AoPS Academy
, 
and 
be points on circle 
divided into three equal parts. Construct three equal circles 
, 
, and 
tangent to internally at points , , and respectively. Let 
be any point on arc 
, and draw tangents 
, 
, and 
to circles , , and respectively. Prove that 
.
, 
, 
, 
be points on circle divided into 
equal parts. Construct equal circles , , , 
tangent to internally at , , , . Let be any point on circle , and draw tangents 
, 
, , 
to circles , , , . If the sum of 
of , , , equals the sum of the remaining 
(where 
), find all such .
denote the set of positive integers
such that for all 
it holds that 
Note: consider 
and 
, ∀x∈[0,1].
and 
and there exists a positive integer 
such that 
. Let 
be minimal polynomial 
over 
. Prove that 
be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let 
denote the event that the number 
has been selected and that it is in the same place in both lines. Prove that the events 
are mutually independent, and 
.
. Calculate the first and second derivative of the function at the point 
.
kg sky diver that jumps out of a plane with no initial velocity and an air resistance of 
. For this example assume that the positive direction is downward.
satisfying ![\[|z -a | + |z + a| = 2|c|\]](http://latex.artofproblemsolving.com/4/b/d/4bdafe46f8d3ed1c8945a1553fc3bc0d23fe11e2.png)
if and only if 
If this condition is fulfilled, what are the smallest and largest values of 
with k and u continuous in the unit square and unit interval can have only the trivial solution. Prove this in detail. Here k(x,y)=sin(xy)
are natural numbers satisfy that: 
and 
are prime numbers. Prove that 
is a perfect squared number
are natural numbers satisfy that: and are prime numbers. Prove that is a perfect squared number
is prime 
is prime
, so we must have 
,
is a prime 
is prime 
, thus 
is a perfect square
be a field. Find the Galois groups
