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.
are positive integers, and smallest side is equal to 
. Determine the area of triangle 
if 
, where 
, 
and 
are lengths of altitudes in triangle from vertices 
, 
and 
, respectively.
be a triangle and let 
be its incircle. Denote by 
and 
the points where is tangent to sides 
and 
, respectively. Denote by 
and 
the points on sides and , respectively, such that 
and 
, and denote by the point of intersection of segments 
and 
. Circle intersects segment at two points, the closer of which to the vertex is denoted by 
. Prove that 
.
, 
be positive integers such that 
is a multiple of 
for all positive integers 
. Prove that 
., point 
lies on side and point 
lies on side . Let and be points on segments 
and 
, respectively, such that 
is parallel to 
. Let 
be a point on line 
, such that lies strictly between and , and 
. Similarly, let 
be the point on line 
, such that lies strictly between and , and 
.
, and are concyclic.
and 
be positive real numbers satisfying 
Prove that
be a random point on the smaller arc of the circumcircle of square 
, and let 
be the intersection point of segments and 
. The feet of the tangents from point 
to the circumcircle of the triangle 
are and , where 
is the center of the square. Prove that points , , and lie on a single circle.
such that:
is strictly increasing,
such that 
for all 
,
, there exists 
such that![\[
f(y) = \frac{f(x) + f(x + 2024)}{2}.
\]](http://latex.artofproblemsolving.com/b/e/2/be26213154bb74bd5a35b8d160011351871bfa9b.png)
Proposed by Pavle Martinović
is written on the board. A move consists of adding either , or 
to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is 
, then in a move a player can't choose the number , but he can choose either or ). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.
there exists an -digit number divisible by 
all of whose digits are odd.
and such that 
, 
, 
, 
are integers such that 
.
and 
of 
such that for each integer 
with 
, we have![\[ n\mid a_i - b_i - c_i
\]](http://latex.artofproblemsolving.com/1/a/e/1ae8d54f068b646ac7a1e43d75e7f26510cfdc95.png)
+1 w
satisfy 
. Determine the minimal and the maximal value of the expression 
be an odd prime congruent to 2 modulo 3. Prove that at most 
members of the set 
are divisible by .
with 
, let its incircle be tangent to sides , 
, and at , 
, and 
, respectively. The internal angle bisector of 
intersects lines 
and 
at 
and 
, respectively. Let 
and 
be distinct points on side such that 
. Finally, let 
be the circumcircle of 
. is tangent to the circumcircle of 
. is tangent to the incircle of .
, then we can use QF to find 
.
is a 36-72-72 triangle. Thus, is the foot of the angle bisector from , and 
, 
. We are done.
and 
.
and passes the point 
,
.