MathDB

1

a_i elements of sequence equal to i

(a_0,...,a_k)

is said to be medaled if, for each

i = 0,...,k

, there are exactly

a_i

elements of the sequence equal to

i

. For example,

(1,2,1,0)

is a medaled seqence. Indicates all medaled sequences

(a_0,...,a_{2015})

.

6

1

n points, each point hase at least 3 points at distance

For what values of

n

is it possible to mark

n

points on the plane so that each point has at least three other points at distance

1

?

3

1

n-m is prime, m, n from 1-2015

The numbers from

1

to

2015

are written on sheets so that if if

n-m

is a prime, then

n

and

m

are on different sheets. What is the minimum number of sheets required?

1

firm 3-digit numbers - Portugal OPM 2015 p1

A number of three digits is said to be firm when it is equal to the product of its unit digit by a number formed by the remaining digits. For example,

153

is firm because

153 = 3 \times 51

. How many firm numbers are there?

4

1

area chasing inside a parallelogram

Let

[ABCD]

be a parallelogram and

P

a point between

C

and

D

. The line parallel to

AD

that passes through

P

intersects the diagonal

AC

in

Q

. Knowing that the area of

[PBQ]

is

2

and the area of

[ABP]

is

6

, determine the area of

[PBC]

. https://cdn.artofproblemsolving.com/attachments/0/8/664a00020065b7ad6300a062613fca4650b8d0.png

2

1

angle <ABC wanted if ABC,ACD,BCD are all isosceles with D between A,B

Let

[ABC]

be a triangle and

D

a point between

A

and

B

. If the triangles

[ABC], [ACD]

and

[BCD]

are all isosceles, what are the possible values of

\angle ABC

?

2015 Portugal MO

Subcontests

a_i elements of sequence equal to i

n points, each point hase at least 3 points at distance

n-m is prime, m, n from 1-2015

firm 3-digit numbers - Portugal OPM 2015 p1

area chasing inside a parallelogram

angle <ABC wanted if ABC,ACD,BCD are all isosceles with D between A,B

2015 Portugal MO

Subcontests

a_i elements of sequence equal to i

n points, each point hase at least 3 points at distance

n-m is prime, m, n from 1-2015

firm 3-digit numbers - Portugal OPM 2015 p1

area chasing inside a parallelogram

angle &lt;ABC wanted if ABC,ACD,BCD are all isosceles with D between A,B

angle <ABC wanted if ABC,ACD,BCD are all isosceles with D between A,B