MathDB

2

2003 green points are contained in a triangle T of area <4

2003

green points on the plane, so all triangles with three green vertices have area less than

1

. Prove that the

2003

green points are contained in a triangle

T

of area less than

4

.

list of numers game

Celia chooses a number

n

and writes the list of natural numbers from

1

to

n

:

1, 2, 3, 4, ..., n-1, n.

At each step, it changes the list: it copies the first number to the end and deletes the first two. After

n-1

steps a single number will be written. For example, for

n=6

the five steps are:

1,2,3,4,5,6 \to 3,4,5,6,1 \to 5,6,1,3 \to 1,3,5 \to 5,1 \to 5

and the number

5

is written. Celia chose a number

n

between

1000

and

3000

and after

n-1

steps the number

1

was written. Determine all the values of

n

that Celia could have chosen. Justify why those values work, and the others do not.

3

2

a\(8b+1) and b \(8a+1)

Find all pairs of positive integers

(a,b)

such that

8b+1

is a multiple of

a

and

8a+1

is a multiple of

b

.

min multiple of 56, ending in 56 with sum of digits 56

Find the smallest positive integer that ends in

56

, is a multiple of

56

, and has the sum of its digits equal to

56

.

1

2

greatest possible no of 4-digit numbers, multiples of 36

Four digits

a, b, c, d

, different from each other and different from zero, are chosen and the list of all the four-digit numbers that are obtained by exchanging the digits

a, b, c, d

is written. What digits must be chosen so that the list has the greatest possible number of four-digit numbers that are multiples of

36

?

abcd 4-digit wanted b=a+2, c=a+2, d=c+2

Pedro writes all the numbers with four different digits that can be made with digits

a, b, c, d

, that meet the following conditions:

a\ne 0 \, , \, b=a+2 \, , \, c=b+2 \, , \, d=c+2

Find the sum of all the numbers Pedro wrote.

5

2

shortest path of movement of an ant on the surface of a cube, returning in start

An ant, which is on an edge of a cube of side

8

, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.

chess knight moves on 2x2 board

We have a

4 \times 4

squared board. We define the separation between two squares as the least number of moves that a chess knight must take to go from one square to the other (using moves of the knight). Three boxes

A, B, C

form a good trio if the three separations between

A

and

B

, between

A

and

C

and between

B

and

C

are equal. Determines the number of good trios that are formed on the board.
Clarification: In each move the knight moves

2

squares in the horizontal direction plus one square in the vertical direction or moves

2

squares in the vertical direction plus one square in the horizontal direction.

2

calculations in a rectangle

Let

ABCD

be a rectangle of sides

AB = 4

and

BC = 3

. The perpendicular on the diagonal

BD

drawn from

A

cuts

BD

at point

H

. We call

M

the midpoint of

BH

and

N

the midpoint of

CD

. Calculate the measure of the segment

MN

.

area of a triangle, right triangle related (May Olympiad 2003 L1)

The triangle

ABC

is right in

A

and

R

is the midpoint of the hypotenuse

BC

. On the major leg

AB

the point

P

is marked such that

CP = BP

and on the segment

BP

the point

Q

is marked such that the triangle

PQR

is equilateral. If the area of triangle

ABC

is

27

, calculate the area of triangle

PQR

.

2003 May Olympiad

Subcontests

2003 green points are contained in a triangle T of area <4

list of numers game

a\(8b+1) and b \(8a+1)

min multiple of 56, ending in 56 with sum of digits 56

greatest possible no of 4-digit numbers, multiples of 36

abcd 4-digit wanted b=a+2, c=a+2, d=c+2

shortest path of movement of an ant on the surface of a cube, returning in start

chess knight moves on 2x2 board

calculations in a rectangle

area of a triangle, right triangle related (May Olympiad 2003 L1)

2003 May Olympiad

Subcontests

2003 green points are contained in a triangle T of area &lt;4

list of numers game

a\(8b+1) and b \(8a+1)

min multiple of 56, ending in 56 with sum of digits 56

greatest possible no of 4-digit numbers, multiples of 36

abcd 4-digit wanted b=a+2, c=a+2, d=c+2

shortest path of movement of an ant on the surface of a cube, returning in start

chess knight moves on 2x2 board

calculations in a rectangle

area of a triangle, right triangle related (May Olympiad 2003 L1)

2003 green points are contained in a triangle T of area <4