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.
be a convex quadrilateral. If 
are the midpoints of the segments 
, and 
, respectively, and there is also a point 
inside the quadrilateral such that, 
and 
. Show that 
.
that satisfy the following condition: for all 
and 
relatively prime to 
we have ![\[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]](http://latex.artofproblemsolving.com/e/8/5/e85c5b249ac8e75b45a525b8786414798a4921fe.png)
by the expression
and 1.
Find the range of 
such that 
such that 
is an integer.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
sports, each awarding one gold medal to a winner. athletes participate, each in all sports. There are also experts, each of whom must predict the number of gold medals each athlete will win. In each prediction, the medal counts must be non-negative integers summing to . An expert is called competent if they correctly guess the number of gold medals for at least one athlete. What is the maximum number 
such that the experts can make their predictions so that at least of them are guaranteed to be competent regardless of the outcome?
has exactly two real roots, one of which is 
where 
and 
are integers, 
and are relatively prime, and 
Find 
to the system of equations![\[\begin{aligned}
\begin{cases}
x^2+y^2-xy-3x+3=0,\\
x^2+y^2+z^2-xy-yz-2zx-3x+3=0.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/4/a/b/4ab8e94e2e39be219c80264aa5ea23456b74b3e8.png)
such that 
and 
, evaluate 
.
by 
and, for 
,![\[a_n = a_{\lfloor n/2 \rfloor} + a_{\lfloor n/3 \rfloor} + \ldots + a_{\lfloor n/n \rfloor} + 1.\]](http://latex.artofproblemsolving.com/a/5/8/a58bbcbb429bcada079bb810b286a7c243f25c41.png)
such that 
.
be a triangle with orthic triangle 
and orthocenter 
and midpoint of 
as 
. If 
then prove that 
passes through pole of wrt 
.
Prove that
with incenter 
,
is a point on 
,the bisector of 
meet 
at 
.Prove that 
for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is .
and 
Prove that
