[/quote]
,
greater than 
,
,
such that 
is a perfect square but none of the integers 
are perfect squares.
satisfying 
and their sum is an even number, consider the quadratic polynomial:![\[
P(x) = x^2 - (m^2 - m + 1)x + (m^2 - n^2 - m)(n^2 + 1).
\]](http://latex.artofproblemsolving.com/d/5/1/d51aaddfddfe06cc91abfdd3476a155770a8a5fd.png)
are positive integers but are not perfect squares.
and 
Prove that
be a set of integers with the following properties:
.
and 
,
.
,
is composite, then all positive divisors of 
and 
are externally tangent at a point 
.
be a line tangent to 
and 
.
and 
be diameters in 
and 
lies on 
and 
are tangent to 
and 
and 
are tangent to 
.
,
times the radius of 
and 
Prove that
be a triangle and let 
and 
denote its incentre and circumcentre respectively. Let 
be the circle through 
and 
which is tangent to the incircle of the triangle 
and 
are defined similarly. The circles 
distinct from 
and 
are defined similarly. Prove that the lines 
and 
are concurrent at a point on the line 
.
.
at 
and 
intersects at a point on line 
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Sharon High School is proud to announce the 11th annual Sharon Math and Science Tournament, to be held in-person at Sharon High School in Sharon, Massachusetts from 8:30 AM to 3:30 PM on Saturday, March 29.
SMST is a math and science tournament held for middle-schoolers passionate about STEM. Teams of students have sizes of 4 to 6, but participating individually is also allowed. The competition is a day-long event with math and science individual and team rounds as well as competitive engineering challenges and hands-on science experiments.
Registration is now open on our website. There is a 15 dollar fee per participant, or 60 dollars per team, paid on the contest day. Pizza will be provided for lunch at no additional cost, and accommodations will be available for dietary restrictions. As we near the competition date, the schedule will be finalized and be available on our website as well.
Having gone to SMST twice myself, I can assure that whether you are an avid math and/or science competitor or a STEM-enthusiast in general, you will have a ton of fun at SMST!
More information is available on our website and feel free to email us with any questions at: smst.sharonschools@gmail.com.
Registration is now open on our website. There is a 15 dollar fee per participant, or 60 dollars per team, paid on the contest day. Pizza will be provided for lunch at no additional cost, and accommodations will be available for dietary restrictions. As we near the competition date, the schedule will be finalized and be available on our website as well.
Having gone to SMST twice myself, I can assure that whether you are an avid math and/or science competitor or a STEM-enthusiast in general, you will have a ton of fun at SMST!
More information is available on our website and feel free to email us with any questions at: smst.sharonschools@gmail.com.
