[/quote]
,
has property 
if :![\[
\begin{cases}
\text{the set } A \text{ has at least } 4 \text{ elements} \\
\text{the element } 1+1 \text{ is invertible}\\
x+x^4=x^2+x^3 \text{ holds for all } x \in A
\end{cases}
\]](http://latex.artofproblemsolving.com/b/6/9/b6993ee76eb3792e08d441f5d5c463247c392c4a.png)
a) Prove that if a ring 
are distinct elements, such that 
and 
are units, then 
is also a unit, but 
is not a unit.
be a positive integer, and 
be two complex numbers such that 
and 
,
.
satisfy the relation 
.
and 
are invertible.
by 
grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place 
counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.
be a positive integer . Prove that there exists infinitely many pairs of positive integers 
such that 
and :
matrix whose entries come from the set 
is called a silver matrix if, for each 
,
-
.
;
.
the cells of a 
table can be filled with the numbers 
such that the following conditions are satisfied:
rectangle, one of the numbers is the arithmetic mean of the other two.
is located in the middle cell of the table.
candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:
She eats a candy of type 
,
followed by one candy type 
(we consider type 
to be type 1, and type 0 to be type 
She chooses two consecutive candies which are the same type, and eats them.
with 
elements. Prove that there exist three elements 
from 
.
be a convex hexagon with 
and 
.
which is equidistant from sides 
and 
.
and 
are the centers of mass of 
and 
,
.
is solvable over 
for every 
.
,
.
be real numbers satisfying
and 
: Prove that :
be collinear points (in order) and 
a point in plane. Consider the disc 
of center 
,
.![$\mathcal{D}\cap [AC]$](http://latex.artofproblemsolving.com/0/5/6/0569d4130943000380a87b72be211965b5ff02ad.png)
is either the empty set or a segment of length at most 
.![$P(X)\in\mathbb{C}[X]$](http://latex.artofproblemsolving.com/9/9/e/99e8cde28b0b9bc9414425ebc30fa2dc0386fb37.png)
be a polynomial of degree ![\[\sup_{x\in[0,1]}|P(x)|\le(2n+1)^{n+1}\int\limits_{0}^{1}|P(x)|\mathrm{d}x.\]](http://latex.artofproblemsolving.com/9/1/6/91647cd75888c5e9d291280c8ae3f199cc6d1bfd.png)
Y by
Give a function
Suppose that 
is a differentiable map from 
into 
with 
and 
. Let 
, prove that 
is differentiable at every 
. If 
, prove that 
.
(Note:
isn't differentiable at (0,0).$
Suppose that 
is a differentiable map from 
into 
with 
and 
. Let 
, prove that 
is differentiable at every 
. If 
, prove that 
.(Note:
isn't differentiable at (0,0).$This post has been edited 2 times. Last edited by Sifan.C.Maths, Mar 31, 2025, 8:28 AM
Reason: .
Reason: .
