2020 CMIMC (Proposals)
by djmathman, Feb 3, 2020, 4:21 AM
I suspect this will be the last hurrah.
Alg/NT 4 (Spring 2016): For all real numbers
, let 
. What is the sum of all possible values of 
, given that 
is an angle satisfying ![\[P(\sin\theta) = P(\cos\theta)?\]](//latex.artofproblemsolving.com/7/9/3/793ef7ed096ec65735190eac48a1fd765dc74bca.png)
C/CS 2 (Winter 2020): David is taking a true/false exam with 9 questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly 5 of the answers are True. In accordance with this, David guesses the answers to all 9 questions, making sure that exactly 5 of his answers are True. What is the probability he answers at least 5 questions correctly?
Geo 6 (Summer 2018): Two circles
and 
have centers at points 
and 
respectively and intersect at points 
and 
in such a way that , , , and all lie on a common circle 
. The tangent to at intersects and again at points 
and 
respectively. Suppose 
and 
. Compute the sum of the radii of and .
Geo 8 (Summer 2019): Let
be an ellipse with foci 
and 
. Parabola 
, having vertex and focus , intersects at two points and . Suppose the tangents to at and intersect on the directrix of . Compute the eccentricity of .
(A parabola is the set of points which are equidistant from a point, called the focus of , and a line, called the directrix of . An ellipse is the set of points such that the sum 
is some constant 
, where and are the foci of . The eccentricity of is defined to be the ratio 
.)
Team 2 (Winter 2020): Find all sets of five positive integers whose mode, mean, median, and range are all equal to
.
Team 7 (Summer 2019): Points and lie on a circle . The tangents to at and intersect at point 
, and point 
is chosen on so that and lie on opposite sides of 
and 
. Let 
meet for the second time at point 
. Given that 
and 
, determine 
.
The trip was lovely and I got to see so many people again! I really do miss CMU; there's a certain atmosphere that permeates CMU undergraduate culture that I haven't been able to find yet at UIUC, and I'm not sure I'll ever truly be able to replicate it. (But UIUC is great in its own ways, too.)
I've also realized (or maybe re-realized?) that math contests are at their peak when they're used both as great bonding events and as showcases for amazing mathematics that can't be found elsewhere (as opposed for being about the competition itself). I've always felt weird about crafting problems completely free of restrictions only to force contestants to solve them under those exact constraints; I wonder (especially after a conversation I had with one of the CMIMC board members during the exam) if there's a way to take these positives and isolate them from the inherently negative parts of competition. I'm not sure.
Alg/NT 4 (Spring 2016): For all real numbers
, let 
. What is the sum of all possible values of 
, given that 
is an angle satisfying ![\[P(\sin\theta) = P(\cos\theta)?\]](http://latex.artofproblemsolving.com/7/9/3/793ef7ed096ec65735190eac48a1fd765dc74bca.png)
C/CS 2 (Winter 2020): David is taking a true/false exam with 9 questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly 5 of the answers are True. In accordance with this, David guesses the answers to all 9 questions, making sure that exactly 5 of his answers are True. What is the probability he answers at least 5 questions correctly?
Geo 6 (Summer 2018): Two circles
and 
have centers at points 
and 
respectively and intersect at points 
and 
in such a way that , , , and all lie on a common circle 
. The tangent to at intersects and again at points 
and 
respectively. Suppose 
and 
. Compute the sum of the radii of and .Geo 8 (Summer 2019): Let
be an ellipse with foci 
and 
. Parabola 
, having vertex and focus , intersects at two points and . Suppose the tangents to at and intersect on the directrix of . Compute the eccentricity of .(A parabola
is the set of points which are equidistant from a point, called the focus of , and a line, called the directrix of . An ellipse is the set of points such that the sum 
is some constant 
, where and are the foci of . The eccentricity of is defined to be the ratio 
.)Team 2 (Winter 2020): Find all sets of five positive integers whose mode, mean, median, and range are all equal to
.Team 7 (Summer 2019): Points
and lie on a circle . The tangents to at and intersect at point 
, and point 
is chosen on so that and lie on opposite sides of 
and 
. Let 
meet for the second time at point 
. Given that 
and 
, determine 
.
The trip was lovely and I got to see so many people again! I really do miss CMU; there's a certain atmosphere that permeates CMU undergraduate culture that I haven't been able to find yet at UIUC, and I'm not sure I'll ever truly be able to replicate it. (But UIUC is great in its own ways, too.)
I've also realized (or maybe re-realized?) that math contests are at their peak when they're used both as great bonding events and as showcases for amazing mathematics that can't be found elsewhere (as opposed for being about the competition itself). I've always felt weird about crafting problems completely free of restrictions only to force contestants to solve them under those exact constraints; I wonder (especially after a conversation I had with one of the CMIMC board members during the exam) if there's a way to take these positives and isolate them from the inherently negative parts of competition. I'm not sure.
This post has been edited 3 times. Last edited by djmathman, Feb 3, 2020, 4:28 AM
~dj
April 2025
November 2024