MathDB

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

Source: Romanian TST 2002

2/5/2011

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

modular arithmeticfloor functioncombinatorics proposedcombinatorics

60% of participants can speak the same language

Source: Romanian TST 2002

2/5/2011

At an international conference there are four official languages. Any two participants can speak in one of these languages. Show that at least

60\%

of the participants can speak the same language.
Mihai Baluna

inductioncombinatorics proposedcombinatorics

If x-y=2,3 or 5 then f(x) is not equal to f(y)

Source: Romanian TST 2002

2/5/2011

Let

f:\mathbb{Z}\rightarrow\{ 1,2,\ldots ,n\}

be a function such that

f(x)\not= f(y)

, for all

x,y\in\mathbb{Z}

such that

|x-y|\in\{2,3,5\}

. Prove that

n\ge 4

.
Ioan Tomescu

functionmodular arithmeticcombinatorics proposedcombinatorics

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