AoPS Academy
and 
Show that
Where 
and 
Show that
Where
+1 w
be positive integers such that 
is not squarefree. Prove that there exist a positive real 
, such that 
and 
are coprime for any positive integer 
.
and 
are in the coordinate plane such that 
. Let 
denote the locus of all points 
in the coordinate plane satisfying 
, and let be the midpoint of 
. Points 
and 
are on such that 
and 
. The value of 
can be expressed in the form 
, where 
and are relatively prime positive integers. Find 
.
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 
.
. The smallest natural numebr such that they leave distinct remainders when divided by is:
Where 
are positive integers and 
is a non-square positive integer. If 
denotes the n-th set of solution to the equation with 
Then, prove that 
, points 
and 
lie on the side 
, with 
. Also, is the midpoint of 
. Show that the centroid of lies on 
.
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 the altitudes of 
. A line passing through 
and parallel to 
intersects the line 
at the point 
. If 
is the orthocenter of , find the angle 
.
\angle ABC= 
\angle CGH=90
Prove that
. Prove that
and 
are cyclic. From 
we have 
=
are concyclic
Now, Since, 
Hence, 
is cyclic 
and 
are cyclic, so the quadrilateral 
is also cyclic. Finally, since 
it will be 
.
finishes our proof, i.e if we show that 
we are done. Firstly we notice that 
lies on 
(
aand we are done 
.
.