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Problem 753

AMC 10/12, early questions

Geometry Difficulty 3.5 Find the answer China Mathematical Competition · China

The number of integral points (i.e., the points whose $x$ - and $y$ -coordinates are both integers) within the area (not including the boundary) enclosed by the right branch of hyperbola $x^2 - y^2 = 1$ and line $x = 100$ is ______.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

By symmetry, we only need to consider the part of the area above the $x$ -axis. Suppose line $y = k$ intercepts the right branch of the hyperbola and line $x = 100$ at points $A_k$ and $B_k$ ( $k = 1, 2, \dots, 99$ ), respectively. Then the number of integral points within the segment $A_k B_k$ is $99 - k$ . Therefore, the number of integral points within the area above the $x$ -axis is
$\sum_{k=1}^{99} (99 - k) = 99 \times 49 = 4851.$
Finally, we obtain the total number of integral points within the whole area as $2 \times 4851 + 98 = 9800$ . $\square$

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