[/quote]
,
are positive real numbers such that 
prove that
girls on one side of a tennis table, and a queue of 
to 
grid. We color the squares with three colors: blue, red, and yellow. Two squares are defined as opposite if they share a vertex but not an edge. A valid coloring is a coloring such that no two squares that are red and blue are opposite each other. Count the number of valid colorings.
,
and 
are in segments 
and 
respectively such that 
.
be the intersection of 
and 
and 
be the intersection of 
and 
.
is a cyclic quadrilateral prove that 
,
boys (with 
)
boys and for any 
boys, there is exactly 
girl who is their common friend. Let 
be the greatest number of girls that can be chosen from the camp to form a group such that every boy is friend with at most 
.
such that 
is a perfect square.
be an 
square matrix with eigenvalues 
(each 
having algebraic multiplicity 
,
)
be a polynomial. It is known that if 
is an eigenvalue of 
is an eigenvalue of 
.
(counting multiplicities)?
(with multiplicity 3),
(with multiplicity 2),
(with multiplicity 1),
(with multiplicity 3), 
(with multiplicity 2), and 
(with multiplicity 1)?
vs. equation 
)
such that
.
is invertible
,
![\[T_1 z = \frac{z + 2}{z + 3}, \qquad T_2 z = \frac z{z + 1},\]](http://latex.artofproblemsolving.com/b/7/2/b72b5e302cecdf2649c43a078aea3aa9a88ef2cc.png)
find 
and 
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I write today to announce my retirement as CEO from Art of Problem Solving. When I founded AoPS 22 years ago, I never imagined that we would reach so many students and families, or that we would find so many channels through which we discover, inspire, and train the great problem solvers of the next generation. I am very proud of all we have accomplished and I’m thankful for the many supporters who provided inspiration and encouragement along the way. I'm particularly grateful to all of the wonderful members of the AoPS Community!
I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.
Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.
And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:
I’m delighted to introduce our new leaders - Ben Kornell and Andrew Sutherland. Ben has extensive experience in education and edtech prior to joining AoPS as my successor as CEO, including starting like I did as a classroom teacher. He has a deep understanding of the value of our work because he’s an AoPS parent! Meanwhile, Andrew and I have common roots as founders of education companies; he launched Quizlet at age 15! His journey from founder to MIT to technology and product leader as our Chief Product Officer traces a pathway many of our students will follow in the years to come.
Thank you again for your support for Art of Problem Solving and we look forward to working with millions more wonderful problem solvers in the years to come.
And special thanks to all of the amazing AoPS team members who have helped build AoPS. We’ve come a long way from here:
