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]

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

Find all triplets of real $(a,b,c)$ that solve the equation $a(a-b-c)+(b^2+c^2-bc)=4c^2\left(abc-\frac{a^2}{4}-b^2c^2\right)$

Solve the following system of equations in real numbers: \[ \begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}. \]

Solve the system of equations \[x^2 + y^2 + z^2 + t^2 = 50;\] \[x^2 - y^2 + z^2 - t^2 = -24;\] \[xy = zt;\] \[x - y + z - t = 0.\]

Let $a_1,\ldots,a_n\ (n>2)$ be real numbers with at most one zero. Solve the system \begin{align*} x_1x_2 &= a_1, \\ x_2x_3 &= a_2, \\ &\ \vdots \\ x_{n-1}x_n &= a_{n-1}, \\ x_nx_1 &\ge a_n. \end{align*}

Find all pair $(x, y)$ in degrees such that \begin{align*} &\sin (x + 150^\circ) = \cos (y - 75^\circ), \\ &\cos x + \sin (y - 225^\circ) + \frac{\sqrt3}{2} = 0. \end{align*}

The parabola $ y \equal{} ax^2 \plus{} bx \plus{} c$ has vertex $ (p,p)$ and $ y$-intercept $ (0, \minus{} p)$, where $ p\neq 0$. What is $ b$? $ \textbf{(A) } \minus{} p \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } p$

Real numbers $a, b, c, x, y, z$ satisfy $$\begin{aligned} \begin{cases} a^2+2bc=x^2+2yz,\\ b^2+2ca=y^2+2zx,\\ c^2+2ab=z^2+2xy.\\ \end{cases} \end{aligned}$$ Prove that $a^2+b^2+c^2=x^2+y^2+z^2$.

Solve the following system for real $x,y,z$ \[ \{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array} \]

Find the nonnegative real numbers $ a,b,c,d $ that satisfy the following system: $$ \left\{ \begin{matrix} a^3+2abc+bcd-6&=&a \\a^2b+b^2c+abd+bd^2&=&b\\a^2b+a^2c+bc^2+cd^2&=&c\\d^3+ab^2+abc+bcd-6&=&d \end{matrix} \right. $$

Solve in $ \mathbb{R}^3 $ the system: $$ \left\{ \begin{matrix} 3^x+4^x=5^y \\8^y+15^y=17^z \\ 20^z+21^z=29^x \end{matrix} \right. $$ [i]Cristinel Mortici[/i]

Find all real solutions of the system of equations $$\begin{cases} x^2 + y^2 + u^2 + v^2 = 4 \\ xu + yv + xv + yu = 0 \\ xyu + yuv + uvx + vxy = - 2 \\ xyuv = -1 \end{cases}$$

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 $a,b,c$ be coprime nonzero integers. Prove that for any coprime integers $u,v,w$ with $au+bv+cw = 0$ there exist integers $m,n, p$ such that $$\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}$$

Solve the system $$\begin{cases} x+\log\left(x+\sqrt{x^2+1}\right)=y \\ y+\log\left(y+\sqrt{y^2+1}\right)=z \\ z+\log\left(z+\sqrt{z^2+1}\right)=x \end{cases}$$

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 $m + n$.

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?

Find all real solutions of the system of equations: \[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$

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

Solve the system of equations $$ \begin{array}{l}<br /> x^2y^2 + x^2z^2 = axyz\\<br /> y^2z^2 + y^2x^2 = bxyz\\<br /> z^2x^2 + z^2y^2 = cxyz.<br /> \end{array}$$

Six integers $a,b,c,d,e,f$ satisfy: $\begin{cases} ace+3ebd-3bcf+3adf=5 \\ bce+acf-ade+3bdf=2 \end{cases}$ Find all possible values of $abcde$ [i]D. Bazyleu[/i]

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.

Let us denote by $<a>$ the distance from $a$ to the nearest integer. (For example, $<1,2> = 0.2$, $<\sqrt3> = 2-\sqrt3$) How many solutions does the system of equations have $$\begin{cases} <19x>=y \\ <96y>=x \end{cases} \,\,\, ?$$

We consider the following system with $q=2p$: \[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\] in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties: [b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$ [b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$ [b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$