Found problems: 744
Real numbers $(x,y)$ satisfy the following equations:
$$(x + 3)(y + 1) + y^2 = 3y$$
$$-x + x(y + x) = - 2x - 3.$$
Find the sum of all possible values of $x$.
[i]Team #1[/i]
Determine all sets of real numbers $x,y,z$ which satisfy the system of equations
$$\begin{cases} xy = z \\ xz =y \\ yz =x \end{cases}$$
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.
The system $\begin{cases} x^2 - y^2 = 0 \\
(x - a)^2 + y^2 = 1 \end{cases}$ generally has four solutions. For which $a$ the number of solutions of the system is equal to three or two?
Suppose that $4$ real numbers $a, b,c,d$ satisfy the conditions $\begin{cases} a^2 + b^2 = 4\\
c^2 + d^2 = 4 \\
ac + bd = 2 \end{cases}$
Find the set of all possible values the number $M = ab + cd$ can take.
Find all $n$-tuples of real numbers $(x_1, x_2, \dots, x_n)$ such that, for every index $k$ with $1\leq k\leq n$, the following holds:
\[ x_k^2=\sum\limits_{\substack{i < j \\ i, j\neq k}} x_ix_j \]
[i]Proposed by Oriol Solé[/i]
Find all positive real solutions $(a, b, c)$ to the following system:
$$
\begin{aligned}
a^2 + \frac{b}{a} &= 8, \\
ab + c^2 &= 18, \\
3a + b + c &= 9\sqrt{3}.
\end{aligned}
$$
Find all triples $(a, b, c)$ of real numbers such that
$$ a^2 + ab + c = 0, $$
$$b^2 + bc + a = 0, $$
$$c^2 + ca + b = 0.$$
In what case does the system of equations
$\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$
have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that
\[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.
Determine all polynomials $a(x)$, $b(x)$, $c(x)$, $d(x)$ with real coefficients satisfying the simultaneous equations
\begin{align*}
b(x) c(x) + a(x) d(x) & = 0 \\
a(x) c(x) + (1 - x^2) b(x) d(x) & = x + 1.
\end{align*}
Determine all triples of real numbers that satisfy the following system of equations:
\[x+y-z=-1\\ x^2-y^2+z^2=1\\ -x^3+y^3+z^3=-1\]
Find all $n$ for which there are real numbers $0 < a_1 \le a_2 \le ... \le a_n$ with
$$\begin{cases} \sum_{k=1}^{n}a_k = 96 \\ \\ \sum_{k=1}^{n}a_k^2 = 144 \\ \\ \sum_{k=1}^{n}a_k^3 = 216 \end{cases}$$
Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.
Find all real solutions of the system :
(a) $$\begin{cases}1-x_1^2=x_2 \\ 1-x_2^2=x_3\\ ...\\ 1-x_{98}^2=x_{99}\\ 1-x_{99}^2=x_1\end{cases}$$
(b)* $$\begin{cases} 1-x_1^2=x_2\\ 1-x_2^2=x_3\\ ...\\1-x_{98}^2=x_{n}\\ 1-x_{n}^2=x_1\end{cases}$$
Let $n$ be a positive integer. Show that there are integers $x_1, x_2, \ldots , x_n$, [i]not all equal[/i], satisfying $$\begin{cases} x_1^2+x_2+x_3+\ldots+x_n=0 \\ x_1+x_2^2+x_3+\ldots+x_n=0 \\ x_1+x_2+x_3^2+\ldots+x_n=0 \\ \vdots \\ x_1+x_2+x_3+\ldots+x_n^2=0 \end{cases}$$ if, and only if, $2n-1$ is not prime.
The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. What is the sum of all possible values of $n$?
$ \textbf{(A)}\ 255
\qquad\textbf{(B)}\ 256
\qquad\textbf{(C)}\ 257
\qquad\textbf{(D)}\ 258
\qquad\textbf{(E)}\ 259
$
Solve the system of equations $\begin{cases} x^3+\frac13 y=x^2+x -\frac43 \\
y^3+\frac14 z=y^2+y -\frac54 \\
z^3+\frac15 x=z^2+z -\frac65 \end{cases}$
Determine $x^2+y^2$ and $x^4+y^4$, when $x^3+y^3=2$ and $x+y=1$
Find all triples $(x,y,z)$ of positive real numbers such that
$$
\begin{cases}
2x^2+y^3=3 \\
3y^2+z^3=4 \\
4z^2+x^3=5 \\
\end{cases}
$$
[i]M. Zorka[/i]
Solve the system of equations $$\begin{cases} \dfrac{1}{x+y}+ x = 3 \\ \\ \dfrac{x}{x+y}=2 \end{cases}$$
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$
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 solutions of the system of equations
$\begin{cases} s(x) + s(y) = x \\ x + y + s(z) = z \\ s(x) + s(y) + s(z) = y - 4 \end{cases}$
where $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digits in the decimal representations of $x, y$, and $z$, respectively.
Determine and justify all solutions $(x,y, z)$ of the system of equations:
$x^2 = y + z$
$y^2 = x + z$
$z^2 = x + y$