MathDB

Weird if it contains exactly one of the distinct elements

Source: VAIMO 2007, P1

1/3/2009

Let

n > 1, n \in \mathbb{Z}

and B \equal{}\{1,2,\ldots, 2^n\}. A subset

A

of

B

is called weird if it contains exactly one of the distinct elements

x,y \in B

such that the sum of

x

and

y

is a power of two. How many weird subsets does

B

have?

combinatorics unsolvedcombinatorics

Representation in ternary system

Source: AIMO 2007, TST 5, P1

1/11/2009

Let

k \in \mathbb{N}

. A polynomial is called

k

-valid if all its coefficients are integers between 0 and

k

inclusively. (Here we don't consider 0 to be a natural number.) a.) For

n \in \mathbb{N}

let

a_n

be the number of 5-valid polynomials

p

which satisfy

p(3) = n.

Prove that each natural number occurs in the sequence

(a_n)_n

at least once but only finitely often. b.) For

n \in \mathbb{N}

let

a_n

be the number of 4-valid polynomials

p

which satisfy

p(3) = n.

Prove that each natural number occurs infinitely often in the sequence

(a_n)_n

.

algebrapolynomialnumber theory unsolvednumber theory

a%b + a%2b + a%3b + ... + a%nb = a + b

Source: AIMO 2007, TST 6, P1

1/11/2009

For a multiple of

kb

of

b

let

a \% kb

be the greatest number such that a \% kb \equal{} a \bmod b which is smaller than

kb

and not greater than

a

itself. Let n \in \mathbb{Z}^ \plus{} . Determine all integer pairs

(a,b)

with: a\%b \plus{} a\%2b \plus{} a\%3b \plus{} \ldots \plus{} a\%nb \equal{} a \plus{} b

algebra unsolvedalgebra

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