[/quote]
,
be a positive integer such that 
,![\[
A = \{a_1, a_2, a_3, \ldots, a_m\}
\]](http://latex.artofproblemsolving.com/1/7/9/179687a134d5495ffff0b44fe763a2bb8b3ddd9c.png)
consist of distinct positive integers not exceeding 2025. Suppose that for every 
,
,
,
as well. Prove that![\[
\frac{a_1 + a_2 + a_3 + \cdots + a_m}{m} \geq 1013.
\]](http://latex.artofproblemsolving.com/c/0/d/c0d190f78d94deac1b947ba57a2de51d0b06f0bd.png)
,
in the array can be changed into either 
or 
so that the row-sums and column-sums remain unchanged. (Note that 
be 
given points in the plane, and let 
be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points 
a non-negative real number 
,
,
.
such that 
and 
are connected by an edge for every 
.
is not fixed and the first selected vertex always has to be 
.![\[
\frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;
\]](http://latex.artofproblemsolving.com/1/7/9/1795db8e4a509dd465f6ff462093ae75b04de2b4.png)
otherwise, Alice wins. Depending on 
,
for which Bobby has a winning strategy.
,
,
,
,
,
be positive integers and let 
.
divides 
and 
.
the point 
is the centre of the excircle opposite the vertex 
This excircle is tangent to the side 
at 
and 
at 
and 
,
and 
meet at 
and 
meet at 
Let 
be the point of intersection of the lines 
and 
be the point of intersection of the lines 
and 
Prove that 
is the circle that is tangent to the line segment 
,
.
be positive reals such that 
.
.
are there such that 
and 
is an integer.
and 
Prove that
Y by Mango247
Let 
be a scalene triangle with incenter 
, and let the midpoint of side 
be 
. Suppose that the feet of the altitudes from to 
and 
are 
and 
respectively, and suppose that there exists a point 
on 
such that 
is a parallelogram. If 
, 
, and the area of can be expressed as 
where 
and 
are positive integers such that is squarefree, find 
.
Aprameya Tripathy
be a scalene triangle with incenter 
, and let the midpoint of side 
be 
. Suppose that the feet of the altitudes from to 
and 
are 
and 
respectively, and suppose that there exists a point 
on 
such that 
is a parallelogram. If 
, 
, and the area of can be expressed as 
where 
and 
are positive integers such that is squarefree, find 
.Aprameya Tripathy
.
are 
.
and 
are perpendicular to 
.
bisects 
in 
.
.
as well. 
(which follows from 
being a parallelogram).
and 
are the feet of the altitudes from 
This follows from the fact that 
and 
separately
This follows from the fact that 
,
Therefore we may extract:
So then:
Except notice that incircle tangencies form equal lengths! In particular, if we denote the third altitude from 
,
and 
,
But we know that 
:
So the area is 
,
and notice that 
So, we randomly do a funny and guess 
which leads to the correct final answer using the extraction :clown:
minutes on this problem and got the same observation but started some strange bash and got a weird radical 
intersect 
So 
lie on 
by Iran. Easy angle chase gives 
and also notice 
so 
Then by some similar triangle ratios
rest just computation. 
.
.
.
However, we also know that 
are colinear so we must have 
By the quadratic formula, 
and WLOG take 
.
.![$$[ABC]=\sqrt{19(19-14)(19-12+\sqrt{11})(19-12-\sqrt{11})}=\sqrt{19(5)(7^2-11)}=19\sqrt{10}.$$](http://latex.artofproblemsolving.com/4/9/4/49432b733f8073bf353c4a09a321d025aa85a01d.png)
.
and 
,
and 
.
.
and 
.
,
,
.
,![$[ABC]=19r=19\sqrt{10}$](http://latex.artofproblemsolving.com/b/a/7/ba72289dc582069422ca3b75c50024c89e1234aa.png)
.
,
and 
is the average of the distances from 
to 
Thus equating the two, 
,
.
,
Thus 
,![$$[ABC] = 133 \cdot \frac{2\sqrt{10}}7 \cdot \frac 12 = 19\sqrt{10}.$$](http://latex.artofproblemsolving.com/b/f/5/bf5ba42c380b0d15c118d8af9021ecf24cee6322.png)
,
.
since 
.
in some order. Plugging into Heron's now gives 
.
again at 
and 
and 
as the projections of 
.
.
.
.
.
where the penultimate equality follows from 
.
to obtain 
.![$[ABC] = r \cdot s = 19 \sqrt{10}$](http://latex.artofproblemsolving.com/8/f/b/8fb9fe061f0ad7e60e3b422d9c0198cf67474dd9.png)
,
.
