , system of equations , system , algebra

Determine all solutions of the system $$\begin{cases} 2x - 5y + 11z - 6 = 0 \\ -x + 3y - 16z + 8 = 0 \\ 4x - 5y - 83z + 38 = 0 \\ 3x + 11y - z + 9 > 0 \end{cases}$$ in which the first three are equations and the last one is a linear inequality.