[/quote]
,
such that there exist infinitely many rational numbers 
satisfying![\[
\left| \frac{q}{p} - \frac{\sqrt{5} - 1}{2} \right| < \frac{\alpha}{p^2}.
\]](http://latex.artofproblemsolving.com/1/1/6/1162c6a7215bff2f9fdaa042a6952f86524e9e92.png)
such that
are all powers of 
.
be a convex quadrilateral. Let 
be the midpoints of 
respectively. If 
concur at a point 
prove that 
,
children standing in a circle. For each girl, it turns out that among the five people following her clockwise, there are more boys than girls. Find the greatest number of girls that can stand in a circle.
so that there exists a unique 
satisfying 
and![\[f(x+g(y))+f(-x-g(-y))=g(x+f(y))+g(-x-f(-y))\]](http://latex.artofproblemsolving.com/d/e/0/de0ea33c3a679ff02762be39e9013b72974e4287.png)
for all 
.
be a triangle with circumcircle 
and incenter 
and let 
be the midpoint of 
.
,
,
are selected on sides 
,
such that 
,
,
.
intersects 
other than 
.
and 
meet on 
interesting if 
is composite and for every divisor 
of 
and 
is also a divisor of 
satisfying the equation:![\[
(x^2y - y^2z)(1 - zx) = (xy^2 - zx^2)(1 - yz).
\]](http://latex.artofproblemsolving.com/d/1/c/d1c11cb351b59ae5e79edc1e1997834b5402638a.png)
![\[
x + y + z = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}.
\]](http://latex.artofproblemsolving.com/c/1/2/c122da8411c836a38fb2ea639338c3fd9749a619.png)
satisfy: 
What is the value of 
Y by CXNJ.AoPS, tofubear, Yrock, Sedro
As far as I know, it seems that no one on HSM has organized an equation marathon before. So why not give it a try? Let's start one!
Some basic rules need to be clarified:
If a problem has not been solved within 
days, then others are eligible to post a new probkem.
Not only simple one-variable equations, but also systems of equations are allowed.
The difficulty of these equations should be no less than that of typical quadratic one-variable equations. If the problem involves higher degrees or more variables, please ensure that the problem is solvable (i.e., has a definite solution, rather than an approximate one).
Please indicate the domain of the solution to the equation (e.g., solve in 
, solve in 
).
Here's an simple yet fun problem, hope you enjoy it
:
Some basic rules need to be clarified:
If a problem has not been solved within 
days, then others are eligible to post a new probkem. Not only simple one-variable equations, but also systems of equations are allowed. The difficulty of these equations should be no less than that of typical quadratic one-variable equations. If the problem involves higher degrees or more variables, please ensure that the problem is solvable (i.e., has a definite solution, rather than an approximate one). Please indicate the domain of the solution to the equation (e.g., solve in 
, solve in 
).Here's an simple yet fun problem, hope you enjoy it
:This post has been edited 2 times. Last edited by steven_zhang123, Jan 21, 2025, 12:37 AM
.
,
.
,![$$\sqrt[3]{15-x^3+3x^2-3x}=2\sqrt{x^2-4x+2}+3-x$$](http://latex.artofproblemsolving.com/e/f/4/ef457a31ebab89cbe07898a85d61030748a74f75.png)
![$\sqrt[3]{a}-\sqrt[3]{b}=\frac{a-b}{\sqrt[3]{a^2}+\sqrt[3]{ab}+\sqrt[3]{b^2}}$](http://latex.artofproblemsolving.com/c/2/e/c2eb20e1113244696a46430f9199291b17151324.png)
and 
of real numbers that satisfy the system of equations![\[\begin{cases}x^3 = 3x - 12y + 50, \\ y^3 = 12y + 3z - 2, \\ z^3 = 27z + 27x. \end{cases}\]](http://latex.artofproblemsolving.com/0/7/9/079062f832872bdf8b57377d8388254ffa379522.png)
;
is a set of solutions to the original equation.
,
contradiction 
.
,
,
gives:
![$x\in [0,\pi]$](http://latex.artofproblemsolving.com/3/e/c/3ecbeef3f695ea04d27c5cafc03c70ec34b52c0f.png)
:
,
in ![$[0,\pi]$](http://latex.artofproblemsolving.com/e/c/e/ece27ba559bd93cdf2416aa0f88fc8d7d85e5791.png)
.
thus 
and using the identity 
we have the solutions 
.
:
for positive values.
or 
.
,
,
is nonnegative. This obviously has no solution since on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
and 
.
,
.
.
.![$x=\sqrt[3]{a}+\sqrt[3]{a^2}$](http://latex.artofproblemsolving.com/5/5/c/55c5e858342befc8d9deee8c809d5923857452c6.png)
when 
and 
,
,
is the set of natural numbers, 
is the positive reals
is obviously too large, 
is obviously too small, so we guess 
and are correct.
,
.
satisfying the equations 
and 
.
,
and 
are monotonically increasing,
,
.
,
!=(x+2)(x+3)(x+4), when x=7, LHS>RHS, when x≥7, (x+1)!/x!=x>6>(x+5)/(x+2), so LHS>RHS,
the system 
.
