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.
of injective functions from the set of real numbers to itself satisfying
and 
. (We say a function 
is injective if 
for any distinct real numbers 
and 
.)
, find the smallest integer 
such that any graph with 
vertices and at least edges must contain at least two disjoint cycles (cycles with no common vertices).
{
} consisting 
distinct positive integers is 
if for every 
the number 
is divisible by the product of all numbers in except 
. Find all integers such that exists a set consisting of 
positive integers.
with 
and 
.
?
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
, prove that is a square.
be the Fermat point of a 
. Prove that the Euler line of the triangles 
, 
, 
are concurrent and the point of concurrence is 
, the centroid of .
, such that for 
? be a cyclic quadrilateral for which 
. Construct two squares 
and 
externally on the sides 
and 
respectively.
meets 
at 
, 
meets 
at 
and 
denotes circumcenter of . Prove that 
and 
meets again on 
.
be a circle inscribed inside a rhombus . Let and 
be variable points on 
and 
respectively, such as 
is always the tangent line to . and the area of triangle 
is the same.
are constructed on diameter 
of a semicircle of radius 
. The centers of the small semicircles divide 
into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?
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?
. Each coefficient 
, and
is equal to either or 
. If 
, what is the value of 
?
.
.
After testing some we see that 
Thus our polynomial is
Plugging in 
gives
.
is equal to 
.
,
,
,
,
,
.
.
.