Difference between revisions of "2019 USAJMO"

(Created page with "If you cliked this u noob cuz it still 2018")
 
(added day 1 problems)
Line 1: Line 1:
If you cliked this u noob cuz it still 2018
+
 
 +
==Day 1==
 +
 
 +
<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.
 +
 
 +
===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.
 +
 
 +
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]]}}

Revision as of 19:39, 18 April 2019

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 $a+b$ bowls arranged in a row, number $1$ through $a+b$, where $a$ and $b$ are given positive integers. Initially, each of the first $a$ bowls contains an apple, and each of the last $b$ bowls contains a pear.

A legal move consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, provided that the difference $i-j$ is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first $b$ bowls each containing a pear and the last $a$ bowls each containing an apple. Show that this is possible if and only if the product $ab$ is even.

Problem 2

Let $\mathbb Z$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f:\mathbb Z\rightarrow\mathbb Z$ and $g:\mathbb Z\rightarrow\mathbb Z$ satisfying \[f(g(x))=x+a\quad\text{and}\quad g(f(x))=x+b\] for all integers $x$.

Problem 3

$(*)$ Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2+BC^2=AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\overline{AB}$ satisfying $\angle APD=\angle BPC$. Show that line $PE$ bisects $\overline{CD}$.


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. AMC logo.png

2019 USAJMO (ProblemsResources)
Preceded by
2018 USAJMO
Followed by
2020 USAJMO
1 2 3 4 5 6
All USAJMO Problems and Solutions