Difference between revisions of "2017 USAMO Problems"
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+ | ==Day 1== | ||
+ | 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 1=== | ||
+ | Prove that there are infinitely many distinct pairs <math>(a,b)</math> of relatively prime positive integers <math>a > 1</math> and <math>b > 1</math> such that <math>a^b + b^a</math> is divisible by <math>a + b.</math> | ||
+ | |||
+ | [[2017 USAMO Problems/Problem 1|Solution]] | ||
+ | |||
+ | ===Problem 2=== | ||
+ | Let <math>m_1, m_2, \ldots, m_n</math> be a collection of <math>n</math> positive integers, not necessarily distinct. For any sequence of integers <math>A = (a_1, \ldots, a_n)</math> and any permutation <math>w = w_1, \ldots, w_n</math> of <math>m_1, \ldots, m_n</math>, define an <math>A</math>-inversion of <math>w</math> to be a pair of entries <math>w_i, w_j</math> with <math>i < j</math> for which one of the following conditions holds: | ||
+ | <cmath>a_i \ge w_i > w_j,</cmath> | ||
+ | <cmath>w_j > a_i \ge w_i,</cmath> or | ||
+ | <cmath>w_i > w_j > a_i.</cmath> | ||
+ | Show that, for any two sequences of integers <math>A = (a_1, \ldots, a_n)</math> and <math>B = (b_1, \ldots, b_n)</math>, and for any positive integer <math>k</math>, the number of permutations of <math>m_1, \ldots, m_n</math> having exactly <math>k</math> <math>A</math>-inversions is equal to the number of permutations of <math>m_1, \ldots, m_n</math> having exactly <math>k</math> <math>B</math>-inversions. | ||
+ | |||
+ | [[2017 USAMO Problems/Problem 2|Solution]] | ||
+ | |||
+ | ===Problem 3=== | ||
+ | (<math>*</math>) Let <math>ABC</math> be a scalene triangle with circumcircle <math>\Omega</math> and incenter <math>I</math>. Ray <math>AI</math> meets <math>\overline{BC}</math> at <math>D</math> and meets <math>\Omega</math> again at <math>M</math>; the circle with diameter <math>\overline{DM}</math> cuts <math>\Omega</math> again at <math>K</math>. Lines <math>MK</math> and <math>BC</math> meet at <math>S</math>, and <math>N</math> is the midpoint of <math>\overline{IS}</math>. The circumcircles of <math>\triangle KID</math> and <math>\triangle MAN</math> intersect at points <math>L_1</math> and <math>L_2</math>. Prove that <math>\Omega</math> passes through the midpoint of either <math>\overline{IL_1}</math> or <math>\overline{IL_2}</math>. | ||
+ | |||
+ | [[2017 USAMO Problems/Problem 3|Solution]] | ||
+ | |||
+ | ==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=== | ||
+ | Let <math>P_1</math>, <math>P_2</math>, <math>\dots</math>, <math>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> red points and <math>n</math> blue points. Let <math>R_1</math>, <math>R_2</math>, <math>\dots</math>, <math>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> travelling counterclockwise around the circle from <math>R_2</math>, and so on, until we have labeled all of the blue points <math>B_1, \dots, B_n</math>. Show that the number of counterclockwise arcs of the form <math>R_i \to B_i</math> that contain the point <math>(1,0)</math> is independent of the way we chose the ordering <math>R_1, \dots, R_n</math> of the red points. | ||
+ | |||
+ | [[2017 USAJMO Problems/Problem 6|Solution]] | ||
+ | |||
+ | ===Problem 5=== | ||
+ | Let <math>\mathbf{Z}</math> denote the set of all integers. Find all real numbers <math>c > 0</math> such that there exists a labeling of the lattice points <math> ( x, y ) \in \mathbf{Z}^2</math> with positive integers for which: | ||
+ | only finitely many distinct labels occur, and | ||
+ | for each label <math>i</math>, the distance between any two points labeled <math>i</math> is at least <math>c^i</math>. | ||
+ | |||
+ | [[2017 USAMO Problems/Problem 5|Solution]] | ||
+ | |||
+ | ===Problem 6=== | ||
+ | Find the minimum possible value of <cmath>\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}</cmath>given that <math>a</math>, <math>b</math>, <math>c</math>, <math>d</math> are nonnegative real numbers such that <math>a+b+c+d=4</math>. | ||
+ | |||
+ | [[2017 USAMO Problems/Problem 6|Solution]] | ||
+ | |||
+ | {{MAA Notice}} | ||
+ | |||
+ | {{USAMO newbox|year=2017|before=[[2016 USAMO Problems]]|after=[[2018 USAMO Problems]]}} |
Latest revision as of 12:49, 22 November 2023
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
Let be a collection of positive integers, not necessarily distinct. For any sequence of integers and any permutation of , define an -inversion of to be a pair of entries with for which one of the following conditions holds: or Show that, for any two sequences of integers and , and for any positive integer , the number of permutations of having exactly -inversions is equal to the number of permutations of having exactly -inversions.
Problem 3
() Let be a scalene triangle with circumcircle and incenter . Ray meets at and meets again at ; the circle with diameter cuts again at . Lines and meet at , and is the midpoint of . The circumcircles of and intersect at points and . Prove that passes through the midpoint of either or .
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
Let , , , be distinct points on the unit circle , other than . Each point is colored either red or blue, with exactly red points and blue points. 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 travelling counterclockwise around the circle from , and so on, until we have labeled all of 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.
Problem 5
Let denote the set of all integers. Find all real numbers such that there exists a labeling of the lattice points with positive integers for which: only finitely many distinct labels occur, and for each label , the distance between any two points labeled is at least .
Problem 6
Find the minimum possible value of given that , , , are nonnegative real numbers such that .
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
2017 USAMO (Problems • Resources) | ||
Preceded by 2016 USAMO Problems |
Followed by 2018 USAMO Problems | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |