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2018-01-01 Intermediate

         

Each of the digits from 0 to 9 appears in the number 2292^{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,x=0, then xx2+x3x4+=0.x - x^2 + x^3 - x^4 + \cdots = 0.

Which of the following is the value of yy2+y3y4+,y - y^2 + y^3 - y^4 + \cdots , where yy is some real number?

A Pythagorean triple is a set of positive integers a<b<ca < b < c such that a2+b2=c2a^2 + b^2 = c^2. Some examples are 32+42=5252+122=13282+152=172.\begin{aligned} 3^2 + 4^2 &= 5^2 \\ 5^2 + 12^2 &= 13^2 \\ 8^2 + 15^2 &= 17^2. \end{aligned} Note that each of these Pythagorean triples contains a multiple of 5.

How many Pythagorean triples {a,b,c}\{a, b, c\} are there for which none of the three numbers is a multiple of 5 and c1000?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}\{ISOKAY\} is a valid set of six letters, but {VETOIF}\{V\color{#D61F06}E\color{#333333}TOI\color{#D61F06}F\color{#333333}\} is not because EE and FF are both in the set.


Note: As always, a "set" is considered unordered. Hence, {ISOKAY}\{ISOKAY\} and {YAKOSI}\{YAKOSI\} and {AIKOSY}\{AIKOSY\} are considered the same set.

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