USA Team Selection Test 2017
TST #1
December 8th, 2016
1
In a sports league, each team uses a set of at most 
signature colors. A set 
of teams is color-identifiable if one can assign each team in one of their signature colors, such that no team in is assigned any signature color of a different team in .
For all positive integers
and , determine the maximum integer 
such that: In any sports league with exactly distinct colors present over all teams, one can always find a color-identifiable set of size at least .
signature colors. A set 
of teams is color-identifiable if one can assign each team in one of their signature colors, such that no team in is assigned any signature color of a different team in .For all positive integers
and , determine the maximum integer 
such that: In any sports league with exactly distinct colors present over all teams, one can always find a color-identifiable set of size at least .
2
Let 
be an acute scalene triangle with circumcenter 
, and let 
be on line 
such that 
. The circle with diameter 
intersects the circumcircle of 
at two points 
and 
, where 
. Points 
, 
, 
, 
are defined analogously.
be an acute scalene triangle with circumcenter 
, and let 
be on line 
such that 
. The circle with diameter 
intersects the circumcircle of 
at two points 
and 
, where 
. Points 
, 
, 
, 
are defined analogously.
- Prove that
, 
, 
are concurrent. - Prove that
, 
, 
are concurrent on the Euler line of triangle .
,
,
are concurrent.
,
,
are concurrent on the Euler line of triangle 
3
Let ![$P, Q \in \mathbb{R}[x]$](//latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
be relatively prime nonconstant polynomials. Show that there can be at most three real numbers 
such that 
is the square of a polynomial.
Alison Miller
![$P, Q \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
be relatively prime nonconstant polynomials. Show that there can be at most three real numbers 
such that 
is the square of a polynomial.Alison Miller
TST #2
January 19th, 2017
1
You are cheating at a trivia contest. For each question, you can peek at each of the 
other contestants' guesses before writing down your own. For each question, after all guesses are submitted, the emcee announces the correct answer. A correct guess is worth 
points. An incorrect guess is worth 
points for other contestants, but only 
point for you, since you hacked the scoring system. After announcing the correct answer, the emcee proceeds to read the next question. Show that if you are leading by 
points at any time, then you can surely win first place.
Linus Hamilton
other contestants' guesses before writing down your own. For each question, after all guesses are submitted, the emcee announces the correct answer. A correct guess is worth 
points. An incorrect guess is worth 
points for other contestants, but only 
point for you, since you hacked the scoring system. After announcing the correct answer, the emcee proceeds to read the next question. Show that if you are leading by 
points at any time, then you can surely win first place.Linus Hamilton
2
Let be a triangle with altitude 
. The 
-excircle touches 
at 
, and intersects the circumcircle at two points 
and 
. Prove that one can select points 
and 
on lines 
and 
such that quadrilateral 
is a rhombus.
Danielle Wang and Evan Chen
be a triangle with altitude 
. The 
-excircle touches 
at 
, and intersects the circumcircle at two points 
and 
. Prove that one can select points 
and 
on lines 
and 
such that quadrilateral 
is a rhombus.Danielle Wang and Evan Chen
3
Prove that there are infinitely many triples 
of positive integers with 
prime, 
, and 
, such that 
is a multiple of 
.
Noam Elkies
of positive integers with 
prime, 
, and 
, such that 
is a multiple of 
.Noam Elkies
