Difference between revisions of "2019 USAJMO"
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+ | ==Day 1== | ||
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+ | <b>Note:</b> 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. | ||
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+ | ===Problem 1=== | ||
+ | There are <math>a+b</math> bowls arranged in a row, number <math>1</math> through <math>a+b</math>, where <math>a</math> and <math>b</math> are given positive integers. Initially, each of the first <math>a</math> bowls contains an apple, and each of the last <math>b</math> bowls contains a pear. | ||
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+ | A legal move consists of moving an apple from bowl <math>i</math> to bowl <math>i+1</math> and a pear from bowl <math>j</math> to bowl <math>j-1</math>, provided that the difference <math>i-j</math> is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first <math>b</math> bowls each containing a pear and the last <math>a</math> bowls each containing an apple. Show that this is possible if and only if the product <math>ab</math> is even. | ||
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+ | ===Problem 2=== | ||
+ | Let <math>\mathbb Z</math> be the set of all integers. Find all pairs of integers <math>(a,b)</math> for which there exist functions <math>f:\mathbb Z\rightarrow\mathbb Z</math> and <math>g:\mathbb Z\rightarrow\mathbb Z</math> satisfying <cmath>f(g(x))=x+a\quad\text{and}\quad g(f(x))=x+b</cmath> for all integers <math>x</math>. | ||
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+ | ===Problem 3=== | ||
+ | <math>(*)</math> Let <math>ABCD</math> be a cyclic quadrilateral satisfying <math>AD^2+BC^2=AB^2</math>. The diagonals of <math>ABCD</math> intersect at <math>E</math>. Let <math>P</math> be a point on side <math>\overline{AB}</math> satisfying <math>\angle APD=\angle BPC</math>. Show that line <math>PE</math> bisects <math>\overline{CD}</math>. | ||
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+ | ==Day 2== | ||
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+ | ===Problem 4=== | ||
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+ | ===Problem 5=== | ||
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+ | ===Problem 6=== | ||
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+ | {{MAA Notice}} | ||
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+ | {{USAJMO newbox|year= 2019 |before=[[2018 USAJMO]]|after=[[2020 USAJMO]]}} |
Revision as of 18:39, 18 April 2019
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
There are bowls arranged in a row, number through , where and are given positive integers. Initially, each of the first bowls contains an apple, and each of the last bowls contains a pear.
A legal move consists of moving an apple from bowl to bowl and a pear from bowl to bowl , provided that the difference is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first bowls each containing a pear and the last bowls each containing an apple. Show that this is possible if and only if the product is even.
Problem 2
Let be the set of all integers. Find all pairs of integers for which there exist functions and satisfying for all integers .
Problem 3
Let be a cyclic quadrilateral satisfying . The diagonals of intersect at . Let be a point on side satisfying . Show that line bisects .
Day 2
Problem 4
Problem 5
Problem 6
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
2019 USAJMO (Problems • Resources) | ||
Preceded by 2018 USAJMO |
Followed by 2020 USAJMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |