Olympiad problems

2015 Denmark MO - Mohr Contest, 4

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

1977 Swedish Mathematical Competition, 6

Tags: inequalities , System , algebra

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

2004 Peru MO (ONEM), 3

Tags: algebra , trigonometry , system of equations , 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.

1971 Czech and Slovak Olympiad III A, 1

Tags: algebra , System , inequalities

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*}

1980 Czech And Slovak Olympiad IIIA, 5

Tags: algebra , inequalities , System

Solve a set of inequalities in the domain of integer numbers: $$3x^2 +2yz \le 1+y^2$$ $$3y^2 +2zx \le 1+z^2$$ $$3z^2 +2xy \le 1+x^2$$

2016 Junior Balkan Team Selection Tests - Moldova, 5

Tags: system of equations , System , algebra

Real numbers $a$ and $b$ satisfy the system of equations $$\begin{cases} a^3-a^2+a-5=0 \\ b^3-2b^2+2b+4=0 \end{cases}$$ Find the numerical value of the sum $a+ b$.

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) $\begin{cases} x+y = 1 \\ (ax+by)^4 \le a^4x+b^4y \end{cases}$

1961 Poland - Second Round, 5

Tags: algebra , System , inequalities

Prove that if the real numbers $ a $, $ b $, $ c $ satisfy the inequalities $$a + b + c> 0,$$ $$ ab + bc + ca > 0$$ $$ abc > 0$$ then $a > 0, b > 0, c > 0$.

2000 Junior Balkan Team Selection Tests - Moldova, 4

Tags: algebra , inequalities , System

Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions: 1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$ 2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$ 3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$

2010 Saudi Arabia BMO TST, 4

Tags: system of equations , System , algebra

Let $a > 0$. If the system $$\begin{cases} a^x + a^y + a^z = 14 - a \\ x + y + z = 1 \end{cases}$$ has a solution in real numbers, prove that $a \le 8$.

1979 Swedish Mathematical Competition, 1

Tags: algebra , system of equations , System

Solve the equations: \[\left\{ \begin{array}{l} x_1 + 2 x_2 + 3 x_3 + \cdots + (n-1) x_{n-1} + n x_n = n \\ 2 x_1 + 3 x_2 + 4 x_3 + \cdots + n x_{n-1} + x_n = n-1 \\ 3 x_1 + 4 x_2 + 5 x_3 + \cdots + x_{n-1} + 2 x_n = n-2 \\ \cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot \\ (n-1) x_1 + n x_2 + x_3 + \cdots + (n-3) x_{n-1} + (n-2) x_n = 2 \\ n x_1 + x_2 + 2 x_3 + \cdots + (n-2) x_{n-1} + (n-1) x_n = 1 \end{array} \right. \]

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 Poland - Second Round, 1

Tags: algebra , system of equations , System

Prove that there are no real numbers $ a, b, c $, $ x_1, x_2, x_3 $ such that for every real number $ x $ $$ ax^2 + bx + c = a(x - x_2)(x - x_3) $$ $$bx^2 + cx + a = b(x - x_3) (x - x_1)$$ $$cx^2 + ax + b = c(x - x_1) (x - x_2)$$ and $ x_1 \neq x_2 $, $ x_2 \neq x_3 $, $ x_3 \neq x_1 $, $ abc \neq 0 $.

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 Mathcenter Contest + Longlist, 7

Tags: algebra , system of equations , System

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]

2005 Denmark MO - Mohr Contest, 2

Tags: parameterization , algebra , 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.

2002 Swedish Mathematical Competition, 3

Tags: parabola , system of equations , System , parameter , algebra , analytic geometry

$C$ is the circle center $(0,1)$, radius $1$. $P$ is the parabola $y = ax^2$. They meet at $(0, 0)$. For what values of $a$ do they meet at another point or points?

2011 Swedish Mathematical Competition, 3

Tags: system of equations , System , algebra

Find all positive real numbers $x, y, z$, such that $$x - \frac{1}{y^2} = y - \frac{1}{z^2}= z - \frac{1}{x^2}$$

2003 Swedish Mathematical Competition, 1

Tags: algebra , min , system of equations , System , inequalities

If $x, y, z, w$ are nonnegative real numbers satisfying \[\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\ \end{array} \right. \] find the smallest possible value of $x$ and the values of $y, z, w$ corresponding to it.

1942 Eotvos Mathematical Competition, 2

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

Let $a, b, c $and $d$ be integers such that for all integers m and n, there exist integers $x$ and $y$ such that $ax + by = m$, and $cx + dy = n$. Prove that $ad - bc = \pm 1$.

2000 Swedish Mathematical Competition, 6

Tags: algebra , System , system of equations

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

2008 Mathcenter Contest, 1

Tags: algebra , system of equations , System

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]

1982 Spain Mathematical Olympiad, 1

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

On the puzzle page of a newspaper this problem is proposed: “Two children, Antonio and José, have $160$ comics. Antonio counts his by $7$ by $7$ and there are $4$ left over. José counts his $ 8$ by $8$ and he also has $4$ left over. How many comics does he have each?" In the next issue of the newspaper this solution is given: “Antonio has $60$ comics and José has $100$.” Analyze this solution and indicate what a mathematician would do with this problem.

1977 Swedish Mathematical Competition, 3

Tags: diophantine , system of equations , System , 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 $x=n+1$, $y=n$, $z=n-1$.

2011 Saudi Arabia Pre-TST, 3.4

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

Find all quadruples $(x,y,z,w)$ of integers satisfying the system of equations $$x + y + z + w = xy + yz + zx + w^2 - w = xyz - w^3 = - 1$$