[/quote]
,
cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number 
.
,
is a perfect square.
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
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 
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than 
is not a perfect square, prove that there exist a student that has attended in at least 
classes.
,
,
,
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
and 
.
and 
Prove that
where 
be a function defined on 
taking values in 
such that
for all 
for all 
;
identical-looking coins in a circle. It is known that exactly two of them are fake. Real coins weigh the same, fake ones too, but they are lighter than the real ones. How can you determine in three weighings on a cup scale without weights whether there are fake coins lying nearby or not??
points of general position marked on the coordinate plane (i.e., points among which there are no three lying on the same straight line). Is there a polynomial of two variables 
a) of degree 
;
be triangle, inscribed in parabola. Tangents in points 
forms triangle 
.
.
)
such that there exist integers 
with ![\[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]](http://latex.artofproblemsolving.com/e/f/a/efaa98e77ac0273284ee3446e4335d8cc4f681cb.png)
Y by RainbowJessa, radian_51, farhad.fritl, dangerousliri
In each cell of a 
board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to 
, and the sum of the numbers in each column is equal to . Define 
to be the largest value in row 
, and let 
. Similarly, define 
to be the largest value in column , and let 
.
What is the largest possible value of
?
Proposed by Paulius Aleknavičius, Lithuania
board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to 
, and the sum of the numbers in each column is equal to . Define 
to be the largest value in row 
, and let 
. Similarly, define 
to be the largest value in column , and let 
.What is the largest possible value of
?Proposed by Paulius Aleknavičius, Lithuania
This post has been edited 1 time. Last edited by Lukaluce, Apr 14, 2025, 12:16 PM
is 
.
columns into 
squares, and put 
in each of the main diagonals of the squares and 
s
.
and 
,
.
.
be the sum of the circles in column 
.
,
as 
is equivalent to 
since each circled number must be at least 
.
rectangle with 
rectangle. Fill the 
,
with ![\[
\frac{R}{C} = \frac{2025 \cdot \frac{1}{45}}{45 \cdot \frac{1}{45} + \frac{2025 - 45}{45}} = \frac{2025}{89}
\]](http://latex.artofproblemsolving.com/c/b/3/cb30fdd6325affe7ccf8b8b9fe3364965db08e44.png)
remains the same and 
remains the same or decreases. 
.
.
.![\[
a_1r_1 + a_2r_2 + \dots + a_ir_i \le N \cdot \left(r_1 + r_2 + r_3 + \dots + \frac{2025 - i}{2025}\right)
\]](http://latex.artofproblemsolving.com/1/d/5/1d586dac5b2dd959c42ceecd6f13cccf31a0d9d2.png)
This is tightest when the 
have 
and the 
have 
.![\[
j + \frac{t}{2025} \le N \cdot \left(\frac{1}{a_1} + \dots + \frac{1}{a_j} + \frac{i-j}{2025} + \frac{2025 - i}{2025}\right)
\]](http://latex.artofproblemsolving.com/1/3/4/134b1b2312664e85d716e7095387c4c850265866.png)
where 
.
,![\[
\frac{j}{N} \le \frac{1}{a_1} + \dots + \frac{1}{a_j} + \frac{2025 - i}{2025}
\]](http://latex.artofproblemsolving.com/c/c/b/ccb98feb86ff336cda077524517d61d7338a5960.png)
which with AM-HM and applying 
becomes![\[
\frac{j}{N} \le \frac{j^2}{a+j} + \frac{a}{2025}
\]](http://latex.artofproblemsolving.com/e/a/1/ea1ea3b356d56851573e1e7569ee2debe1e97438.png)
with 
.
as the only root. If 
,![\[
\frac{j}{N} \le \frac{j^2}{45j} + \frac{44j}{2025} \iff
\frac{89j}{2025} \le \frac{45j}{2025j} + \frac{44j}{2025}
\]](http://latex.artofproblemsolving.com/9/e/b/9eb5abf9121eeae979999aa3df959f986e75b35d.png)
which holds. If 
,![\[
\frac{j}{N} \le \frac{j^2}{2025} + \frac{2025-j}{2025} \iff
89j \le j^2 + 2025 - j \iff (j-45)^2 \ge 0
\]](http://latex.artofproblemsolving.com/7/0/9/709b85997c99d6f948062143604dd238d5b5bc8c.png)
which holds.
,
(everything should be divided by 
)
columns that contains scorers. Note that the non-scorers in a column with more than 
scorers should be distributed to the scorers to maximize the ratio, the converse is true (distribute among non-scorers if there aren't enough scorers).![\[\frac{n^2a+n^2-s}{n^2-a+\frac{n^2a^2}{s}}\le\frac{n^2}{2n-1}.\]](http://latex.artofproblemsolving.com/c/6/e/c6ecd4d4270253776f6af21ed254ba252c09f0d1.png)
After more bashing, we realize that this is optimized when 
,
the number of yellow cells it has.
and all we want is for this to be in terms of 
is trivially true by double counting so now we will split sum of 
and therefore 
And now here we need 
and thus 
but also trivially 
which should lead to picking 
for some 
,
and therefore 
so our pick here will be 
to get rid of each 
in which case we get that 
this so that we don't have the need to summon each 
so tracing back the equality this gives that in each of the columns where 
we have that the only nonzero cells are the ones in yellow, also from the AM-GM we have that 
if 
'
.
.
,
in each such column, we guarantee that each heavy column carries the minimal possible maximum to serve 45 rows. (A short optimization shows that if one designates 
columns to serve as the targets for high entries and distributes the mass evenly among them so that every heavy row gets exactly 
in the column it uses, then one may obtain a ratio of![\[ \frac{n}{\,k+r-1}\,, \]](http://latex.artofproblemsolving.com/b/5/c/b5c9448f65ff9408fbdbb06068b0f8da03f56b39.png)
with the constraint 
.
;
,![\[ \frac{2025}{45+45-1}=\frac{2025}{89}\,. \]](http://latex.artofproblemsolving.com/d/d/8/dd8e174440dcffbb58f3bc3039935de45724c965.png)
)
block with the “flat” matrix on the remaining cells.)
rows into 
