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be positive integers satisfying ![\[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\]](http://latex.artofproblemsolving.com/b/8/c/b8c3fe14640a2dcb30b755b2347721da5d080a78.png)
Determine all possible values of 
.
+1 w
over 
is linear
is called 
if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
for every 
.
cut out 
congruent regular 
gons from a sheet of paper and placed these gons on the table such that some parts of each of these gons may be covered by others. We say that a vertex of one of the afore-mentioned gons is 
if it is not in the interior of another gon that is placed on top of it. For any 
determine the minimum possible number of visible vertices.
cannot completely cover a square of side length 
, if the three smaller squares do not overlap in their interiors (i.e., they may touch at edges or corners, but no part of one lies over another).
denote the set of all positive integers that can be expressed as a sum of three or fewer powers of 
. Let 
be the smallest positive integer that cannot be expressed in the form 
, where 
. Find the remainder when is divided by 
.
have angles 
and 
such that 
and 
. Moreover, suppose that the product of the side lengths of the triangle is equal to its area. Let 
denote the circumcircle of 
. Let 
intersect 
at 
, 
intersect 
at 
, and 
intersect 
at 
. If the area of 
can be written as 
for relatively prime integers 
and 
and squarefree 
, find the sum of all prime factors of , counting multiplicities (so the sum of prime factors of 
is 
), given that has 
divisors.
be real numbers satisfying the following two conditions:
Find the possible maximum and minimum values of
fair coins are flipped. If of them land heads, find the expected value of 
.
and 
are two parallel lines a distance 
apart. Given and 
, construct a triangle , with 
on , and 
and 
on , such that the inradius is , and angle 
. Calculate angles and in terms of , and . If the incircle touches the side 
at , find a relation between 
and 
and are two parallel lines a distance apart. Given and , construct a triangle , with on , and and on , such that the inradius is , and angle . Calculate angles and in terms of , and . If the incircle touches the side at , find a relation between and 
is given, A is one of the 2 intersections of the line L and with circle 
Tangents to the incircle from A meet the line M at B, C. The circumcircle of the 
passes through the A-excenter 
of the 
and also through its reflection 
in the perpendicular bisector of the side BC. Power of D to this circumcircle is 
The incircle (I, r) and A-excircxle 
are centrally similar with center A and coefficient 
where d is the A-altitude of the 
Hence 
But
with roots
.
Let 
.
Let 
.
Let 
.
.
.
.
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
