MathDB

N tennis player

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005

In a tennis tournament where

n

players

A_1,A_2,\dots,A_n

take part, any two players play at most one match, and k \leq \frac {n(n \minus{} 1)}{2}

2

matches are played. The winner of a match gets

1

point while the loser gets

0

. Prove that a sequence

d_1,d_2,\dots,d_n

of nonnegative integers can be the sequence of scores of the players (

d_i

being the score of

A_i

) if and only if (i)\ \ d_1 \plus{} d_2 \plus{} \dots \plus{} d_n \equal{} k, and

(ii)\ \text{for any} X\subset\{A_1,\dots,A_n\}

, the number of matches between the players in

X

is at most

\sum_{A_j\in X}d_j

combinatorics proposedcombinatorics

A1,...an

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005

Let

n

be a positive integer. Suppose

S

is a set of ordered n-\mbox{tuples} of nonnegative integers such that, whenever

(a_1,\dots,an)\in S

and

b_i

are nonnegative integers with

b_i\le a_i

, the

n-\text{tuple}

(b_1,\dots,b_n)

is also in

S

. If

h_m

is the number of elements of

S

with the sum of components equal to

m

, prove that

h_m

is a polynomial in

m

for all sufficiently large

m

.

algebrapolynomialalgebra proposed

Power of 2

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005

Does there exist a natural number

N

which is a power of

2

, such that one can permute its decimal digits to obtain a different power of

2

?

number theory proposednumber theory

2 circles

Source: Romanian IMO Team Selection Test TST 1999, problem 12; 17-th Iranian Math. Olympiad 1999/2000

5/1/2004

Two circles intersect at two points

A

and

B

. A line

\ell

which passes through the point

A

meets the two circles again at the points

C

and

D

, respectively. Let

M

and

N

be the midpoints of the arcs

BC

and

BD

(which do not contain the point

A

) on the respective circles. Let

K

be the midpoint of the segment

CD

. Prove that

\measuredangle MKN = 90^{\circ}

.

geometrycircumcircleparallelogramratiogeometric transformationreflectiontrigonometry

Semi circle

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005

Let us denote

\prod = \{(x, y) | y > 0\}

. We call a semicircle in

\prod

with center on the

x-\text{axis}

a semi-line. Two intersecting semi-lines determine four semi-angles. A bisector of a semi-angle is a semi-line that bisects the semi-angle. Prove that in every semi-triangle (determined by three semi-lines) the bisectors are concurrent.

geometry proposedgeometry

Sequence

Source: 17-th Iranian Mathematical Olympiad 1999/2000

12/14/2005

A sequence of natural numbers

c_1, c_2,\dots

is called perfect if every natural number

m

with

1\le m \le c_1 +\dots+ c_n

can be represented as

m =\frac{c_1}{a_1}+\frac{c_2}{a_2}+\dots+\frac{c_n}{a_n}

Given

n

, find the maximum possible value of

c_n

in a perfect sequence

(c_i)

.

inductionnumber theory proposednumber theory

1

Problems(6)