[/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
Given polynomial x^3 − 18x^2 + 95x − 150, what is the harmonic mean of the roots?
A. 30/19
B. 290/95
C. 90/19
D. 133/95
E. NOTA
Am I supposed to used the sum of the roots, the sum of the products of the roots twice at a time, and the product of the roots to form a system of equations to find the roots? I tried that and I'm stuck. Is there a trick to this problem? Or should I just find the roots via rational root theorem? Could someone please help me? Thanks.
A. 30/19
B. 290/95
C. 90/19
D. 133/95
E. NOTA
Am I supposed to used the sum of the roots, the sum of the products of the roots twice at a time, and the product of the roots to form a system of equations to find the roots? I tried that and I'm stuck. Is there a trick to this problem? Or should I just find the roots via rational root theorem? Could someone please help me? Thanks.
are roots of equation 
,
,
,
are the roots, then the harmonic mean is 
.
and 
without finding the individual roots themselves.
