[/quote]
,
be a positive integer. We start with 
and 
if 
then find the coefficient of 
in 
in terms of 
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
and 
Prove that
be an acute triangle and let 
be the midpoint of 
.
passing through 
and 
and 
at points 
and 
respectively. Let 
be the point such that 
is a parallelogram. Suppose that 
.
such that![\[f(x+y)+f(y+z)+f(z+x)\ge 3f(x+2y+3z)\]](http://latex.artofproblemsolving.com/9/8/a/98ab64d17bc7d6bf7533b0874177fc963405c1d8.png)
.
satisfying the following property: if 
,
and 
are integers such that 
,
such that 
.
be functions 
such that for all reals 
,
Prove that either 
is the identity function or 
is periodic.
and ![\[ f(mn) = f(m) + f(n) + f(m)f(n) \]](http://latex.artofproblemsolving.com/e/f/f/eff10a32cffaec742403ebfc46ff09bc8ef7b7c5.png)
for all nonnegative integers 
and 
be the set of real numbers. Determine all functions ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](http://latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers 
and 
.
![\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]](http://latex.artofproblemsolving.com/e/b/4/eb4e0d6580d9d3d6a627e39837edb3d1943d4596.png)
such that for all 
the following equality holds ![\[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \]](http://latex.artofproblemsolving.com/f/3/a/f3ac860e90d671f9fb592a6bd55c102a3000daec.png)
where 
is greatest integer not greater than 
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 number of circles in column 
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 
,
occurences of 
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 
.
.
columns that have 
numbers in those columns and the sum in each row is 
.
because they are contained in 
.
'
.
,
.
which will get us to 
which yields the needed answer.
