Looks difficult number mock#5
by Physicsknight, Apr 14, 2025, 1:51 PM
Consider 
be distinct prime numbers. Let ![$\alpha_i=\sqrt{a}, \,\mathrm{A}=\mathbb{Q}[\alpha_1,\alpha_2,\hdots,\alpha_b]$](//latex.artofproblemsolving.com/0/e/0/0e06919fe05d9f17760890bb2aebb9cfa85d54ce.png)
Let
be distinct prime numbers. Let ![$\alpha_i=\sqrt{a}, \,\mathrm{A}=\mathbb{Q}[\alpha_1,\alpha_2,\hdots,\alpha_b]$](http://latex.artofproblemsolving.com/0/e/0/0e06919fe05d9f17760890bb2aebb9cfa85d54ce.png)
Let
- Prove that
![$[\mathrm{A}:\mathbb{Q}]=2^b$](//latex.artofproblemsolving.com/b/e/1/be12fe0e1becd8a9dcddd1eece210f2df8ecd026.png)
- Prove that
![$\mathrm{A}=\mathbb{Q}[\gamma];$](//latex.artofproblemsolving.com/f/6/3/f639e791853133a444708bf47517457379d347e8.png)
and deduce that the minimum polynomial 
of 
over 
has degree 
- Prove that factors in
![$\mathbb{Z}_a[X]$](//latex.artofproblemsolving.com/e/9/9/e994e6e9083e52f44aef280b4346f6994a9d9f02.png)
into a product of polynomials of degree 
either of degree 
![$[\mathrm{A}:\mathbb{Q}]=2^b$](http://latex.artofproblemsolving.com/b/e/1/be12fe0e1becd8a9dcddd1eece210f2df8ecd026.png)
![$\mathrm{A}=\mathbb{Q}[\gamma];$](http://latex.artofproblemsolving.com/f/6/3/f639e791853133a444708bf47517457379d347e8.png)
and deduce that the minimum polynomial 
of 
over 
has degree 
![$\mathbb{Z}_a[X]$](http://latex.artofproblemsolving.com/e/9/9/e994e6e9083e52f44aef280b4346f6994a9d9f02.png)
into a product of polynomials of degree 
either of degree 
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
,
such that 
and 
for all reals 
.
.
be a point on side 
,
be a point on the extension of 
such that 
.
be the circumcenter of 
,
be a point on the side extension of 
satisfying 
.
such that the following equation holds for every real ![\[ f(f(x) + y) = \lfloor x + f(f(y)) \rfloor. \]](http://latex.artofproblemsolving.com/3/f/b/3fbbb68e9f9c104fe0f701c8ad07e7fbbf583f1e.png)
Note: 
denotes the greatest integer not greater than 
.
be a positive integer, and let 
denote the integers modulo 
satisfying 
for all 
.
such that
holds for all 
.
.
,
or fewer groups, such that each group has total value at most 
.
be a function from the set of integers to the set of positive integers. Suppose that, for any two integers 
and 
is divisible by 
.
,
is divisible by 
.
that satisfy the following condition: for all 
and 
relatively prime to 
we have ![\[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]](http://latex.artofproblemsolving.com/e/8/5/e85c5b249ac8e75b45a525b8786414798a4921fe.png)
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