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 
, and 
. What is the sum 
?
through 
, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?
such that 
is also prime.
vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?
with orthocenter 
.Let 
and 
be the altitudes of triangle.Let 
and 
be reflections of points 
across the line 
, respectively.Let 
and 
be the midpoints of 
and 
, respectively.Let 
and 
.Prove that 
and 
are collinear.
such that 
.
let be the midpoint of 
. Let 
be a circle inside of and is tangent to 
at , respectively. The tangents from to meet at 
such that 
and 
lie on the same side of 
. Let 
and 
. If 
prove that 
is tangent to . and 
be the incircle and circumcircle of the acute triangle , respectively. Draw a square 
so that all of its sides are tangent to , and , are both on . Extend 
and 
, intersecting at and 
, respectively. Prove that 
and 
intersects on .
in which 
and 
. Let 
be a point on the side and 
a point on the line such that 
. Prove that 
.
is a positive integer such that the product of all its positive divisors is 
. Find all such less than 
.
be a cyclic quadrilateral inscribed in circle whose diagonals meet at . Lines 
and 
meet at . Segment 
intersects at . Lines 
and meet at , and lines 
and meet at . Prove that 
and concur with the tangent to at . be an integer greater than . For a positive integer 
, let 
. Suppose that there exists a 
-element set 
such that is an -element subset of 
; shares at most one common element; is contained in exactly two elements of . in terms of .
, 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.
, the perpendicular from 
onto meets at 
, the perpendicular from 
onto meets at . The length of 
is , the length of 
is 
, the length of 
is 
and the length of is 
. Find .
