Found problems: 744
Real numbers $a, b, c, d, e$, have the property $$|a - b| = 2|b -c| = 3|c - d| = 4|d- e| = 5|e - a|.$$ Prove they are all equal.
Find all solutions of the following system of $n$ equations in $n$ variables:
\[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\]
where $a$ is a given number.
Solve the system of equations:
$$\begin{cases} x + [y] + \{z\}=3.9 \\
y + [z] + \{x\}= 3.5 \\
z + [x] + \{y\}= 2.
\end{cases}$$
Solve system of equation $$8(x^3+y^3+z^3)=73$$ $$2(x^2+y^2+z^2)=3(xy+yz+zx)$$ $$xyz=1$$ in set $\mathbb{R}^3$
Find all real $x,y,z$ such that $\frac{x-2y}{y}+\frac{2y-4}{x}+\frac{4}{xy}=0$ and $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$.
Suppose $x$ and $y$ are nonzero real numbers simultaneously satisfying the equations
$x + \frac{2018}{y}= 1000$ and $ \frac{9}{x}+ y = 1$.
Find the maximum possible value of $x + 1000y$.
Find all solutions in positive integers to: $$\begin{cases} x_1^4 + 14x_1x_2 + 1 = y_1^4 \\ x_2^4 + 14x_2x_3 + 1 = y_2^4 \\ ... \\ x_n^4 + 14x_nx_1 + 1 = y_n^4 \end{cases}$$
Positive numbers $ x_1, x_2, . . ., x_{2009}$ satisfy the equalities
$$x^2_1 - x_1x_2 +x^2_2 =x^2_2 -x_2x_3+x^2_3=x^2_3 -x_3x_4+x^2_4= ...= x^2_{2008}- x_{2008}x_{2009}+x^2_{2009}=
x^2_{2009}-x_{2009}x_1+x^2_1$$. Prove that the numbers $ x_1, x_2, . . ., x_{2009}$ are equal.
Determine all triples $(x,y,z)$ of real numbers such that the following system of equations holds true:
\begin{align*}2^{\sqrt[3]{x^2}}\cdot 4^{\sqrt[3]{y^2}}\cdot 16^{\sqrt[3]{z^2}}&=128\\
\left(xy^2+z^4\right)^2&=4+\left(xy^2-z^4\right)^2\mbox{.}\end{align*}
Find all real solutions $x, y$ of the system of equations
$$\begin{cases} x + \dfrac{3x-y}{x^2 + y^2} = 3 \\ \\ y-\dfrac{x + 3y}{x^2 + y^2} = 0 \end{cases}$$
Let $k > 1, n > 2018$ be positive integers, and let $n$ be odd. The nonzero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and satisfy $$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$
Find:
a) the product $x_1 x_2 \ldots x_n$ as a function of $k$ and $n$
b) the least value of $k$, such that there exist $n,x_1,x_2,\ldots,x_n$ satisfying the given conditions.
Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that
$a + b + c = 3k + 1$,
$ab + bc + ca = 3k^2 + 2k$.
Slovakia
Let $x = p, y = q, z = r, w = s$ be the unique solution of the system of linear equations \[ x + a_i \cdot y + a^2_i \cdot z + a^3_i \cdot w = a^4_i, i = 1,2,3,4. \] Express the solutions of the following system in terms of $p,q,r$ and $s:$ \[ x + a^2_i \cdot y + a^4_i \cdot z + a^6_i \cdot w = a^8_i, i = 1,2,3,4. \] Assume the uniquness of the solution.
Let $a, b, c, d$ be real numbers satisfying the system of equation
$\[(a+b)(c+d)=2 \\
(a+c)(b+d)=3 \\
(a+d)(b+c)=4\]$
Find the minimum value of $a^2+b^2+c^2+d^2$.
Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations
\[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \]
Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply:
$$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$
Determine all trios of integers $(x, y, z)$ which are solution of system of equations
$\begin{cases} x - yz = 1 \\ xz + y = 2 \end{cases}$
Are there four real numbers $a, b, c, d$ for every three positive real numbers $x, y, z$ with the property $ad + bc = x$, $ac + bd = y$, $ab + cd = z$ and one of the numbers $a, b, c, d$ is equal to the sum of the other three?
Suppose that $x, y$, and $z$ are positive real numbers satisfying
$$\begin{cases} x^2 + xy + y^2 = 64 \\
y^2 + yz + z^2 = 49 \\
z^2 + zx + x^2 = 57 \end{cases}$$
Then $\sqrt[3]{xyz}$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
The sequence of Fibonnaci's numbers if defined from the two first digits $f_1=f_2=1$ and the formula $f_{n+2}=f_{n+1}+f_n$, $\forall n \in N$.
[b](a)[/b] Prove that $f_{2010} $ is divisible by $10$.
[b](b)[/b] Is $f_{1005}$ divisible by $4$?
Albanian National Mathematical Olympiad 2010---12 GRADE Question 4.
Let $ X: \equal{} \{x_1,x_2,\ldots,x_{29}\}$ be a set of $ 29$ boys: they play with each other in a tournament of Pro Evolution Soccer 2009, in respect of the following rules:
[list]i) every boy play one and only one time against each other boy (so we can assume that every match has the form $ (x_i \text{ Vs } x_j)$ for some $ i \neq j$);
ii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the win of the boy $ x_i$, then $ x_i$ gains $ 1$ point, and $ x_j$ doesn’t gain any point;
iii) if the match $ (x_i \text{ Vs } x_j)$, with $ i \neq j$, ends with the parity of the two boys, then $ \frac {1}{2}$ point is assigned to both boys.
[/list]
(We assume for simplicity that in the imaginary match $ (x_i \text{ Vs } x_i)$ the boy $ x_i$ doesn’t gain any point).
Show that for some positive integer $ k \le 29$ there exist a set of boys $ \{x_{t_1},x_{t_2},\ldots,x_{t_k}\} \subseteq X$ such that, for all choice of the positive integer $ i \le 29$, the boy $ x_i$ gains always a integer number of points in the total of the matches $ \{(x_i \text{ Vs } x_{t_1}),(x_i \text{ Vs } x_{t_2}),\ldots, (x_i \text{ Vs } x_{t_k})\}$.
[i](Paolo Leonetti)[/i]
Find all real pairs $x,y$ such that
\begin{align*}
x-|y+1|&=1, \\
x^2+y&=10.
\end{align*}
Given integer $ n$ larger than $ 5$, solve the system of equations (assuming $x_i \geq 0$, for $ i=1,2, \dots n$):
\[ \begin{cases} \displaystyle x_1+ \phantom{2^2} x_2+ \phantom{3^2} x_3 + \cdots + \phantom{n^2} x_n &= n+2, \\ x_1 + 2\phantom{^2}x_2 + 3\phantom{^2}x_3 + \cdots + n\phantom{^2}x_n &= 2n+2, \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n &= n^2 + n +4, \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n &= n^3 + n + 8. \end{cases} \]
Let $a, b, c, d$ be distinct non-zero real numbers satisfying the following two conditions:
$ac = bd$ and $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 4$.
Determine the largest possible value of the expression $\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}$.
Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties:
(1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$;
(2) the degree of $f$ is less than $n$.
[i](Hungary) Géza Kós[/i]