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2017-07-17 Advanced

         

If \(N\) is divisible by \(1,2,3,\ldots, 13\), then \(N\) must also be divisible by 14 and 15.

Using this same idea, what is the smallest integer \(M\) such that the following statement is true?

If \(N\) is divisible by \(1,2,3,\ldots,M\), then \(N\) must also be divisible by \(M+1,M+2,M+3,\) and \(M+4\).

Suppose there exists a triangle with integer side lengths such that its perimeter (in units) and area (in square units) have the same numerical value, \(p\).

What is the sum of all possible values of \(p?\)

As shown in the diagram below, three overlapping circles with distinct radii define six points. The lengths of 5 line segments (in blue) connecting the points are known to us: 4, 6, 8, 3, 5.

What is the length of the remaining segment (in yellow), to 3 decimal places?

\[ \LARGE{ \require{enclose} \begin{array}{rll} \phantom{0}\ \mathrm{\large7} \ \mathrm{x} \ \mathrm{x} \ \mathrm{x} && \\[-1pt] \mathrm{x} \ \mathrm{x} \ \mathrm{x} \
\enclose{longdiv}{ \mathrm{x} \ \mathrm{x} \ \mathrm{x} \ \mathrm{x} \ \mathrm{x} \ \mathrm{x} }\kern-.2ex \\[-1pt] \underline{ \mathrm{x} \ \mathrm{x} \ \mathrm{x} \ \phantom 0 \ \phantom 0 \ \phantom 0 } \\[-1pt] { \mathrm{x} \ \mathrm{x} \ \mathrm{x} \ \mathrm{\large7} \ \phantom0 \ \phantom0 \ }\kern-.2ex \\[-1pt] \underline{ \mathrm{x} \ \mathrm{x} \ \mathrm{x} \ \mathrm{x} \ \phantom 0 \ \phantom 0 } && \\[-1pt] { \mathrm{x} \ \mathrm{x} \ \mathrm{x} \ \mathrm{x} }\kern-.2ex \\[-1pt] \underline{ \mathrm{x} \ \mathrm{x} \ \mathrm{x} \ \mathrm{x} } && \\[-1pt] \mathrm{\large7} \end{array} } \]

The above is a long division with most of the digits of any number hidden, except for the three 7's. Given that each of 0, 1, 2, ..., 9 was used at least once for the hidden digits, figure out all of the digits hiding and submit your answer as the value of the dividend (the 6-digit number being divided).

\(\)
Details and Assumptions:

  • Each \(\mathrm X\) represents a single-digit integer.
  • The leading (leftmost) digit of a number cannot be 0.

Find all integer solutions to the equation \(y^2=x^3-p^3\), where \(p\) is a prime number, \(y \neq 0\), \(3 \nmid y\), and \(p \nmid y\). Enter your answer as \(\sum(x+p)\).


Note: Actually, you are going to find integer points on a family of Mordell curves. A Mordell curve is an elliptic curve of the form \(y^2 = x^3 + k\), where \(k\) is a fixed non-zero integer.

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