[/quote]
,
of positive integers such that ![\[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]](http://latex.artofproblemsolving.com/e/c/7/ec778edf97100f3b19350d8a817317ebf1f391bd.png)
be a triangle with circumcenter 
and orthocenter 
.
the altitudes of 
.
lies on line 
such that 
.
lies on the circumcircle of 
are concurrent. 
meets 
at 
.
are concyclic.
,
,
lie on sides 
,
,
,
and 
are cyclic. Line 
meets the circumcircle of 
.
denote the reflection of 
.
,
be the circle with center 
and radius 
.
meets 
.
and 
.
is tangent to the circumcircle of the triangle 
.
and circumcircle 
centred at 
meets 
;
intersect at 
intersect at 
of the triangle 
lies on 
are collinear. Prove that 
.
the feet of the altitudes lying on 
respectively. One of the intersection points of the line 
The lines 
and 
meet at point 
Prove that 
is composite. 
- all divisors of 
are all divisors for some 
(except 
)
be a positive integer, with divisors 
.
is always less than 
,
we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.
at 
is the in-centre of 
.
and 
and 
intersect at 
lie on a circle.
to 
.
to 
are positioned around the sides of 
and 
and 
and 
are the reflections of 
and 
respectively with the circumcircle of 
and 
lies on 
.
,
.
in 
respectively and the line through 
in 
respectively. Prove that point 
lie on a circle.
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, and Anghel David Andrei, Romania
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, and Anghel David Andrei, Romania
This post has been edited 2 times. Last edited by Lukaluce, Apr 29, 2025, 10:37 AM
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)
.
.
.![\[
a_1r_1 + a_2r_2 + \dots + a_ir_i \le N \cdot \left(r_1 + r_2 + r_3 + \dots + \frac{2025 - i}{2025}\right)
\]](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.
