Difference between revisions of "2017 USAJMO Problems"
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[[2017 USAJMO Problems/Problem 2|Solution]] | [[2017 USAJMO Problems/Problem 2|Solution]] | ||
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===Problem 3=== | ===Problem 3=== | ||
(<math>*</math>) Let <math>ABC</math> be an equilateral triangle and let <math>P</math> be a point on its circumcircle. Let lines <math>PA</math> and <math>BC</math> intersect at <math>D</math>; let lines <math>PB</math> and <math>CA</math> intersect at <math>E</math>; and let lines <math>PC</math> and <math>AB</math> intersect at <math>F</math>. Prove that the area of triangle <math>DEF</math> is twice the area of triangle <math>ABC</math>. | (<math>*</math>) Let <math>ABC</math> be an equilateral triangle and let <math>P</math> be a point on its circumcircle. Let lines <math>PA</math> and <math>BC</math> intersect at <math>D</math>; let lines <math>PB</math> and <math>CA</math> intersect at <math>E</math>; and let lines <math>PC</math> and <math>AB</math> intersect at <math>F</math>. Prove that the area of triangle <math>DEF</math> is twice the area of triangle <math>ABC</math>. | ||
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[[2017 USAJMO Problems/Problem 4|Solution]] | [[2017 USAJMO Problems/Problem 4|Solution]] | ||
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===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>. | (<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]] | [[2017 USAJMO Problems/Problem 5|Solution]] | ||
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===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. | 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]] | [[2017 USAJMO Problems/Problem 6|Solution]] | ||
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{{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 19:32, 20 April 2017
Contents
[hide]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.
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 |