[/quote]
,
contestant of EGMO are named 
.
with 
.
has at least 
and 
. Prove that
Let 
. Prove that
numbers 
and 
numbers 
are written in some order.
.
and 
. Prove that
,
is a prime number. Show that if ![\[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]](http://latex.artofproblemsolving.com/5/d/f/5df80f885e63852e4f53a6145dafd18d63dc3526.png)
is divisible by 
.
reals, and 
.
?
be a triangle with 
.
be the orthocenter of 
is the midpoint of 
and point 
lies on the side 
such that 
.
.
be a regular polygon, and let 
be its set of vertices. Each point in 
.
,
.
-
such that
;
and 
,
and 
denote the product and sum of the digits of 
be a point inside a triangle 
meet 
,
meet 
at 
,
meet 
at 
.
be the point such that 
,
be the point such that 
,
be the point such that 
.
cannot all lie strictly inside the circumcircle of triangle 
.
is such that 
is a parallelogram. Prove that 
.
be an acute-angled triangle with 
.
be the midpoint of the side 
and the points 
is perpendicular to the common chord of the circumscribed circles of triangle 
.
boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?
Y by Adventure10, Mango247
Let 
be real numbers. Four solutions on the complex plane of the quartic equation in 
form the four vertices of a quadrilateral. Find the condition such that the area of the quadrilateral is less than 1, and also sketch the domain of the set of the points 
satisfying the condition.
be real numbers. Four solutions on the complex plane of the quartic equation in 
form the four vertices of a quadrilateral. Find the condition such that the area of the quadrilateral is less than 1, and also sketch the domain of the set of the points 
satisfying the condition.
we obtain:
(i)
so that four distinct complex values of 
(ii).
)![$A = 2Re\{z^{+}\} \cdot 2Im\{z^{+}\} = 4\sqrt{b}\cos\left[\frac{1}{2}\arctan\left(\frac{\sqrt{4b-a^2}}{a}\right)\right]\sin\left[\frac{1}{2}\arctan\left(\frac{\sqrt{4b-a^2}}{a}\right)\right]$](http://latex.artofproblemsolving.com/f/b/5/fb52da2df5d20cff04549ee1d97097caeb2b2195.png)
;![$A = 2\sqrt{b}\sin\left[\arctan\left(\frac{\sqrt{4b-a^2}}{a}\right)\right] = 2\sqrt{b}\sin\left[\arcsin\left(\frac{\sqrt{4b-a^2}}{\sqrt{a^2+4b-a^2}}\right)\right] =\sqrt{4b-a^2}$](http://latex.artofproblemsolving.com/6/5/8/658038a5ba0205e3e40f368e95bbcc31c2d722a7.png)
(iv).
,
are plotted as:
