Let be the radius of the lower circle.
We label the square as , the centre of the upper circle as , and the centre of the lower circle as . We call the point at which the upper circle touches the top line, the point at which the lower circle touches the bottom line, the point at which the two circles touch, and the point at which the lower circle touches the square.
We join , , , , , and .
We need to use two facts about circles:
- When the centre of a circle is joined to a point of tangency, the resulting line segment is perpendicular to the tangent line. Thus, is perpendicular to the top line, is perpendicular to the bottom line, and is perpendicular to .
- When two circles are tangent, the line segment joining their centres passes through the point of tangency between the circles. Thus, is a straight line segment.
Extend downwards to meet at . Since and are perpendicular to the parallel lines and is perpendicular to , then is perpendicular to .
Extend to meet at . Using a similar argument, is perpendicular to .
We focus on , which is right-angled at .
Since the radius of the upper circle is 65, then we know that .
Since the radius of the lower circle is , then we know that .
Thus, .

Since the top and bottom lines are parallel, and are perpendicular to these lines, and is parallel to these lines, then equals the distance between the lines.
Thus, and so .
Next, we note that , since is a rectangle.
Now is right-angled and has .
Also, .
But equals the difference between the distance between the lines and the side length of the square, or .
Thus, .
Using the Pythagorean Theorem in , we obtain and so .
Since , then .
Thus, .
Finally, using the Pythagorean Theorem in , we obtain By the quadratic formula, .
Since must be less than the distance between the two lines, which is 400, then . Of the given answers, this is closest to 153.
Answer: (C)
This solution is written in as mathematically complete a way as possible. Of course, while doing a multiple choice contest, those who attempt and/or complete this problem would be unlikely to work through all of these details.
Step 1: Using information about decimal equivalents to rewrite fractions
Consider a real number whose decimal equivalent is of the form for some integers and and digits .
(Note that if , then .)
Then for some positive integer .
We demonstrate this in a specific example and leave the full algebraic derivation to the end of this solution. If , then Suppose that a fraction has and positive integers with . Then .
Suppose now that a fraction of positive integers with has the property that there is a sequence of consecutive digits of length 6 in its decimal equivalent that repeats consecutively and indefinitely. That is, suppose that for some integer and digits .
From above, for some positive integer . (Note that .)
Step 2: Using further conditions given to analyze
Continuing from above, .
Since we are also told that is in lowest terms, then and have no common divisors larger than 1, and so must be a divisor of .
Note that .
Since is a divisor of , then cannot contain any prime factors other than .
Since is not divisible by the square of any positive integer, then it cannot be divisible by the square of any prime number.
Thus, must be a divisor of .
Step 3: Consolidating current progress
We now know that any fraction satisfying the properties
- and are positive integers with ,
- is in lowest terms,
- is not divisible by the square of any integer larger than 1, and
- the decimal equivalent of includes a sequence of consecutive digits of length 6 that repeats consecutively and indefinitely,
can be written in the form for some positive integer with .
(We note that we have not yet determined whether the shortest sequence of consecutive digits that repeats consecutively and indefinitely has length 6.)
Step 4: Every fraction with can be written as a fraction satisfying these four bullets
Each is between 0 and 1, can be written in lowest terms and has denominator not divisible by the square of any positive integer larger than 1, so any equivalent fraction in lower (or lowest terms) shares this property as factors of the denominator will only be removed in reducing, not added.
Furthermore, for some non-negative integers and with and . ( and are the quotient and remainder, respectively, when is divided by 999999.)
Writing for some digits (some or all possibly 0), then so every can be written as a decimal with a repeating sequence of length 6.
Also, each is different and so will produce a different .
Therefore, the number of such fractions (which is 1111109) will equal the number of fractions that satisfy the four bullets above.
Notice that we have not yet checked to see if the sequence of digits of length 6 is the shortest such sequence.
Step 5: Considering shorter possible lengths
Since 6 is to be the length of the shortest sequence of repeating digits after the decimal point, then there can be no sequence of repeating digits of length 1, 2, 3, 4, or 5.
Using a similar approach to the first derivation above, we see that thus cannot be written in any of the forms or or or or .
Using a similar approach to the analysis of prime factors above, we see that cannot be written in any of the forms or or or or .
It is possible that an with properties as above including a repeating sequence of length 6 in its decimal equivalent can also be written with a repeating sequence of length 1, 2 or 3.
This is because, for example, which all have sequences of length 6 that repeat.
It is not possible for an with a repeating sequence of length 6 in its decimal equivalent to also be written with a repeating sequence of length 4 or 5, without a repeating sequence of length 1 or 2.
This is because if, for example, there is a repeating sequence of length 4, then for some positive integer and so and so which tells us that is divisor of and so for some positive integer .
Therefore, the decimal equivalent to has a sequence of length 2 that repeats, so any fraction with a decimal equivalent that has a repeating sequence of length 4 will be dealt with among those with sequences of length 1, 2 or 3.
In a similar way, we can rule out decimal equivalents with sequences of length 5 that repeat.
Therefore, we need to (carefully) remove fractions from our count that have decimal equivalents with sequences of lengths 1, 2 or 3 that repeat.
Step 6: Considering overlap
We have 1111109 fractions to consider, each of which satisfies the four bullets above.
Since the decimal equivalent of cannot have sequences of lengths 1, 2 or 3 that repeat, then cannot be written in any of the forms or or for positive integers , and with , , and .
Let be the set of the 1009 fractions of the form , be the set of the 329 fractions of the form , and be the set of the 29 fractions of the form .
Any fraction of the form is also of the form (since ) and is also of the form (since ).
Therefore, every fraction in is also in and in . In set notation and .
Furthermore, any fraction that is in both and is also in :
Suppose that a fraction can be written in both forms and .
Then or and so .
Since is a thus a multiple of 11 and is not divisible by the prime number 11, then is a multiple of 11.
This means that for some positive integer , and so is in .
In set notation, .
Since and and , then ; in other words, the set of fractions in each of and is precisely the set of fractions .
Step 7: Final counting
We start with 1111109 fractions, as above, and want to remove all of the fractions in , and .
Since each fraction in is in and , it is enough to remove those and only.
The total number of fractions in and (that is, in ) equals the number of fractions in plus the number of fractions in minus the number of fractions in their overlap (that is, in ). This is because any fraction in the overlap is “counted twice" when include all fractions in and all fractions in .
Therefore, we need to remove fractions from the set of 1111109.
Therefore, , the number of fractions having the desired properties, is Since has 7 digits, then . The sum of the squares of the digits of is .
Step 8: General algebraic derivation from Step 1
Consider a real number whose decimal equivalent is of the form for some integers and and digits . Then
Answer: (E)