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I have N+1 numbers. If I sum any N of them, the sum is divisible by N.
If one of the numbers is divisible by N, must all of the other numbers also be divisible by N?
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If a periodic non-zero function f(x) exists such that 3f(x)=f(x−1)+f(x+1), which of these options could be its fundamental period?
Note: A function g(x) is periodic if there exists some T such that g(x)=g(x+T) for all x. The fundamental period is the minimum positive T such that this property holds.
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n¡=n(n−1)(n−2)...21
If n¡ is defined as above, what are the last two digits of the number 2018¡?
Note: ¡ is the factorial notation ! turned upside down. Note that ¡ keeps exponentiating while ! keeps multiplying.
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n=1∑∞(2n4n)−1=X+Yπ−ZZ2ln(21+Z)
The equation above holds true for rational numbers X, Y, and Z. Find XYZ.
Note: (⋅⋅) is the binomial coefficient. The first few terms of the series are as follows:
n=1∑∞(2n4n)−1=(24)1+(48)1+(612)1+⋯=61+701+9241+⋯.
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Let A⊂Z be a non-trivial set of integers (i.e. A=∅ and A=Z). We call such a set A a "remarkable set of type N" if it has the following properties:
If a is an element of A, then −a is also an element of A.
If a is an element of A, then a+N is also an elements of A.
If a,b are elements of A (not necessarily different), then a+2b is also an element of A.
How many (non-trivial) remarkable sets of type 18 are there?
Bonus: Generalize. If N = 2e2⋅3e3⋅5e5⋯ is the prime factorization of N, how many non-trivial remarkable sets of type N exist? What do they look like?
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