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- Feb 14, 2012

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- Feb 14, 2012

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- Feb 14, 2012

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I must mention here that the idea of this problem came from Professor Gregory Galperin of Eastern Illinois State University, U.S.A. and I feel so grateful and thankful to Professor Gregory Galperin for approving me to show his solution to this problem at this site.

We know that the center of the ellipse is the point (19, 98). The two vertical lines \(\displaystyle x=0 \) and \(\displaystyle x=2\cdot19=38\) and the two horizontal lines \(\displaystyle y=0\) and \(\displaystyle y=2\cdot98=196\) divide the ellipse into nine regions, as shown below.

Then, in terms of the areas marked by S, T, U and V, we find that \(\displaystyle A_1=U+V+S+T\), \(\displaystyle A_2=S+T\), \(\displaystyle A_3=S\), and \(\displaystyle A_4=V+S\), and therefore

\(\displaystyle A_1-A_2+A_3-A_4=(U+V+T+S)-(S+T)+S-(V+S)=U.\)

Hence the answer is \(\displaystyle 38\cdot196=7448\).

Note:

This solution was published in the book by G. Berzsenyi "