Olympiad problems

2006 Denmark MO - Mohr Contest, 2

Determine all sets of real numbers $(x,y,z)$ which fulfills $$\begin{cases} x + y =2 \\ xy -z^2= 1\end{cases}$$

2016 Hanoi Open Mathematics Competitions, 13

Tags: algebra , System , inequalities

Find all triples $(a,b,c)$ of real numbers such that $|2a + b| \ge 4$ and $|ax^2 + bx + c| \le 1$ $ \forall x \in [-1, 1]$.

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 $x = y = z= 1$?

1996 Denmark MO - Mohr Contest, 2

Tags: algebra , system of equations , System

Determine all sets of real numbers $x,y,z$ which satisfy the system of equations $$\begin{cases} xy = z \\ xz =y \\ yz =x \end{cases}$$

2014 NZMOC Camp Selection Problems, 7

Tags: algebra , system of equations , System

Determine all pairs of real numbers $(k, d)$ such that the system of equations $$\begin{cases} x^3 + y^3 = 2 \\ kx + d = y\end{cases}$$ has no solutions $(x, y)$ with $x$ and $y$ real numbers.

2021 Dutch BxMO TST, 2

Tags: system of equations , System , algebra

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

2010 Saudi Arabia Pre-TST, 4.4

Tags: system of equations , System , algebra

Find all pairs $(x, y)$ of real numbers that satisfy the system of equations $$\begin{cases} x^4 + 2z^3 - y =\sqrt3 - \dfrac14 \\ y^4 + 2y^3 - x = - \sqrt3 - \dfrac14 \end{cases}$$

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

2017 OMMock - Mexico National Olympiad Mock Exam, 3

Tags: number theory , Squares , System

Let $x, y, z$ be positive integers such that $xy=z^2+2$. Prove that there exist integers $a, b, c, d$ such that the following equalities are satisfied: \begin{eqnarray*} x=a^2+2b^2\\ y=c^2+d^2\\ z=ac+2bd\\ \end{eqnarray*} [i]Proposed by Isaac Jiménez[/i]

2014 Hanoi Open Mathematics Competitions, 13

Tags: inequalities , System , algebra

Let $a, b,c$ satises the conditions $\begin{cases} 5 \ge a \ge b \ge c \ge 0 \\ a + b \le 8 \\ a + b + c = 10 \end{cases}$ Prove that $a^2 + b^2 + c^2 \le 38$

1986 Swedish Mathematical Competition, 4

Tags: system of equations , System , algebra

Prove that $x = y = z = 1$ is the only positive solution of the system \[\left\{ \begin{array}{l} x+y^2 +z^3 = 3\\ y+z^2 +x^3 = 3\\ z+x^2 +y^3 = 3\\ \end{array} \right. \]

2022 IFYM, Sozopol, 1

Tags: algebra , system of equations , System

Find all triples of complex numbers $(x, y, z)$ for which $$(x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = x^2(y + z) + y^2(z + x ) + z^2(x + y) = 0$$

2004 Denmark MO - Mohr Contest, 4

Tags: algebra , system of equations , System

Find all sets $x,y,z$ of real numbers that satisfy $$\begin{cases} x^3 - y^2 = z^2 - x \\ y^3 -z^2 =x^2 -y \\z^3 -x^2 = y^2 -z \end{cases}$$

2004 Cuba MO, 1

Tags: algebra , system of equations , System

Determine all real solutions to the system of equations: $$x_1 + x_2 +...+ x_{2004 }= 2004$$ $$x^4_1+ x^4_2+ ... + x^4_{2004} = x^3_1+x^3_2+... + x^3_{2004}$$

1976 Swedish Mathematical Competition, 2

Tags: algebra , system of equations , System

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

1993 IMO, 6

Tags: invariant , polynomial , combinatorics , algorithm , System , IMO , IMO 1993

Let $n > 1$ be an integer. In a circular arrangement of $n$ lamps $L_0, \ldots, L_{n-1},$ each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \ldots .$ If $L_{j-1}$ ($j$ is taken mod $n$) is ON then $Step_j$ changes the state of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If $L_{j-1}$ is OFF then $Step_j$ does not change anything at all. Show that: (i) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again, (ii) If $n$ has the form $2^k$ then all the lamps are ON after $n^2-1$ steps, (iii) If $n$ has the form $2^k + 1$ then all lamps are ON after $n^2 - n + 1$ steps.

1979 Dutch Mathematical Olympiad, 2

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

Solve in $N$: $$\begin{cases} a^3=b^3+c^3+12a \\ a^2=5(b+c) \end{cases}$$

2021 Dutch BxMO TST, 2

Tags: system of equations , System , algebra

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

2017 QEDMO 15th, 7

Tags: algebra , system of equations , System

Find all real solutions $x, y$ of the system of equations $$\begin{cases} x + \dfrac{3x-y}{x^2 + y^2} = 3 \\ \\ y-\dfrac{x + 3y}{x^2 + y^2} = 0 \end{cases}$$

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.

1993 Denmark MO - Mohr Contest, 3

Tags: algebra , system of equations , System

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

1926 Eotvos Mathematical Competition, 1

Tags: diophantine , Diophantine equation , system of equations , System , number theory

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 $x, y,z,t$.

2010 Saudi Arabia Pre-TST, 2.1

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

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

2019 Durer Math Competition Finals, 13

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

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 $k$?

1974 Swedish Mathematical Competition, 5

Tags: system of equations , System , algebra

Find the smallest positive real $t$ such that \[\left\{ \begin{array}{l} x_1 + x_3 = 2t x_2 \\ x_2 + x_4 = 2t x_3 \\ x_3 + x_5=2t x_4 \\ \end{array} \right. \] has a solution $x_1$, $x_2$, $x_3$, $x_4$, $x_5$ in non-negative reals, not all zero.