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 an acute-angled triangle with 
, let 
be its circumcentre, and let 
be a point on the segment 
. The line through perpendicular to intersects the lines 
and 
at 
and 
respectively. The circumcircles of triangles 
and intersect again at 
.
and 
then 
is tangent to the circle 
, 
, 
, 
be distinct positive integers, 
. Prove that there exist distinct indices 
and 
such that 
does not divide any of the numbers 
, 
, , 
.
written on the first page and 
empty pages. Suppose in each of the next seconds, Alice can flip to the next page, and write down the sum or product of two numbers (possibly the same) which are already written in her grimoire.
be the largest possible number such that for any 
, Alice can write down the number 
on the last page of her grimoire. Prove that there exists a positive integer 
such that for all 
, we have that ![\[n^{0.99n}\leqslant F(n)\leqslant n^{1.01n}.\]](http://latex.artofproblemsolving.com/5/7/e/57e2b7e3d97564665d8de28a3afd74bdcf74ba5b.png)
, let 
and 
be a pair of isogonal conjugate points. The line 
intersects 
at 
, and the line 
intersects at 
. Let the circumcircle of 
and the circumcircle of 
intersect again at 
(other than ). Prove that the line 
bisects 
.
is right angled (for clarity, 
). Define 
as a point on line 
such that 
is parallel to . Additionally, since 
, 
meaning 
, 
, and 
. Also, since 
, by SSS, we have 
meaning 
. Since 
, we have 
or 
and we are done.
from the reals to the reals such that![\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]](http://latex.artofproblemsolving.com/e/b/4/eb4e0d6580d9d3d6a627e39837edb3d1943d4596.png)
.
is called 
if
is a perfect square, and
, 
is the smallest positive integer such that 
is a perfect square., there exists a positive integer such that 
for all integers 
. be a scalene triangle with incircle 
. Denote by the midpoint of arc in the circumcircle of , and by the point where the 
-excircle touches . Suppose the circumcircle of 
meets again at 
and intersects at two points 
, 
.
or 
is tangent to .
, with 
, for which the edges 
are
with 
, for which the edges 
are all coloured blue. be a positive integer. Alice and Bob play the following game. Alice considers a permutation 
of the set ![$[n]=\{1,2, \dots, n\}$](http://latex.artofproblemsolving.com/8/f/e/8fea25e7cf72aabc60f48c3b30f4d4818a084ae3.png)
and keeps it hidden from Bob. In a move, Bob tells Alice a permutation 
of ![$[n]$](http://latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
, and Alice tells Bob whether there exists an ![$i \in [n]$](http://latex.artofproblemsolving.com/e/6/7/e67206e47d5d5875105b339f177a3ab8f5151a5e.png)
such that 
. Bob wins if he ever tells Alice the permutation . Prove that Bob can win the game in at most 
moves.
denote the set of all real numbers. Find all functions 
such that
for all real numbers 
, 
.
for which there exist infinitely many positive integers 
such that ![\[ \frac{a^m+a-1}{a^n+a^2-1} \]](http://latex.artofproblemsolving.com/2/8/a/28a68e409451bbdcaee7c1286341f6aeea9ffdad.png)
is itself an integer.
be a cyclic quadrilateral who interior angle at 
is 
degrees. Show that if 
, then 
. Does the converse hold?
,
then prove that 
