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 triangle with incenter 
, and suppose the incircle is tangent to 
and 
at 
and 
. Denote by 
and 
the reflections of and over . Let 
be the intersection of 
with 
, and let 
be the midpoint of . Prove that 
and 
are perpendicular.
, 
and 
. Point 
lies on 
such that 
, and point is the intersection of the angle bisector of 
with side 
. What is the angle 
in degrees?
and 
by 
and 
with the recurrence relations 
and 
for all nonnegative integers 
Let 
and 
be the remainders when 
and 
are divided by 
respectively. Compute 
and 
Prove that
Let and 
Prove that
Let and 
Prove that
be positive integers such that 
and 
. Let 
be the set of values attained by 
as 
runs through the positive integers. Show that is the set of all positive divisors of some positive integer.
board where we identify the squares with pairs 
where 
and 
denote the row and column number of that square, respectively.
and places a pawn at the bottom left corner (i.e. on 
) and makes the following moves. In his 
-th move, he moves the pawn from to either 
or 
if is odd and to either 
and 
if is even. Here all the numbers are taken modulo 
. Find the number of pairs 
that Calvin could have picked such that he can make moves so that the pawn covers all the squares on the board without being on any square twice.
. The candidate's score is calculated as follows:
after a correct answer, the score is increased by 
; after a wrong answer, the score is divided by 
. questions and answers all of them, how many different scores are possible?
and 
with the same length, where 
denotes the correct answer and 
denotes the wrong answer, both result in the same score of 1.
that satisfies the following condition: For every finite set of points on the plane, if for any points from this set there exist two lines containing all the points, then there exist two lines containing all points from the set.
be a circle, and 
are two fixed points on . Given a third point 
on , let 
and 
denote the feet of the altitudes from 
and 
, respectively, in the triangle . Prove that there exists a fixed circle 
such that 
is tangent to regardless of the choice of the point .
Choose a natural number 
such that 
. If both numbers and 
are on the board, erase both.
Choose a natural number such that 
. If both numbers and 
are on the board, erase both.
be real numbers satisfying 
Prove that 
and determine all cases of equality.
, and the width of each step increases from one to 
. Is it true that the segment from the lower left point of the stairs to the upper right point of the stairs doesn’t cross the stairs?
, and the width of each step increases from one to . Is it true that the segment from the lower left point of the stairs to the upper right point of the stairs doesn’t cross the stairs?
,
Since this function's concavity is strictly negative, a segment connecting 2 points on the curve does not intersect the curve again because of Jensen's.