Each of the digits from 0 to 9 appears in the number \(2^{29}\) *once* except for one. Which digit is missing?

\(\)

**Hint:** Divisibility Rules

How effective are gold bullets compared to lead bullets of the same shape and size?

**Assume** that the bullets do not change shape and that the bullets leave the gun at the same velocity.

If \(x=0, \) then \(x - x^2 + x^3 - x^4 + \cdots = 0.\)

Which of the following is the value of \[y - y^2 + y^3 - y^4 + \cdots , \] where \(y\) is some real number?

A **Pythagorean triple** is a set of positive integers \(a < b < c\) such that \(a^2 + b^2 = c^2\). Some examples are
\[\begin{align}
3^2 + 4^2 &= 5^2 \\
5^2 + 12^2 &= 13^2 \\
8^2 + 15^2 &= 17^2.
\end{align}\]
Note that each of these Pythagorean triples contains a multiple of 5.

How many Pythagorean triples \(\{a, b, c\}\) are there for which none of the three numbers is a multiple of 5 and \(c \leq 1000?\)

Using an ordered alphabet of 26 letters, how many ways are there to choose a set of six different letters such that no two letters in the set are adjacent in the alphabet?

For instance, \(\{ISOKAY\}\) is a valid set of six letters, but \(\{V\color{red}E\color{black}TOI\color{red}F\color{black}\}\) is not because \(E\) and \(F\) are both in the set.

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