MathDB

1

n rare coins of denominations 1,2,.., n into 5 boxes unter 4 conditions

A collector of rare coins has coins of denominations

1, 2,..., n

(several coins for each denomination). He wishes to put the coins into

5

boxes so that: (1) in each box there is at most one coin of each denomination; (2) each box has the same number of coins and the same denomination total; (3) any two boxes contain all the denominations; (4) no denomination is in all

5

boxes. For which

n

is this possible?

5

1

double angle in a triangle given, equal segments wanted

ABC

is a triangle with

AB < AC

and

\angle A = 2 \angle C

.

D

is the point on

AC

such that

CD = AB

. Let L be the line through

B

parallel to

AC

. Let

L

meet the external bisector of

\angle A

at

M

and the line through

C

parallel to

AB

at

N

. Show that

MD = ND

.

4

1

f(n,m)=a_1-a_2+a_3 -a_4+ ...+a _{2001} & a_{k+1} is residue of a_k^2 mod n

For positive integers

n, m

define

f(n,m)

as follows. Write a list of

2001

numbers

a_i

, where

a_1 = m

, and

a_{k+1}

is the residue of

a_k^2

mod \, n

(for

k = 1, 2,..., 2000

). Then put

f(n,m) = a_1-a_2 + a_3 -a_4 + a_5- ... + a_{2001}

. For which

n \ge 5

can we find m such that

2 \le m \le n/2

and

f(m,n) > 0

?

3

1

Mexican cyclic quadrilateral

ABCD

is a cyclic quadrilateral.

M

is the midpoint of

CD

. The diagonals meet at

P

. The circle through

P

which touches

CD

at

M

meets

AC

again at

R

and

BD

again at

Q

. The point

S

on

BD

is such that

BS = DQ

. The line through

S

parallel to

AB

meets

AC

at

T

. Show that

AT = RC

.

2

1

some colored balls (at least 3 different colors) and at least 3 boxes ...

Given some colored balls (at least three different colors) and at least three boxes. The balls are put into the boxes so that no box is empty and we cannot find three balls of different colors which are in three different boxes. Show that there is a box such that all the balls in all the other boxes have the same color.

1

7-digit numbers , multiples of 21 and which have each digit 3 or $7

Find all

7

-digit numbers which are multiples of

21

and which have each digit

3

or

7

.

2001 Mexico National Olympiad

Subcontests

n rare coins of denominations 1,2,.., n into 5 boxes unter 4 conditions

double angle in a triangle given, equal segments wanted

f(n,m)=a_1-a_2+a_3 -a_4+ ...+a _{2001} & a_{k+1} is residue of a_k^2 mod n

Mexican cyclic quadrilateral

some colored balls (at least 3 different colors) and at least 3 boxes ...

7-digit numbers , multiples of 21 and which have each digit 3 or $7

2001 Mexico National Olympiad

Subcontests

n rare coins of denominations 1,2,.., n into 5 boxes unter 4 conditions

double angle in a triangle given, equal segments wanted

f(n,m)=a_1-a_2+a_3 -a_4+ ...+a _{2001} &amp; a_{k+1} is residue of a_k^2 mod n

Mexican cyclic quadrilateral

some colored balls (at least 3 different colors) and at least 3 boxes ...

7-digit numbers , multiples of 21 and which have each digit 3 or $7

f(n,m)=a_1-a_2+a_3 -a_4+ ...+a _{2001} & a_{k+1} is residue of a_k^2 mod n