Difference between revisions of "2017 USAJMO Problems"
Liopleurodon (talk | contribs) (→Problem 6) |
|||
Line 23: | Line 23: | ||
==Day 2== | ==Day 2== | ||
+ | |||
+ | Note: For any geometry problem whose statement begins with an asterisk (<math>*</math>), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction. | ||
+ | |||
===Problem 4=== | ===Problem 4=== | ||
+ | Are there any triples <math>(a,b,c)</math> of positive integers such that <math>(a-2)(b-2)(c-2) + 12</math> is prime that properly divides the number <math>a^2 + b^2 + c^2 + abc - 2017</math>? | ||
+ | [[2017 USAJMO Problems/Problem 4|Solution]] | ||
===Problem 5=== | ===Problem 5=== | ||
+ | (<math>*</math>) Let <math>O</math> and <math>H</math> be the circumcenter and the orthocenter of an acute triangle <math>ABC</math>. Points <math>M</math> and <math>D</math> lie on side <math>BC</math> such that <math>BM = CM</math> and <math>\angle BAD = \angle CAD</math>. Ray <math>MO</math> intersects the circumcircle of triangle <math>BHC</math> in point <math>N</math>. Prove that <math>\angle ADO = \angle HAN</math>. | ||
+ | [[2017 USAJMO Problems/Problem 5|Solution]] | ||
===Problem 6=== | ===Problem 6=== | ||
+ | Let <math>P_1, \ldots, P_{2n}</math> be <math>2n</math> distinct points on the unit circle <math>x^2 + y^2 = 1</math> other than <math>(1,0)</math>. Each point is colored either red or blue, with exactly <math>n</math> of them red and exactly <math>n</math> of them blue. Let <math>R_1, \ldots, R_n</math> be any ordering of the red points. Let <math>B_1</math> be the nearest blue point to <math>R_1</math> traveling counterclockwise around the circle starting from <math>R_1</math>. Then let <math>B_2</math> be the nearest of the remaining blue points to <math>R_2</math> traveling counterclockwise around the circle from <math>R_2</math>, and so on, until we have labeled all the blue points <math>B_1, \ldots, B_n</math>. Show that the number of counterclockwise arcs of the form <math>R_i \rightarrow B_i</math> that contain the point <math>(1,0)</math> is independent of the way we chose the ordering <math>R_1, \ldots, R_n</math> of the red points. | ||
+ | |||
+ | [[2017 USAJMO Problems/Problem 6|Solution]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
{{USAJMO newbox|year= 2017 |before=[[2016 USAJMO]]|after=[[2018 USAJMO]]}} | {{USAJMO newbox|year= 2017 |before=[[2016 USAJMO]]|after=[[2018 USAJMO]]}} |
Revision as of 20:32, 20 April 2017
Contents
Day 1
Note: For any geometry problem whose statement begins with an asterisk (), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Problem 1
Prove that there are infinitely many distinct pairs of relatively prime positive integers and such that is divisible by
Problem 2
Consider the equation
(a) Prove that there are infinitely many pairs of positive integers satisfying the equation.
(b) Describe all pairs of positive integers satisfying the equation.
Problem 3
() Let be an equilateral triangle and let be a point on its circumcircle. Let lines and intersect at ; let lines and intersect at ; and let lines and intersect at . Prove that the area of triangle is twice the area of triangle .
Day 2
Note: For any geometry problem whose statement begins with an asterisk (), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Problem 4
Are there any triples of positive integers such that is prime that properly divides the number ?
Problem 5
() Let and be the circumcenter and the orthocenter of an acute triangle . Points and lie on side such that and . Ray intersects the circumcircle of triangle in point . Prove that .
Problem 6
Let be distinct points on the unit circle other than . Each point is colored either red or blue, with exactly of them red and exactly of them blue. Let be any ordering of the red points. Let be the nearest blue point to traveling counterclockwise around the circle starting from . Then let be the nearest of the remaining blue points to traveling counterclockwise around the circle from , and so on, until we have labeled all the blue points . Show that the number of counterclockwise arcs of the form that contain the point is independent of the way we chose the ordering of the red points.
Solution The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
2017 USAJMO (Problems • Resources) | ||
Preceded by 2016 USAJMO |
Followed by 2018 USAJMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |