AoPS Academy
be positive reals such that 
. Prove that ![\[(a-b)^2 + a + b \geq 2\]](http://latex.artofproblemsolving.com/c/f/a/cfa01a5b54a72afe6746f547390e65dbbb82ea64.png)
and 
we define 
as the set of all products of an element of and an element of . For example, if 
and 
then 
We call a set 
of positive integers good if there do not exist sets 
of positive integers, each with at least two elements and such that the sets 
and are the same. Prove that the set of perfect powers greater than or equal to 
is good.
, 
, no two elements are equal, but any two or three of these sets may have common elements. A perfect power is an integer of the form 
, where 
and 
are integers.)
baskets, each containing at least one candy. The total number of candies is 
. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad can take all the candies from 
consecutive non-empty baskets, while Sevinch can take all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least 
candies, regardless of the initial distribution of candies and Sevinch's actions.
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
, parallel to 
, intersects the external angle bisector of 
at point 
. Prove that the angles 
, 
, 
, 
, 
can be divided into two groups, so that the angles in each group have a sum of 
.
and 
.
with multiplication of matrixes operation is making an isomorphic-group structure with 
.
.
for which there exists positive integers 
, 
, and 
satisfying![\[
2a^n+3b^n=4c^n.
\]](http://latex.artofproblemsolving.com/0/2/9/0295405b01494edc05ce9867b2ec48a4e069bf26.png)
be a real function such that![\[ \lim_{x \rightarrow +\infty} \frac{f(x)}{e^x}=1\]](http://latex.artofproblemsolving.com/7/f/e/7fe200614381665ef11aed463fbbee51f8461bb3.png)
for all sufficiently large 
. Prove that ![\[ \lim_{x \rightarrow +\infty} \frac{f'(x)}{e^x}=1.\]](http://latex.artofproblemsolving.com/0/9/6/096b5b71111efdd7565d60424240e20ee3251382.png)
and 
with complex coefficients have the same set of numbers for their zeros but possibly different multiplicities. The same is true for the polynomials
Prove that 
be a linear transformation on linear space 
satisfying:
but 
, and 
is the eigensubspace of eigenvalue 
. Prove that can be decomposed to 
-cyclic subspace's direct sum. be a square matrix of odd order (at least 
) whose entries are odd integers. Prove that if is invertible, then it is not possible for all the minors of the entries of a row to have equal absolute values.
be then -th fibbiance number. As gets bigger and bigger, we have,
my question is dose,
My reservations about this is that 
and 
so 
. So, if the limit holds, does that mean that if is a set and 
is a set, for each 
that 
we can have, for 
we can have,
for some ?
such that 
for all 
order, and if the constituents of its first, second, ..., rows are the first figurate numbers of the first, second, ..., orders respectively, show that it's value is 
satisfying the conditions:
for all real numbers 
with ![$ t\in [0,1] . $](http://latex.artofproblemsolving.com/1/2/a/12ac756f5dc034618c1566c89e80dcf52761f105.png)
for any real numbers 
such that 
and 
and any 
real numbers 
the following inequality holds.
satisfying the conditions:for all real numbers with for any real numbers such that and and any real numbers the following inequality holds.
