MathDB

1

tiles 3x1 and 2-1 in 7x 7board

7\times 7

board divided into

49

boxes. Mateo places a coin in a box.
a) Prove that Mateo can place the coin so that it is impossible for Emi to completely cover the

48

remaining squares, without gaps or overlaps, using

15

3\times1

rectangles and a cubit of three squares, like those in the figure. https://cdn.artofproblemsolving.com/attachments/6/9/a467439094376cd95c6dfe3e2ad3e16fe9f124.png b) Prove that no matter which square Mateo places the coin in, Emi will always be able to cover the 48 remaining squares using

14

3\times1

rectangles and two cubits of three squares.

4

1

products of 50x50 numbers on 50x50 board

There is a

50\times 50

grid board.. Carlos is going to write a number in each box with the following procedure. He first chooses

100

distinct numbers that we denote

f_1,f_2,f_3,…,f_{50},c_1,c_2,c_3,…,c_{50}

among which there are exactly

50

that they are rational. Then he writes in each box (

i,j)

the number

f_i \cdot c_j

(the multiplication of

f_i

by

c_j

). Determine the maximum number of rational numbers that the squares on the board can contain.

5

1

30 colors for 2018 points on plane

In the plane you have

2018

points between which there are not three on the same line. These points are colored with

30

colors so that no two colors have the same number of points. All triangles are formed with their three vertices of different colors. Determine the number of points for each of the

30

colors so that the total number of triangles with the three vertices of different colors is as large as possible.

2

1

n numbered kinghts and a round table with n chairs

There are

n

knights numbered

1

to

n

and a round table with

n

chairs. The first knight chooses his chair, and from him, the knight number

k+1

sits

k

places to the right of knight number

k

, for all

1 \le k\le n-1

(occupied and empty seats are counted). In particular, the second knight sits next to the first. Find all values of

n

such that the

n

gentlemen occupy the

n

chairs following the described procedure.

1

r= p mod 210, r =a^2+b^2

Let

p

a prime number and

r

the remainder of the division of

p

by

210

. It is known that

r

is a composite number and can be written as the sum of two non-zero perfect squares. Find all primes less than

2018

that satisfy these conditions.

6

1

exists circle passing through E,F and tangent to CB,CD, parallelogram

Let

ABCD

be a parallelogram. An interior circle of the

ABCD

is tangent to the lines

AB

and

AD

and intersects the diagonal

BD

at

E

and

F

. Prove that exists a circle that passes through

E

and

F

and is tangent to the lines

CB

and

CD

.

2018 Argentina National Olympiad

Subcontests

tiles 3x1 and 2-1 in 7x 7board

products of 50x50 numbers on 50x50 board

30 colors for 2018 points on plane

n numbered kinghts and a round table with n chairs

r= p mod 210, r =a^2+b^2

exists circle passing through E,F and tangent to CB,CD, parallelogram