[/quote]
,
be a positive integer and let 
be an irrational number. Show that for every integer 
,
such that 
.
be positive integers with 
,
of the set of 
”
”
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
of positive integers that satisfy
be a monic polynomial with complex coefficients such that there exist a polynomial 
with complex coefficients for which ![\[P(x^2-2)=P(x)Q(x).\]](http://latex.artofproblemsolving.com/a/6/2/a62b06890b6bbcc5c9634f6de8c7ae5fe9f37828.png)
Determine all complex numbers that could be the root of 
,
such that, if we let 
,
we have 
;
is divisible by 
each have a prime factor not dividing 
is irreducible for every natural number 
.
be positive real numbers satisfying![\[
a^2 + c^2 = b(a + c).
\]](http://latex.artofproblemsolving.com/9/c/0/9c0730d15b1177ee23d3f65bc2fe8b2cc85d0c62.png)
Let![\[
m = \min \left( \frac{a^2 + ab + b^2}{ab + bc + ca} \right).
\]](http://latex.artofproblemsolving.com/e/b/6/eb6f340defa4b38e35292062b202d138b9de843b.png)
Find the value of 
.
be a function such that
for all 
.
.
be real numbers such that 
Prove that 
Where 
.
such that for all 
Y by
The shapes in the image consist of six unit cubes. Which of the following 3D objects can be filled up with the aforementioned shapes:
a) a cube with side length
, from which one edge has been removed (i.e. three layers of the shape 
)?
b) a rectangular prism of size
, from which two edges of length have been removed from one of the 
sides (i.e. three layers of the shape 
)?
We can use each of shapes at most once, no two shapes can overlap, nor protrude from the 3D object and every unit cube of the 3D object must be covered by a unit cube of one of the constituent shapes.
Proposed by Ilija Jovčeski
a) a cube with side length
, from which one edge has been removed (i.e. three layers of the shape )?b) a rectangular prism of size
, from which two edges of length have been removed from one of the 
sides (i.e. three layers of the shape )?We can use each of shapes at most once, no two shapes can overlap, nor protrude from the 3D object and every unit cube of the 3D object must be covered by a unit cube of one of the constituent shapes.
Proposed by Ilija Jovčeski
This post has been edited 1 time. Last edited by Lukaluce, Apr 14, 2025, 1:02 PM
