such that 
and exist 
odd numbers 
satisfied all conditions below:![$\sqrt[k]{a_1}+\sqrt[k]{a_2} + ... + \sqrt[k]{a_n} = \sqrt[k]{b_1}+\sqrt[k]{b_2} + ... + \sqrt[k]{b_m} $](http://latex.artofproblemsolving.com/c/9/e/c9e051a562376b2b9e02e82d46a41126874cdf60.png)
for all positive number 
.
(
take on exactly 
distinct values?
be one of the numbers 
.
,
never has remainder 
.
where each 
is chosen from 
.
,
,
are either all the same or all different. How many different lines are there?
,
,
. Compute the second largest integer that appears in a 
-
has side length 
.
,
to be a point on the segment 
uniformly at random, assuming the convention that 
for all integers 
.
?
.
,
,
,
all positive. The plane also intersects all three coordinate axes with intercepts greater than zero (i.e. there exist positive numbers 
,
,
such that 
,
,
all lie on this plane). Find, in terms of 
,
centimeters per second, while the child uniformly stretches the rubber band at a rate of one meter per second. The rubber band is infinitely stretchable and the ant and child are immortal. Compute the time in seconds, if it exists, at which the ant reaches the right end of the rubber band. If the ant never reaches the right end, answer 
.
is the minimum distance between the point and any point on the line segment. Finally, we say that a lattice point 
is nearby a line segment if the distance between 
.
,
in the permutation. What is the probability that a random permutation of the first 
integers is valid?
and 
,
.
,
,
.
,
implies 
?
and 
such that the following limit is finite and nonzero: 
.
.
,
and 
.
-
,
denote the ratio between the area of a sheet with dimension a×b and the area of the horizontal cross-section of the box with dimension 
after the sheet has been folded in halves along each dimension until it occupies the largest possible area that will still fit in the box (because Sean is picky, the sheet must be placed with sides parallel to the box’s sides). Compute the smallest value of b/a that maximizes the expectation 
.
where 
Y by
Let 
be a triangle in which all angles are less than 
and 
. Consider a point 
inside 
such that 
. Let the line 
intersect the side 
in 
, and the line 
intersect the side 
in 
. Prove that the lines 
and 
have a common point 
, and 
.
be a triangle in which all angles are less than 
and 
. Consider a point 
inside 
such that 
. Let the line 
intersect the side 
in 
, and the line 
intersect the side 
in 
. Prove that the lines 
and 
have a common point 
, and 
.
