Difference between revisions of "2019 USAJMO"

Latest revision as of 19:44, 18 April 2019

The test was held on April 17th and 18th, 2019. The first link will contain the full set of test problems. The rest will contain each individual problem and its solution.

2019 USAJMO Problems

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+The test was held on April 17th and 18th, 2019. The first link will contain the full set of test problems. The rest will contain each individual problem and its solution.
-==Day 1==
+[[2019 USAJMO Problems]]
+* [[2019 USAJMO Problems/Problem 1]]
+* [[2019 USAJMO Problems/Problem 2]]
+* [[2019 USAJMO Problems/Problem 3]]
+* [[2019 USAJMO Problems/Problem 4]]
+* [[2019 USAJMO Problems/Problem 5]]
+* [[2019 USAJMO Problems/Problem 6]]
-<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.
+== See Also ==
+* [[Mathematics competitions]]
-===Problem 1===
+* [[Mathematics competition resources]]
-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.
+* [[Math books]]
+* [[USAJMO]]
-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.
-===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>.
-===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>.
-==Day 2==
-===Problem 4===
-===Problem 5===
-===Problem 6===
-{{MAA Notice}}
 {{USAJMO newbox|year= 2019 |before=[[2018 USAJMO]]|after=[[2020 USAJMO]]}}

2019 USAJMO (Problems • Resources)
Preceded by 2018 USAJMO	Followed by 2020 USAJMO
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Difference between revisions of "2019 USAJMO"

Latest revision as of 19:44, 18 April 2019

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