MathDB

Choosing signs + or - to make the sum equal to 0

Source: Romanian TST 2002

February 5, 2011

modular arithmeticfloor functioncombinatorics proposedcombinatorics

Problem Statement

For any positive integer

n

, let

f(n)

be the number of possible choices of signs

+\ \text{or}\ -

in the algebraic expression

\pm 1\pm 2\ldots \pm n

, such that the obtained sum is zero. Show that

f(n)

satisfies the following conditions: a)

f(n)=0

for

n=1\pmod{4}

or

n=2\pmod{4}

. b)

2^{\frac{n}{2}-1}\le f(n)\le 2^n-2^{\lfloor\frac{n}{2}\rfloor+1}

, for

n=0\pmod{4}

or

n=3\pmod{4}

.
Ioan Tomsecu

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