Found problems: 744
Prove that if the equations
$x^3+mx-n = 0$
$nx^3-2m^2x^2 -5mnx-2m^3-n^2 = 0$
have one root in common ($n \ne 0$), then the first equation has two equal roots, and find the roots of the equations in terms of $n$.
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.
As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$.
[asy]
size(200);
pair A=origin, B=(14,0), C=(9,12), D=foot(A, B,C), E=foot(B, A, C), F=foot(C, A, B), H=orthocenter(A, B, C);
draw(F--C--A--B--C^^A--D^^B--E);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("84", centroid(H, C, E), fontsize(9.5));
label("35", centroid(H, B, D), fontsize(9.5));
label("30", centroid(H, F, B), fontsize(9.5));
label("40", centroid(H, A, F), fontsize(9.5));[/asy]
Find the largest integer $k$ with the following property: Whenever real numbers $x_1,x_2,\dots,x_{2024}$ satisfy
\[x_1^2=(x_1+x_2)^2=\dots=(x_1+x_2+\dots+x_{2024})^2,\]
at least $k$ of them are equal.
Solve the system of equations for real $x$ and $y$: \begin{eqnarray*} 5x \left( 1 + \frac{1}{x^2 + y^2}\right) &=& 12 \\ 5y \left( 1 - \frac{1}{x^2+y^2} \right) &=& 4 . \end{eqnarray*}
If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$?
$ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30} $
Find positive integers $a$, $b$, and $c$ such that
\[\sqrt{a}+\sqrt{b}+\sqrt{c}=\sqrt{219+\sqrt{10080}+\sqrt{12600}+\sqrt{35280}}.\]
Prove that your solution is correct. (Warning: numerical approximations of the values do not constitute a proof.)
Prove that solution of equation $y=x^2+ax+b$ and $x=y^2+cy+d$ it belong a circle.
If the function $f$ is defined by
\[ f(x)=\frac{cx}{2x+3} , ~~~x\neq -\frac 32 , \] satisfies $x=f(f(x))$ for all real numbers $x$ except $-\frac 32$, then $c$ is
$\text{(A)} \ -3 \qquad \text{(B)} \ - \frac{3}{2} \qquad \text{(C)} \ \frac{3}{2} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$
Find all triplets of positive real numbers $(x,y,z)$ that satisfy the following system of equations:
$$ \begin{cases}
x+y^2+z^3=3\\
y+z^2+x^3=3\\
z+x^2+y^3=3.
\end{cases}$$
Consider the system of equations $\begin{cases} x^2 - 3y + p = z, \\
y^2 - 3z + p = x, \\
z^2 - 3x + p = y \end{cases}$ with real parameter $p$.
a) For $p \ge 4$, solve the considered system in the field of real numbers.
b) Prove that for $p \in (1, 4)$ every real solution of the system satisfies $x = y = z$.
(Jaroslav Svrcek)
Let $x, y$ and $z$ be real numbers such that: $x^2 = y + 2$, and $y^2 = z + 2$, and $z^2 = x + 2$.
Prove that $x + y + z$ is an integer.
Find all solutions of the system
$\sqrt[3]{\frac{yz^4}{x^2}}+2wx=0 $
$\sqrt[3]{\frac{xz^4}{y}}+5wy=0 $
$\sqrt[3]{\frac{xy}{x}}+7wz^{-1/3}=0$
$x^{12}+\frac{125}{4}y^5+\frac{343}{2}z^4=16$
where $x, y, z \ge 0$ and $w \in R$
[hide=PS] I attached the system, in case I have any typos[/hide]
Solve following system equations:
\[\left\{ \begin{array}{c}
3x+4y=26\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\sqrt{x^2+y^2-4x+2y+5}+\sqrt{x^2+y^2-20x-10y+125}=10\ \end{array}
\right.\ \ \]
Suppose $x$ and $y$ are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then $x^2+y^2$ can be written in the form $\sqrt[m]{n}$, where $m$ and $n$ are positive integers and $m$ is as small as possible. Find $m+n$.
In $1960$, the oldest of three brothers has an age that is the sum of the of his younger siblings. A few years later, the sum of the ages of two of brothers is double that of the other. A number of years have now passed since $1960$, which is equal to two thirds of the sum of the ages that the three brothers were at that year, and one of them has reached $21$ years. What is the age of each of the others two?
A cart of mass $m$ moving at $12 \text{ m/s}$ to the right collides elastically with a cart of mass $4.0 \text{ kg}$ that is originally at rest. After the collision, the cart of mass $m$ moves to the left with a velocity of $6.0 \text{ m/s}$. Assuming an elastic collision in one dimension only, what is the velocity of the center of mass ($v_{\text{cm}}$) of the two carts before the collision?
$\textbf{(A) } v_{\text{cm}} = 2.0 \text{ m/s}\\
\textbf{(B) } v_{\text{cm}}=3.0 \text{ m/s}\\
\textbf{(C) } v_{\text{cm}}=6.0 \text{ m/s}\\
\textbf{(D) } v_{\text{cm}}=9.0 \text{ m/s}\\
\textbf{(E) } v_{\text{cm}}=18.0 \text{ m/s}$
Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\
c + d = ab \end{cases}$ .
The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
Find all sets $x,y,z$ of real numbers that satisfy
$$\begin{cases} x^3 - y^2 = z^2 - x \\ y^3 -z^2 =x^2 -y \\z^3 -x^2 = y^2 -z \end{cases}$$
The systems of equations
\[\left\{ \begin{array}{l}
2x_1 - x_2 = 1 \\
-x_1 + 2x_2 - x_3 = 1 \\
-x_2 + 2x_3 - x_4 = 1 \\
-x_3 + 3x_4 - x_5 =1 \\
\cdots\cdots\cdots\cdots\\
-x_{n-2} + 2x_{n-1} - x_n = 1 \\
-x_{n-1} + 2x_n = 1 \\
\end{array} \right.
\]
has a solution in positive integers $x_i$. Show that $n$ must be even.
Find all positive integers $a,b,c$ such that $$\begin{cases} a^2 = 4(b+c) \\ a^3 -2b^3 -4c^3 =\frac12 abc \end {cases}$$
Let $S(b)$ be the number of nonuples of positive integers $(a_1, a_2, \ldots , a_9)$ satisfying $3b-1=a_1+a_2+\ldots+a_9$ and $b^2+1=a_1^2+\ldots+a_9^2$. Prove that for all $\epsilon>0$, there exists $C_{\epsilon}>0$ such that $S(b)\leq C_{\epsilon}b^{3+\epsilon}$.
Are there positive real numbers $x$, $y$ and $z$ such that
$ x^4 + y^4 + z^4 = 13\text{,} $
$ x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,} $
$ x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?} $
[i]Proposed by Matko Ljulj.[/i]
Let $x,y,z$ be pairwise distinct real numbers such that $x^2-1/y = y^2 -1/z = z^2 -1/x$.
Given $z^2 -1/x = a$, prove that $(x + y + z)xyz= -a^2$.
(I. Voronovich)