2021 May Olympiad

Part of May Olympiad

Subcontests

(5)

2

numbers 1-13 on vertices of 13-gon

At each vertex of a

13

-sided polygon we write one of the numbers

1,2,3,…, 12,13

, without repeating. Then, on each side of the polygon we write the difference of the numbers of the vertices of its ends (the largest minus the smallest). For example, if two consecutive vertices of the polygon have the numbers

2

11

9

is written on the side they determine. a) Is it possible to number the vertices of the polygon so that only the numbers

3, 4

5

are written on the sides? b) Is it possible to number the vertices of the polygon so that only the numbers

3, 4

6

are written on the sides?

equal areas by 2 straight cuts in a convex quadr. 2021 May Olympiad L1 p4

Facundo and Luca have been given a cake that is shaped like the quadrilateral in the figure. https://cdn.artofproblemsolving.com/attachments/3/2/630286edc1935e1a8dd9e704ed4c813c900381.png They are going to make two straight cuts on the cake, thus obtaining

4

portions in the shape of a quadrilateral. Then Facundo will be left with two portions that do not share any side, the other two will be for Luca. Show how they can cut the cuts so that both children get the same amount of cake. Justify why cutting in this way achieves the objective.

2

100 numbers and 1 perfect cube

Prove that there are

100

distinct positive integers

n_1,n_2,\dots,n_{99},n_{100}

\frac{n_1^3+n_2 ^3+\dots +n_{100}^3}{100}

is a perfect cube.

Sum of 16 < Sum of 10

36

consecutive positive integers in a white paper(in ascending order), next he computes the sum of digits of each one of

36

numbers(in the order) and writes the first

16

results in a red paper and the last

10

results in a blue paper. Determine if Bob can choose the

36

integers, such that the sum of the numbers in the red paper is less than or equal to sum of the numbers in the blue paper.

2

60, 30, 90 and + angles

ABC

be a triangle and

D

is a point inside of the triangle, such that

\angle DBC=60^{\circ}

\angle DCB=\angle DAB=30^{\circ}

M

N

be the midpoints of

AC

BC

, respectively. Prove that

\angle DMN=90^{\circ}

max no of Tuesday the 13th in a year (2021 May Olympiad L1 p3)

In a year that has

365

days, what is the maximum number of "Tuesday the

13

th" there can be?
Note: The months of April, June, September and November have

30

days each, February has

28

and all others have

31

2

special and common numbers

N

be a positive integer; a divisor of

N

is called common if it's great than

1

and different of

N

. A positive integer is called special if it has, at least, two common divisors and it is multiple of all possible differences between any two of their common divisors. Find all special integers.

2 colours in a 2x8 board (2021 May Olympiad L1 p2)

2 \times 8

squared board, you want to color each square red or blue in such a way that on each

2 \times 2

sub-board there are at least

3

boxes painted blue. In how many ways can this coloring be done?
Note. A

2 \times 2

board is a square made up of

4

squares that have a common vertex.

2

Split in sum groups

On a board the numbers

1,2,3,\dots,98,99

are written. One has to mark

50

of them, such that the sum of two marked numbers is never equal to

99

100

. How many ways one can mark these numbers?

distance between collinear points (2021 May Olympiad L1 p1)

In a forest there are

5

A, B, C, D, E

that are in that order on a straight line. At the midpoint of

AB

there is a daisy, at the midpoint of

BC

there is a rose bush, at the midpoint of

CD

there is a jasmine, and at the midpoint of

DE

there is a carnation. The distance between

A

E

28

m; the distance between the daisy and the carnation is

20

m. Calculate the distance between the rose bush and the jasmine.