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.![$P \in \mathbb Q[x,y]$](http://latex.artofproblemsolving.com/a/3/8/a384cbf38fd3a7f812b22baaeb0860e7f6eb26c0.png)
with 
and ![$\forall (a,b) \in \mathbb Z^2 \cap [0,d]^2, P(a,b) \in \mathbb Z$](http://latex.artofproblemsolving.com/a/6/5/a65ecb4bb2d9b5e6096f322c2de8bd11b945ed6a.png)
.
?
have,where 
and 
are natural numbers.
inscribed in 
with 
being the midpoint of 
. The tangents at 
of intersect at 
. Let 
be the projection of 
onto 
. On the perpendicular bisector of , take a point 
that is not on and different from M. Circle 
intersects 
at 
. Lines 
and 
intersect at 
. Prove that 
is an isosceles triangle.
be a positive integer greater or equal to 
and let 
, 
, ..., 
be a sequence of non-negative real numbers. Find the maximum value of 
in terms of .
. For each positive integer 
, we define its twist 
as follows: write as 
, where 
are non-negative integers and 
, then 
. For the positive integer , consider the infinite sequence 
where 
and 
is the twist of 
for each positive integer 
.
if and only if the remainder when is divided by 
is either or 
.
every next number (from the fifth number) is the unit number of the sum of the four numbers preceding it. Is there any cases that we get the numbers 
and 
in this series?
, the diagonals 
and 
meet at the point 
. We know that 
and 
. If we have 
prove 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?
such that 
.
is a positive integer such that the product of all its positive divisors is 
. Find all such less than 
.
be a square and the point of the segment ![$[BD]$](http://latex.artofproblemsolving.com/3/2/6/3261e689901bce018ecdef47b9bc60a78ead3746.png)
such that 
. We define the point as the point of intersection of lines 
and 
. Find the value of the angle 
.
lies only on the one diagonal , it says 
and not 
as the intersection works, since 
for sq side length .
of the arc AC of the circle (B, BA) has this property. Best regards, sunken rock
,
?
,
?
degrees?
,
as well, and since 
,
and 
are congruent through 
congruency. From there, you have that 
;
degrees.