AoPS Academy
be 
unit vectors on a plane, where is an odd number. Prove that there exist 
for 
such that 
st for every two subsets 
of 
, there exists a polynomial 
with integer coefficients st either 
or 
where the equations are considered mod n.
is the set of reminders of 
where 
mod n.
,
,
, 
, 
is not greater than 
.
and 
be points on a plane such that 
, where is a positive integer. Let 
be the set of all points 
such that 
, where 
is a real number. The path that traces is continuous, and the value of is minimized. Prove that is rational for all positive integers 
to the cross product 
reveals that a negative determinant [of 
] corresponds to a matrix which reverses the direction of the cross product of two vectors.[/quote]
and 
, but am confused on how to prove this statement for general vectors 
and 
although its supposed to be a quick comparison.
be a function defined in any real numbers 
with 
Prove that on the 
plane, the line 
passes through the fixed point which isn't on the 
axis in regardless of the value of if only if is a linear function in .
be a positive integer not divisible by 3. Prove that there are infinitely many positive integers such that 
and 
are divisible by , where 
is the sum of digits of 
.
. Prove that
.
.
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
with 
where 
is any integer with 
and 