AoPS Academy
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 .
foot radius. Bruno completes 
laps every hour, while Brutus completes 
laps every hour. If they start at the same point but run in opposite directions, how far along the track’s circumference (in feet) from the starting point are they when they meet for the sixth time? Note: Do not count the moment they start running as a meeting point.
satisfies that the triangles 
, 
, 
, 
, 
and 
are congruent. Show that 
.
are obtuse, then in 
, there must be 
. Consider the other pairs of congruent triangles similarly, we can get that all inner angles are equal. Moreover, they all equal to 
.
or , we have 
. Consider the other pairs of congruent triangles similarly, we can get 
.
are equilateral, and some of them have the same side length as labeled. From
. Similarly, 
. Now 
So 
.
, then triangles 
leads to 
, false.
and 
Prove that
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
.
.
and 
.
