[/quote]
,
and 
.
![$$a+b+\frac{3}{2}c\geq \frac{9}{8}\sqrt [3]{\frac{3}{2}}-\frac{3}{16}$$](http://latex.artofproblemsolving.com/2/2/e/22e6ec2f98777db80e39333542664e903663bf81.png)
![$$a+b+\frac{8}{5}c\geq \frac{9\sqrt [3]{25}-4}{20}$$](http://latex.artofproblemsolving.com/f/c/a/fca343ee701eeff4c309b31beb3426ec96b49b55.png)
Let 
.
![$$a+b+\frac{27}{25}c\geq\frac{35\sqrt [3]5-9}{50}$$](http://latex.artofproblemsolving.com/6/9/5/695cc35986a391fdc15e4863d361ce373fbbf64c.png)
is a triangle with circumcenter 
and 
is a circle tangent to 
and internally to 
The tangent
to 
If 
are the contact points of 
respectively, prove that 
,
and 
,
an obtuse triangle can be formed?
.
is an integer, and let 
be 
integers such that 
for any two integers 
and 
satisfying 
.
.
of the form 
with 
,
Determine all ordered pairs of polynomials 
with integer coefficients such that 
.
grid which has rows and columns numbered from 
to 
;
.
,
satisfying 
or 
are not in 
be real numbers such that 
and 
Find the value of 
and 
Find the value of 
for which there are infinitely many positive integers 
is not divisible by 
.
Equality holds when 
Equality holds when 
,
is a regular polygon with 
diagonals. Randomly select a triangle whose three vertices are the three vertices of polygon 
be a sequence of positive integers with 
,
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
.
for some positive integer 
such that 
.
problems in a meeting. For each 
,
so that the chair can ensure that the longest idle time for any leader does not exceed 
Y by Adventure10
Two players, 
(first player) and 
, take alternate turns in playing a game using 2016 chips as follows: the player whose turn it is, must remove 
chips from the remaining pile of chips, where 
. No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?
(first player) and 
, take alternate turns in playing a game using 2016 chips as follows: the player whose turn it is, must remove 
chips from the remaining pile of chips, where 
. No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?
are losing positions" for all 
,
has winning strategy.
be a function that sends a nonnegative integer 
if the second player wins starting from 
if and only if 
using induction.
,
since A can remove 
chips, which is losing for B.
since A can remove 
since A can remove 
chips to leave B as the first player starting from 
since A can remove 
since A can remove 
chips to leave B as the first player starting from 
where 
.
.
,
and at most 
chips, all of which are winning for B.
since A can remove 
chips, which is losing for B.
since A can remove 
chip, which is losing for B.
since A can remove 
since A can remove 
since A can remove 
,
,
