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.
is the center of the inscribed circle, point 
lies on the side of 
, with 
. Prove that the distance from point to line 
is equal to the diameter of the circle inscribed in triangle 
from the set of non-negative integers to itself such that 
, whenever 
, are non-negative integers satisfying 
.
be the set of real numbers. Let 
be a function such that ![\[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\]](http://latex.artofproblemsolving.com/5/d/3/5d3476fd9195d085d0030d6036a9fde1cf6756a7.png)
for every 
. Assume that the inequality is strict for some 
.
for every 
or 
for every .
be an integer. Ion draws a regular 
-gon and all its diagonals. On every diagonal and edge, Ion writes a positive integer, such that for any triangle formed with the vertices of the -gon, one of the numbers on its edges is the sum of the two other numbers on its edges. Determine the smallest possible number of distinct values that Ion can write.
be an acute-angled triangle and 
, and 
be the feet of the altitudes from 
, and 
, respectively. On the rays 
, and 
, we have points 
, and 
respectively, lying outside of , such that![\[\frac{A_1A_2}{AA_1} = \frac{B_1B_2}{BB_1} = \frac{C_1C_2}{CC_1}.\]](http://latex.artofproblemsolving.com/d/f/a/dfa94e10638971d53066c20c4589a9c125059911.png)
If the intersections of 
and 
, 
and 
, and 
and 
are 
, and 
respectively, prove that 
, and 
have a common point.
such that
, 
, and 
, let 
denote the greatest common divisor of 
.
is an integer.
satisfying the equation![\[ 4x^2 - 3y^2 = 1 \]](http://latex.artofproblemsolving.com/d/2/8/d284c2c3a7d7fbe12a7988a4534df16e7adc8947.png)
such that 
.
and 
are standing in front of the room in which there is . They know that there is a chessboard in the room and that on every square there is a coin. Every coin is black on one side and white on the other side and is flipped randomly. enters the room and then points at exactly one square on the chessboard. After that, must flip exactly one coin of his choice on the chessboard to the other side and leave. Finally, enters the room ( and haven't met again after entered the room) and he has to guess which square did point at. and have that will make this happen every time?
such that for any real value of 
we have:
be a triangle with incenter 
and circumcircle 
with circumcenter 
. The incircle touches 
at 
respectively. 
is another intersection point of external bisector of 
with , and 
is 
incircle touch point to . Let 
be points lie on . 
intersect 
at 
. Assume that 
. Suppose that 
and 
are concyclic, and 
are concurrent.
are concurrent. 
, tangent line to at and 
are concurrent.
of triangle 
touch the side 
at the point 
. Let 
intersect at 
. Let 
and 
intersect , respectively, at points 
and 
other than 
. Prove that 
and 
intersect at one point.
of triangle touch the side at the point . Let intersect at . Let and intersect , respectively, at points and other than . Prove that and intersect at one point.
such that 
coprime, 
and 
.
