[/quote]
,
by 
by 
inches 
South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude 
S.In the very close vicinity of this parallel how much are East and West distances exaggerated as compared to North and South distances?
board, there is initially a tree of height 
.
.
such that the gardener can ensure there are eventually 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that the equation
be a polynomial with integer coefficients such that 
.
is divisible by 
.
such that for every positive real 
,
satisfying![\[
\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.
\]](http://latex.artofproblemsolving.com/4/4/4/4443cf7166cc65e67e2e504a826c4ac5f91aab2b.png)
be a positive integer and let 
be a strictly increasing sequence of 
be a subset of 
such that the value of![\[
\left|1-\sum_{i \in X} a_{i}\right|
\]](http://latex.artofproblemsolving.com/2/5/1/2515e074de1782f38453e1fce8ab7c2ec7cfe4c0.png)
is minimised. Prove that there exists a strictly increasing sequence of 
with sum equal to 2 such that![\[
\sum_{i \in X} b_{i}=1.
\]](http://latex.artofproblemsolving.com/3/7/4/37410820fc4c546f11d57212cfce53a7ec4941e3.png)
be a scalene triangle with circumcircle 
,
at 
.
meets 
and 
.
intersects the 
-
,
,
.
passes through either 
be positive real numbers such that : 
. Prove that : 
and 
be relatively prime positive integers, 
,
be chosen so that
is a minimum. Prove that
![$P, Q \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
be relatively prime nonconstant polynomials. Show that there can be at most three real numbers 
such that 
is the square of a polynomial.
be a permutation of the integers 
Call 
a big integer if 
for all 
Find the mean number of big integers over all permutations on the first 
Y by
Q) let 
be one dimensional vector space over field 
then find all linear mapps possible on
? generalize it for n dimensional ?
Q)let
be given by 
for a fixed i. then show that 
is a linear map? generalize it.
Q)let
be a basis of .Define 
by 
if 
then show that is a linear map.
Q)let
be arbitary functions. let 
be defined by 
, when 
is linear?
how to solve help
be one dimensional vector space over field 
then find all linear mapps possible on
? generalize it for n dimensional ?Q)let
be given by 
for a fixed i. then show that 
is a linear map? generalize it.Q)let
be a basis of .Define 
by 
if 
then show that is a linear map.Q)let
be arbitary functions. let 
be defined by 
, when 
is linear?how to solve help
