Olympiad problems

2015 Swedish Mathematical Competition, 4

Solve the system of equations $$ \left\{\begin{array}{l} x \log x+y \log y+z \log x=0\\ \\ \dfrac{\log x}{x}+\dfrac{\log y}{y}+\dfrac{\log z}{z}=0 \end{array} \right. $$

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 $x=n+1$, $y=n$, $z=n-1$.

2003 Swedish Mathematical Competition, 1

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

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 $ad - bc = \pm 1$.

2016 Junior Balkan Team Selection Tests - Moldova, 5

Tags: system , algebra , system of equations

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 $a+ b$.

1999 Romania National Olympiad, 1

Tags: system , algebra

Solve the system $$\begin{cases} \displaystyle 4^{-x}+27^{-y}= \frac{5}{6} \\ \displaystyle 27^y-4^x \le 1 \\ \displaystyle \log_{27}y-\log_4 x \ge \frac{1}{6} \end{cases}.$$

2022 Azerbaijan National Mathematical Olympiad, 4

Tags: system of equations , system , algebra

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$

2002 Swedish Mathematical Competition, 5

Tags: system , algebra , system of equations

The reals $a, b$ satisfy $$\begin{cases} a^3 - 3a^2 + 5a - 17 = 0 \\ b^3 - 3b^2 + 5b + 11 = 0 .\end{cases}$$ Find $a+b$.

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 $a^2 + b^2 + c^2 \le 38$

2021 Dutch IMO TST, 2

Tags: system , algebra , system of equations

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$

2014 NZMOC Camp Selection Problems, 7

Tags: system of equations , system , algebra

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.

1979 Swedish Mathematical Competition, 1

Tags: system , system of equations , algebra

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

2005 Denmark MO - Mohr Contest, 5

Tags: algebra , system , system of equations

For what real numbers $p$ has the system of equations $$\begin{cases} x_1^4+\dfrac{1}{x_1^2}=px_2 \\ \\ x_2^4+\dfrac{1}{x_2^2}=px_3 \\ ... \\ x_{2004}^4+\dfrac{1}{x_{2004}^2}=px_{2005} \\ \\ x_{2005}^4+\dfrac{1}{x_{2005}^2}=px_{1}\end{cases}$$ just one solution $(x_1,x_2,...,x_{2005})$, where $x_1,x_2,...,x_{2005}$ are real numbers?

2017 District Olympiad, 2

Tags: equation , logarithm , system

Solve in $ \mathbb{Z} $ the system: $$ \left\{ \begin{matrix} 2^x+\log_3 x=y^2 \\ 2^y+\log_3 y=x^2 \end{matrix} \right. . $$

1961 Poland - Second Round, 5

Tags: inequalities , system , algebra

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

1972 Poland - Second Round, 1

Tags: algebra , system , system of equations

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 $.

1980 Czech And Slovak Olympiad IIIA, 5

Tags: system , algebra , inequalities

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

2002 Swedish Mathematical Competition, 3

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

$C$ 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?

2017 OMMock - Mexico National Olympiad Mock Exam, 3

Tags: square , system , number theory

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]

1969 German National Olympiad, 4

Tags: system of equations , system , logarithm , algebra

Solve the system of equations: $$|\log_2(x + y)| + | \log_2(x - y)| = 3$$ $$xy = 3$$

2010 Saudi Arabia Pre-TST, 4.4

Tags: system , system of equations , algebra

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

1996 Denmark MO - Mohr Contest, 2

Tags: system , algebra , system of equations

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

1933 Eotvos Mathematical Competition, 1

Tags: algebra , system , system of equations

Let $a, b,c$ and $d$ be rea] numbers such that $a^2 + b^2 = c^2 + d^2 = 1$ and $ac + bd = 0$. Determine the value of $ab + cd$.

2000 Junior Balkan Team Selection Tests - Moldova, 4

Tags: algebra , system , inequalities

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) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$

2017 NZMOC Camp Selection Problems, 8

Tags: algebra , system , system of equations

Find all possible real values for $a, b$ and $c$ such that (a) $a + b + c = 51$, (b) $abc = 4000$, (c) $0 < a \le 10$ and $c \ge 25$.