Difference between revisions of "2019 USAJMO Problems"

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===Problem 1===
 
===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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There are <math>a+b</math> bowls arranged in a row, numbered <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.
 
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.

Revision as of 22:56, 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, numbered $1$ 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.

Solution

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 $x$.

Solution

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 $\overline{CD}$.

Solution


Day 2

Problem 4

Problem 5

Problem 6

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2019 USAJMO (ProblemsResources)
Preceded by
2018 USAJMO
Followed by
2020 USAJMO
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All USAJMO Problems and Solutions