Found problems: 744
Find all triples $(a, b, c)$ of positive integers such that
$a^2 + b^2 = n\cdot lcm(a, b) + n^2$
$b^2 + c^2 = n \cdot lcm(b, c) + n^2$
$c^2 + a^2 = n \cdot lcm(c, a) + n^2$
for some positive integer $n$.
If $x, y, z \in \mathbb{R}$ are solutions to the system of equations
$$\begin{cases}
x - y + z - 1 = 0\\
xy + 2z^2 - 6z + 1 = 0\\
\end{cases}$$
what is the greatest value of $(x - 1)^2 + (y + 1)^2$?
Real numbers $x$ and $y$ satisfy the system of equalities
$$\begin{cases} \sin x + \cos y = 1 \\ \cos x + \sin y = -1 \end{cases}$$
Prove that $\cos 2x = \cos 2y$.
Show that the system of equations
\begin{align*}
\lfloor x\rfloor^2+\lfloor y\rfloor &=0, \\
3x+y &=2,
\end{align*}
has infinitely many solutions and all these solutions satisfy bounds
\begin{align*}
0<\ &x <4, \\
-9\le\ &y\le 1.
\end{align*}
Ten gamblers started playing with the same amount of money. Each turn they cast (threw) five dice. At each stage the gambler who had thrown paid to each of his 9 opponents $\frac{1}{n}$ times the amount which that opponent owned at that moment. They threw and paid one after the other. At the 10th round (i.e. when each gambler has cast the five dice once), the dice showed a total of 12, and after payment it turned out that every player had exactly the same sum as he had at the beginning. Is it possible to determine the total shown by the dice at the nine former rounds ?
Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:
\begin{align*}
abc&=70,\\
cde&=71,\\
efg&=72.
\end{align*}
For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is:
${{ \textbf{(A)}\ -1221 \qquad\textbf{(B)}\ -21.5 \qquad\textbf{(C)}\ -20.5 \qquad\textbf{(D)}\ 3 }\qquad\textbf{(E)}\ 3.5 } $
If $$x+y+z=x^2+y^2+z^2=x^3+y^3+z^3=1 \ \ with \ \ x,y,z\in \mathbb{R},$$ prove that at least one of $x,y,z$ is equal to zero.
Find all real numbers $x,y$ satisfying the following system of equations
\begin{align*}
x^4 +y^4 & =17(x+y)^2 \\
xy & =2(x+y).
\end{align*}
Let $\alpha,\beta,\gamma$ be angles of a triangle. Determine all real triplets $x,y,z$ satisfying the system
\begin{align*}
x\cos\beta+\frac1z\cos\alpha &=1, \\
y\cos\gamma+\frac1x\cos\beta &=1, \\
z\cos\alpha+\frac1y\cos\gamma &=1.
\end{align*}
Find all real solutions of $$ \begin{matrix}
\sqrt{9+x_1}+ \sqrt{9+x_2}+...+ \sqrt{9+x_{100}}=100\sqrt{10}\\
\sqrt{16-x_1}+ \sqrt{16-x_2}+...+ \sqrt{16-x_{100}}=100\sqrt{15}
\end{matrix}$$
Let $f(x)=x^2-6x+5$. On the plane $(x, y)$ draw a set of points $M(x, y)$ whose coordinates satisfy the inequalities $$\begin{cases} f(x)+f(y)\le 0
\\ f(x)-f(y)\ge 0
\end{cases}$$
Let $a$ and $b$ be real numbers such that $$\begin{cases} a^6 - 3a^2b^4 = 3 \\ b^6 - 3a^4b^2 = 3\sqrt2.\end{cases}$$ What is the value of $a^4 + b^4$ ?
Let $n$ be a natural number. Find all solutions $x$ of the system of equations $$\left\{\begin{matrix} sinx+cosx=\frac{\sqrt{n}}{2}\\tg\frac{x}{2}=\frac{\sqrt{n}-2}{3}\end{matrix}\right.$$ On interval $\left[0,\frac{\pi}{4}\right).$
Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations:
\[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.
Find all real solutions:
$$\begin{cases}a^3=2024bc \\ b^3=2024cd \\ c^3=2024da \\ d^3=2024ab \end{cases}$$
There are real numbers $x,y,h$ and $k$ that satisfy the system of equations $$x^2 + y^2 - 6x - 8y = h$$ $$x^2 + y^2 - 10x + 4y = k$$
What is the minimum possible value of $h+k$?
$
\textbf{(A) }-54 \qquad
\textbf{(B) }-46 \qquad
\textbf{(C) }-34 \qquad
\textbf{(D) }-16 \qquad
\textbf{(E) }16 \qquad
$
For which real $a$ are there distinct reals $x$, $y$ such that $$\begin{cases} x = a - y^2 \\ y = a - x^2 \,\,\, ? \end {cases}$$
Two positive integers have the sum $2002$. Can $2002$ divide their product?
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]
Solve the system $\begin{cases}
10x_1 + 3x_2 + 4x_3 + x_4 + x_5 = 0 \\
11x_2 + 2x_3 + 2x_4 + 3x_5 + x_6 = 0 \\
15x_3 + 4x_4 + 5x_5 + 4x_6 + x_7 = 0 \\
2x_1 + x_2 - 3x_3 + 12x_4 - 3x_5 + x_6 + x_7 = 0 \\
6x_1 - 5x_2 + 3x_3 - x_4 + 17x_5 + x_6 = 0 \\
3x_1 + 2x_2 - 3x_3 + 4x_4 + x_5 - 16x_6 + 2x_7 = 0\\
4x_1 - 8x_2 + x_3 + x_4 + 3x_5 + 19x_7 = 0 \end{cases}$
The check for a luncheon of 3 sandwiches, 7 cups of coffee and one piece of pie came to $ \$3.15$. The check for a luncheon consisting of 4 sandwiches, 10 cups of coffee and one piece of pie came to $ \$4.20$ at the same place. The cost of a luncheon consisting of one sandwich, one cup of coffee, and one piece of pie at the same place will come to
$ \textbf{(A)}\ \$1.70 \qquad
\textbf{(B)}\ \$1.65 \qquad
\textbf{(C)}\ \$1.20 \qquad
\textbf{(D)}\ \$1.05 \qquad
\textbf{(E)}\ \$0.95$
For some positive numbers $A, B, C$ and $D$, the system of equations
$$\begin{cases} x^2 + y^2 = A \\ |x| + |y| = B \end{cases}$$
has $m$ solutions, while the system of equations
$$\begin{cases} x^2 + y^2 +z^2= X\\ |x| + |y| +|z| = D \end{cases}$$
has $n$ solutions. If $m > n > 1$, find $m$ and $n$.
( G Galperin)
(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers,
does $a = b = c$ necessarily hold?
(b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$ positive real numbers,
does $a = b = c$ necessarily hold?
Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$. For a given positive integer $n$, $n\geq 2$ consider the system of equations: \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivalent if there exists a real number $\lambda \neq 0$, $x_1=\lambda y_1$, $\ldots$, $x_n= \lambda y_n$. How many nonequivalent solutions does the system have?
[i]Mircea Becheanu[/i]