AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
denote 
. For any real numbers 
prove that![\[ \prod_{i=1}^k F_i(x_{k-i+1},x_{k-i+2},\ldots,x_k) \geq \prod_{i=1}^k F_i(x_i,x_i,\ldots,x_i)\]](http://latex.artofproblemsolving.com/a/4/d/a4d515f1102b31cff69d6823df7a58482aec1171.png)
. Yael and Ziad play a game as follows, involving a monic polynomial of degree 
. With Yael going first, they take turns to choose a strictly positive real number as the value of one of the coecients of the polynomial. Once a coefficient is assigned a value, it cannot be chosen again later in the game. So the game rounds, until Ziad assigns the final coefficient. Yael wins if 
for some real
. Otherwise, Ziad wins. Decide who has the winning strategy.
and 
intersect at two points 
and 
. Let 
be the line tangent to these circles at 
and 
, respectively, so that lies closer to than . Let 
be the line parallel to and passing through the point , with 
on and 
on . Lines 
and 
meet at 
; lines 
and meet at 
; lines 
and meet at 
. Show that 
.
be a positive integer that is not divisible by 
. Furthermore, let 
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 
. The line through 
perpendicular to 
is denoted by 
. Furthermore, let 
be any point on . The circle 
with center and radius 
intersects 
once more at point 
and the circumcircle once more at point 
. Prove that the points 
and 
lie on a straight line.
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 
. Prove the inequality
When does equality hold? such that 
is a perfect square.
as the set of all numbers of the form 
for some nonnegative 
and 
. Find (with proof) all pairs 
such that 
and 
.
, Elbek writes 
. 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 
. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:
. Determine the largest 
such that the gardener can ensure there are eventually majestic trees on the board, no matter how the lumberjack plays.
such that the equation
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
be a rectangle with angle between two diagonal 
.
we have rectangle 
.![$[H\cap H_{x}]$](http://latex.artofproblemsolving.com/6/b/3/6b394579e1d80251a265f0701cc86646fdbb0643.png)
minimum, where ![$[S]$](http://latex.artofproblemsolving.com/6/0/0/600580b5de8084177c6fd276063e6f870b313abe.png)
is area of 
and 
(like in the diagram). Then we are looking for the minimum of ![$ [WXYZ]$](http://latex.artofproblemsolving.com/b/1/3/b132ea1d52066bd54b8dd717c14619f4e134cd40.png)
.![\begin{align*}[WXYZ]&=AB\cdot WX\\
&\ge AB\cdot A'B'\\
&= AB^2\end{align*}](http://latex.artofproblemsolving.com/9/a/5/9a577e9d4a83056c56056130c85f0fdcdc6ffc10.png)
is the perpendicular distance between the parallels 
and 
,
;
is a rotation of 
,
.
,
.
is between 
condition.