and 
Prove that
and 
.
Equality holds when 
or 
or 
with 
.
lies inside the rectangle such that 
is equilateral. Given that 
and the midpoint of 
are on the same line, compute the length of 
.
such that![\[\arctan\frac {1}{3} + \arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.\]](http://latex.artofproblemsolving.com/4/7/6/476f930a87b85c9e1e2e2b56aa2160dcc516e3a0.png)
chessboard made up of 
squares. All of the squares are 
The probability, that when a fair sided coin is tossed, it lands heads up and it lies completely within a square, is 
Part 3: where 
can be expressed as 
Find, with proof, the value of 
Find 
Find 
in terms of 
and 
Let 
and the midpoints of 
are 
respectively. 
intersects the circumcircle of 
at 
and 
respectively. Find the area triangle 
and 
Let 
and 
Let 
in terms of 
and 
,
,
is the Euler's totient theorem, which calculates the number of integers less than 
.
for 
.
be the remainder 
leaves when you divide it by 
.![$\lfloor \frac{x + \sqrt[r]{r^2} + r \ln(r)}{r} \rfloor$](http://latex.artofproblemsolving.com/c/d/3/cd3f7d917b0ec944ec3017e1f737c538e1edd10e.png)
.
unless of your age can be written as 
.
Y by checkmatetang, dhusb45, Chess88, lion11202, spartan168, MathArt4, El_Ectric, MSTang, chessdude2015, Altamira140, ythomashu, Sullymath, Ancy, librian2000, whatshisbucket, kbird, math101010, janabel, thkim1011, bluecarneal, eed7573, 2_confirm_, Not_a_Username, inavda, alchemis80, nihao4112, MathSlayer4444, blue8931, love_emojis, mathlogin, chgon, Benq, Aldwyn22, W.Sun, ethanproz, theorist2, iNomOnCountdown, hzbest, mathonator, rlybd5, phi_ftw1618, pi37, dantx5, CaptainFlint, ak1234, Kwow, thegabster37, wtasfias, ShineBunny, xwang1, abk2015, Geek21, EpicSkills32, MAAIsCorrupt, thatindiankid55, MATH1945, dank1728, rkm0959, mcmcphie, navi_09220114, blizzard10, sas4, speulers_theorem, Pi-rate, Adventure10
Edited to reflect schedule changes -- see second post below for more details.
The 2016 AMC 10B/12B is on Wednesday, February 17.
In order to protect the integrity of the contests, we have locked the Contests forum for the duration of the contests. You should not discuss ANY aspect of the contests until this forum is reopened (we expect this to be on Sunday morning). This means: don't discuss the contests anywhere on the site, and don't discuss any aspect of the contests, including how well you did, whether you thought it was easy or hard, or anything about the content.
Please join us for the 2016 AMC 10B/12B Math Jam on Monday, February 22 at 7 PM Eastern (4 PM Pacific), where will we discuss problems and solutions from the contests.
The 2016 AMC 10B/12B is on Wednesday, February 17.
In order to protect the integrity of the contests, we have locked the Contests forum for the duration of the contests. You should not discuss ANY aspect of the contests until this forum is reopened (we expect this to be on Sunday morning). This means: don't discuss the contests anywhere on the site, and don't discuss any aspect of the contests, including how well you did, whether you thought it was easy or hard, or anything about the content.
Please join us for the 2016 AMC 10B/12B Math Jam on Monday, February 22 at 7 PM Eastern (4 PM Pacific), where will we discuss problems and solutions from the contests.
This post has been edited 1 time. Last edited by DPatrick, Feb 16, 2016, 11:10 PM
Reason: AMC schedule change
Reason: AMC schedule change
