[/quote]
,
and 
intersect at two points 
and 
.
be the line tangent to these circles at 
and 
,
be the line parallel to 
on 
on 
and 
meet at 
;
and 
;
and 
.
.
be a positive integer that is not divisible by 
.
be a positive integer. Prove that none of the numbers of the form 
is a square number.
different bus lines in a city, each stopping at exactly 
stations and running in both directions. Nevertheless, for every two different stations there is always a bus line connecting these two stations. Determine the maximum number of stations in this city.
be an isosceles triangle with 
and circumcircle 
.
is denoted by 
.
be any point on 
with center 
intersects 
and the circumcircle 
.
and 
be a positive integer. Furthermore, let ![$x_1, x_2,\ldots, x_n \in [0, 2]$](http://latex.artofproblemsolving.com/b/2/a/b2a6731961601353e26224c285a32923fbb54eb3.png)
be real numbers subject to 
.
When does equality hold?
such that 
is a perfect square.
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
as the set of all numbers of the form 
for some nonnegative 
and 
.
such that 
and 
.
,
. Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other.
- Greatest common divisor of numbers 
and 
board, there is initially a tree of height 
.
.
such that the gardener can ensure there are eventually 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that the equation
be a polynomial with integer coefficients such that 
.
is divisible by 
.
such that for every positive real 
,
satisfying![\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]](http://latex.artofproblemsolving.com/4/4/4/4443cf7166cc65e67e2e504a826c4ac5f91aab2b.png)
Y by
Let 
and 
be points on a circle with center 
.
Let
be a point on segment 
.
Let
be one of the intersections of the circle with center and passing through and the circle with diameter 
.
Prove that
.
and 
be points on a circle with center 
.Let
be a point on segment 
.Let
be one of the intersections of the circle with center and passing through and the circle with diameter 
.Prove that
.This post has been edited 1 time. Last edited by xeroxia, Sep 11, 2024, 4:21 AM
intersects 
at another point 
,
intersects 
,
(Because 
,
we have 
.
we have 
.
.
