AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
matrix whose entries come from the set 
is called a silver matrix if, for each 
, the 
-th row and the -th column together contain all elements of 
. Show that:
;
.
the cells of a 
table can be filled with the numbers 
such that the following conditions are satisfied: appears exactly once.
rectangle, one of the numbers is the arithmetic mean of the other two.
is located in the middle cell of the table.
types of candy, numerated type 1, type 2, ... type n. Initially she takes 
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 
, and in its position in the row she places one candy 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. for which Celeste can leave the table empty for any value of 
and any configuration of candies on the table.
be a positive integer. Given is a subset 
of 
with 
elements. Prove that there exist three elements 
from such that 
.
be a convex hexagon with 
and 
.
which is equidistant from sides 
and 
.
and 
are the centers of mass of 
and 
, show that 
.
be real numbers satisfying
be a positive integer. On the table, we have 
ornaments in different colours, not necessarily of each colour. Prove that we can hang the ornaments on Christmas trees in such a way that there are exactly ornaments on each tree and the ornaments on every tree are of at most 
different colours.
on each of six problems, and their total score is the sum of the six scores (out of 
). Given two students and 
, we write 
if there are at least five problems on which scored strictly higher than .
such that the following statement is true: for every integer 
, given students 
, \dots, 
satisfying 
, the total score of is always at most points more than the total score of .
written on the board and then Beto erases them and writes one of the following numbers:
The game ends when only two numbers are left on the board and Ana cannot play. If the sum of the final numbers is a multiple of 3, Beto wins. Otherwise, Ana wins. ¿Who has a winning strategy?
be a fixed positive integer and consider an 
grid of real numbers. Determine the greatest possible number of cells in the grid such that the entry in is both strictly greater than the average of 's column and strictly less than the average of 's row. be a positive integer . Prove that there exists infinitely many pairs of positive integers 
such that 
and :
and a constant 
, it is given that 
and 
are also friendly polynomials. Prove that 
.
and a constant , it is given that and are also friendly polynomials. Prove that .
and 
are distinct, and without loss of generality, let have degree d higher than . Every root of 
and 
is a root of 
, and and cannot share any roots. If has a root 
with multiplicity , then has multiplicity 
in 
. Therefore, the number of distinct roots of equals 
minus the number of roots of 
where 
. Then the number of distinct roots of equals 
minus the roots of 
with or 
. The value is at least 
, creating a contradiction.
and 
: Prove that :
and 
and![\[P=a_1^2+a_2^2+...+a_{2024}^2-(a_1a_3+a_2a_4+...+a_{2022}a_{2024})\]](http://latex.artofproblemsolving.com/d/b/b/dbb88435dcbba1f5f43f99cdf1ee42aeea3f24ea.png)
Find maximum 
