and 
be two different primes. Prove that![\[\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\left\lfloor\frac{3p}{q}\right\rfloor+\ldots +\left\lfloor\frac{(q-1)p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1) \]](http://latex.artofproblemsolving.com/9/0/3/903486996aa105ac1a28795e55596680d0884e3c.png)
make the equation ![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
be inscribed in the circumcircle 
and circumscribed about the incircle 
,
.
,
,
at 
,
,
,
,
,
,
,
,
(distinct from 
)
intersects 
(distinct from 
be the midpoint of the arc 
of 
and 
at 
and 
,
at 
.
.
and 
Prove that
and 
.
Equality holds when 
or 
or 
with 
.
is equilateral. Given that 
and the midpoint of 
are on the same line, compute the length of 
![\[\arctan\frac {1}{3} + \arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.\]](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 
and 
respectively. Find the area triangle 
and 
Let 
and 
Let 
in terms of 
and 
Y by Adventure10, Mango247, and 1 other user
In each cell of a matrix 
a number from a set 
is written --- in the first row numbers 
, in the second 
and so on. Exactly 
of them have been chosen, no two from the same row or the same column. Let us denote by 
a number chosen from row number 
. Show that:
![\[ \frac{1^2}{a_1}+\frac{2^2}{a_2}+\ldots +\frac{n^2}{a_n}\geq \frac{n+2}{2}-\frac{1}{n^2+1}\]](//latex.artofproblemsolving.com/a/d/5/ad5ae6861da79a08d32093f63827016868856a3a.png)
a number from a set 
is written --- in the first row numbers 
, in the second 
and so on. Exactly 
of them have been chosen, no two from the same row or the same column. Let us denote by 
a number chosen from row number 
. Show that:![\[ \frac{1^2}{a_1}+\frac{2^2}{a_2}+\ldots +\frac{n^2}{a_n}\geq \frac{n+2}{2}-\frac{1}{n^2+1}\]](http://latex.artofproblemsolving.com/a/d/5/ad5ae6861da79a08d32093f63827016868856a3a.png)
where 
is a permutation of the set 
,
is constant, and its 
.![\[ LHS \geq \frac {(1 + 2 + ... + n)^2}{a_1 + a_2 + ... + a_n} = \frac {n(n + 1)^2}{2(n^2 + 1)} = \frac {n + 2}2 - \frac 1{n^2 + 1}\]](http://latex.artofproblemsolving.com/1/2/9/12942f8d5061ac25ab557b09a5440da839416757.png)
.
more? I don't understand permutations much.
,
