AoPS Academy
such that 
.
.
are of the form 
, where 
, 2, 3, . For each 
, let 
be the greatest common divisor of 
and 
. Find the maximum value of as ranges through the positive integers.
,![\[
1^2 + 2^2 + 3^2 + \ldots + k^2 = \frac{k(k + 1)(2k + 1)}{6}
\]](http://latex.artofproblemsolving.com/b/5/d/b5d49e25bd8d8a211b72372ff967558e1a8365ce.png)
Find the smallest positive integer such that 
is divisible by 200.
, 
, and 
. Show that![\[
a^2 + b^2 + c^2 + 4abc < \frac{1}{2}.
\]](http://latex.artofproblemsolving.com/8/c/5/8c5321801770649d5a6c6133b9cb162b8182882c.png)
are positive real number such that 
. Prove that
, where 
always occurs at 
be a positive integer. Find the smallest value of such that:
is divisible by 
.
?
? is about 
the intersection of 
's angle bisector and 
. We'll prove 
.
, we know that 
must be an isosceles trapezoid (the two opposite angle bisectors are symmetric) where 
and 
is cyclic, then 
, so 
.
is the perpendicular bisector of 
, so 
must be isosceles. By 
, 
.
, so 
is the inner angle bisector of 
. We conclude that 
.
.
and 
Prove that
Let 
Prove that
and 
Prove that
numebr on his list? Write its last two digits as your answer.
grid of square tiles. Robert chooses four of these squares uniformly at random. The probability that the centers of these four squares form a square themselves is 
where 
are coprime naturals. Write the largest 
digit odd factor of 
.
.
which touches one face of the cube and passes through all 
vertices of the face opposite to it. Find the integer closest to 
are distinct numbers in AP such that 
are also in AP. Find the smallest possible value of 
(in degrees).
that satisfy 
is isosceles, so 
is a trapezoid.