MathDB

Prove that if A is a B_3-set, then A = B

Source: IMO Shortlist 2000, A6

8/10/2008

A nonempty set

A

of real numbers is called a

B_3

-set if the conditions

a_1, a_2, a_3, a_4, a_5, a_6 \in A

and a_1 \plus{} a_2 \plus{} a_3 \equal{} a_4 \plus{} a_5 \plus{} a_6 imply that the sequences

(a_1, a_2, a_3)

and

(a_4, a_5, a_6)

are identical up to a permutation. Let

A = \{a_0 = 0 < a_1 < a_2 < \cdots \}

,

B = \{b_0 = 0 < b_1 < b_2 < \cdots \}

be infinite sequences of real numbers with D(A) \equal{} D(B), where, for a set

X

of real numbers,

D(X)

denotes the difference set \{|x\minus{}y|\mid x, y \in X \}. Prove that if

A

is a

B_3

-set, then A \equal{} B.

algebranumber theoryCombinatorial Number TheorySequenceIMO Shortlist

Nice problem

Source: IMO Shortlist 2000, G6

11/2/2005

Let

ABCD

be a convex quadrilateral. The perpendicular bisectors of its sides

AB

and

CD

meet at

Y

. Denote by

X

a point inside the quadrilateral

ABCD

such that \measuredangle ADX \equal{} \measuredangle BCX < 90^{\circ} and \measuredangle DAX \equal{} \measuredangle CBX < 90^{\circ}. Show that \measuredangle AYB \equal{} 2\cdot\measuredangle ADX.

geometrycircumcircleperpendicular bisectorconvex quadrilateralIMO Shortlist

Almost every integer is sum of distinct perfect squares

Source: IMO Shortlist 2000, N6

8/10/2008

Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.

number theoryAdditive Number TheorySum of SquaresPerfect SquareIMO Shortlist

Number of ideal subsets, n+p and n+q belong to S

Source: IMO Shortlist 2000, C6

8/10/2008

Let

p

and

q

be relatively prime positive integers. A subset

S

of

\{0, 1, 2, \ldots \}

is called ideal if

0 \in S

and for each element

n \in S,

the integers n \plus{} p and n \plus{} q belong to

S.

Determine the number of ideal subsets of

\{0, 1, 2, \ldots \}.

modular arithmeticnumber theorycombinatoricsAdditive Number TheoryAdditive combinatoricsIMO ShortlistFrobenius

6

Problems(4)