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?
be the number of positive integers 
such that divides 
and 
divides
where 
denotes the number of positive integers that divide , including and . Find the remainder when is divided by 
.
let 
denote the intersection of diagonals 
and 
, and let 
denote the intersection of diagonals 
and 
. Prove that if 
and 
, then 
bisects angle 
.x table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a 
x square which contains an odd number of unit squares white colored.
be a scalene triangle with circumcenter 
, incenter 
, and incircle 
. Let touch the sides 
, 
, and 
at points 
, 
, and 
respectively. Let 
be the projection of to 
. The line 
intersects the circumcircle of again at point 
. The circumcircles of 
and 
intersect again at points 
and 
respectively. Prove that the lines 
, 
, and 
are concurrent.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.
is called 
if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo for every 
.
th power.
are and , respectively. The excenter and exradius relative to vertex 
is 
and 
, respectively. Let 
denote the circumradius. Prove that if 
, then 
.
-gon, and at each vertex there is a black or a white cap. (Caps of the same colour are indistinguishable from each other.) Under one of the caps dollars are hidden, and there is nothing under the other caps. The host rotates the table, and then Ádám chooses a cap, and take what is underneath. Ádám's accomplice, Béla is working at the company behind this game. Béla is responsible for the placement of the dollars under the caps, however, the colors of the caps are chosen by a different collegaue. After placing the money under a cap, Béla
is divisible by , then at least one of them must be a multiple of . Then because 
is a multiple of (because it's a multiple of 
), then we must have that both are divisible by .
. Prove that a is divisible by c and b is divisible by c
and 
is a prime such that: 
and 
and 
isn't divisible by c, then we have: 
and 
which are wrong.
and 
,
for some 
and 
where 
and 
,
and 
for some 
for some 
makes both conditions filled.
and 
