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.
have real roots and real coefficients with 
. What is the maximum value of 
?
distinct circles in a plane, with radius 
.
circles, which form a set 
, which satisfy the following., they intersect with each other or, they don't intersect with each other.
satisfy:
.
with incentre 
, let 
and 
by the midpoints of 
and 
respectively, and 
and 
be the feet of the altitudes from 
and to the respective sides. Denote by 
the line being tangent tot he circumcircle of triangle and passing through 
, and denote by 
the reflection of in 
. Let 
by the intersection of 
and , and let 
be the intersection of 
and . Defined 
analogously. If 
is the intersection of 
and 
, prove that 
.
with two altitudes 
and 
.Let 
be the point on the line such that 
.The lines 
and 
intersect at point 
, and 
is the point on segment 
such that 
.Suppose that bisects .Prove that triangle 
is isosceles.
s such that:
such that 
. Prove that
of positive integers with 
satisfying 
is
be a positive integer and let 
and be sets of integers satisfying the following conditions:
, 
and is a subset of 
, 
iff 
.
such that,
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?
, 
, and 
be complex numbers such that they satisfy these equations:
If the maximum of 
can be expressed as 
for positive integers , 
, and , find the minimum possible value of 
.
, , and be complex numbers such that they satisfy these equations:
If the maximum of can be expressed as for positive integers , , and , find the minimum possible value of .
and 
and 
.
; multiplying: 
.
.
,
.
have roots 
.
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
such that:
From here we must have,
Observe that thus far the RHS of the divisibility statement has been strictly positive, following from the fact that 
for 
.
,
.
.
substitution as @above, we get 
.
and let 
.
,
.
giving the answer of 
.