MathDB

Determine a_2003

Source:

11/3/2010

A sequence

(a_n)

is defined as follows:

a_1 = p

is a prime number with exactly

300

nonzero digits, and for each

n \geq 1, a_{n+1}

is the decimal period of

1/a_n

multiplies by

2

. Determine

a_{2003}.

algebra unsolvedalgebra

very hard(Russia 2003)(help me)

Source: V. Dolnikov, P. Karasev

8/5/2009

A finite set of points

X

and an equilateral triangle

T

are given on a plane. Suppose that every subset

X'

of

X

with no more than

9

elements can be covered by two images of

T

under translations. Prove that the whole set

X

can be covered by two images of

T

under translations.

analytic geometryemailgeometrygeometric transformationhomothetycombinatorics unsolvedcombinatorics

If H1H2 passes through X, then it also passes through Y

Source:

11/3/2010

Let

B

and

C

be arbitrary points on sides

AP

and

PD

respectively of an acute triangle

APD

. The diagonals of the quadrilateral

ABCD

meet at

Q

, and

H_1,H_2

are the orthocenters of triangles

APD

and

BPC

, respectively. Prove that if the line

H_1H_2

passes through the intersection point

X \ (X \neq Q)

of the circumcircles of triangles

ABQ

and

CDQ

, then it also passes through the intersection point

Y \ (Y \neq Q)

of the circumcircles of triangles

BCQ

and

ADQ.

geometrycircumcirclegeometry proposed

The inscribed sphere of a tetrahedron

Source: Serbia and Montanagro 2003 \grade 11\second day

8/3/2005

The inscribed sphere of a tetrahedron

ABCD

touches

ABC,ABD,ACD

and

BCD

at

D_1,C_1,B_1

and

A_1

respectively. Consider the plane equidistant from

A

and plane

B_1C_1D_1

(parallel to

B_1C_1D_1

) and the three planes defined analogously for the vertices

B,C,D

. Prove that the circumcenter of the tetrahedron formed by these four planes coincides with the circumcenter of tetrahedron of

ABCD

.

geometry3D geometryspheretetrahedroncircumcircleradical axisgeometry unsolved

Find the greatest natural number N

Source:

11/4/2010

Find the greatest natural number

N

such that, for any arrangement of the numbers

1, 2, \ldots, 400

in a chessboard

20 \times 20

, there exist two numbers in the same row or column, which differ by at least

N.

symmetryceiling functiongeometryrectanglecombinatorics unsolvedcombinatorics

palindromic word

Source: Serbia and Montanagro 2004

8/7/2005

Ana and Bora are each given a sufficiently long paper strip, one with letter

A

written , and the other with letter

B

. Every minute, one of them (not necessarily one after another) writes either on the left or on the right to the word on his/her strip the word written on the other strip. Prove that the day after, one will be able to cut word on Ana's strip into two words and exchange their places, obtaining a palindromic word.

inductionblogscombinatorics unsolvedcombinatorics

4

Problems(6)