MathDB

2

9 always remainder by 1-10 May Olympiad (Olimpiada de Mayo) 2019 L2 P1

A positive integer is called piola if the

9

is the remainder obtained by dividing it by

2, 3, 4, 5, 6, 7, 8, 9

and

10

and it's digits are all different and nonzero. How many piolas are there between

1

and

100000

?

two-digit square

Find all the two-digit numbers

\overline{ab}

that squared give a result where the last two digits are

\overline{ab}

.

2

Chess tournament

More than five competitors participated in a chess tournament. Each competitor played exactly once against each of the other competitors. Five of the competitors they each lost exactly two games. All other competitors each won exactly three games. There were no draws in the tournament. Determine how many competitors there were and show a tournament that verifies all conditions.

min sum of 2-products in top row May Olympiad (Olimpiada de Mayo) 2019 L2 P2

There is a board with

2020

squares in the bottom row and

2019

in the top row, located as shown shown in the figure. https://cdn.artofproblemsolving.com/attachments/f/3/516ad5485c399427638c3d1783593d79d83002.png In the bottom row the integers numbers from

1

to

2020

are placed in some order. Then in each box in the top row records the multiplication of the two numbers below it. How can they place the numbers in the bottom row so that the sum of the numbers in the top row be the smallest possible?

5

2

max triangles from regular n-gon May Olympiad (Olimpiada de Mayo) 2019 L2 P5

We consider the

n

vertices of a regular polygon with

n

sides. There is a set of triangles with vertices at these

n

points with the property that for each triangle in the set, the sides of at least one are not the side of any other triangle in the set. What is the largest amount of triangles that can have the set?
[hide=original wording]Consideramos los n vértices de un polígono regular de n lados. Se tiene un conjunto de triángulos con vértices en estos n puntos con la propiedad que para cada triángulo del conjunto, al menos uno de sus lados no es lado de ningún otro triángulo del conjunto. ¿Cuál es la mayor cantidad de triángulos que puede tener el conjunto?

Can $Ana$ complete the second row so that $Beto$ does not achieve his goal?

There is a board with three rows and

2019

columns. In the first row are written the numbers integers from

1

to

2019

inclusive, ordered from smallest to largest. In the second row,

Ana

writes those same numbers but ordered at your choice. In each box in the third row write the difference between the two numbers already written in the same column (the largest minus the smallest).

Beto

have to paint some numbers in the third row so that the sum of the numbers painted is equal to the sum of the numbers in that row that were left unpainted. Can

Ana

complete the second row so that

Beto

does not achieve his goal?

3

2

BQ = 2QC, CR = 2RA, < PRQ = 90^o , equal angles wanted

On the sides

AB, BC

and

CA

of a triangle

ABC

are located the points

P, Q

and

R

respectively, such that

BQ = 2QC, CR = 2RA

and

\angle PRQ = 90^o

. Show that

\angle APR =\angle RPQ

.

Determine the maximum number of distinct numbers the Gus list can contain.

Gus has to make a list of

250

positive integers, not necessarily distinct, such that each number is equal to the number of numbers in the list that are different from it. For example, if

15

is a number from the list so the list contains

15

numbers other than

15

. Determine the maximum number of distinct numbers the Gus list can contain.

4

2

Number Theory

Find the smallest positive integer

N

of two or more digits that has the following property: If we insert any non-null digit

d

between any two adjacent digits of

N

we obtain a number that is a multiple of

d

.

divide a square paper into 3 parts, by 2 straight cuts, create an obtuse

You have to divide a square paper into three parts, by two straight cuts, so that by locating these parts properly, without gaps or overlaps, an obtuse triangle is formed. Indicate how to cut the square and how to assemble the triangle with the three parts.

2019 May Olympiad

Subcontests

9 always remainder by 1-10 May Olympiad (Olimpiada de Mayo) 2019 L2 P1

two-digit square

Chess tournament

min sum of 2-products in top row May Olympiad (Olimpiada de Mayo) 2019 L2 P2

max triangles from regular n-gon May Olympiad (Olimpiada de Mayo) 2019 L2 P5

Can $Ana$ complete the second row so that $Beto$ does not achieve his goal?

BQ = 2QC, CR = 2RA, < PRQ = 90^o , equal angles wanted

Determine the maximum number of distinct numbers the Gus list can contain.

Number Theory

divide a square paper into 3 parts, by 2 straight cuts, create an obtuse

2019 May Olympiad

Subcontests

9 always remainder by 1-10 May Olympiad (Olimpiada de Mayo) 2019 L2 P1

two-digit square

Chess tournament

min sum of 2-products in top row May Olympiad (Olimpiada de Mayo) 2019 L2 P2

max triangles from regular n-gon May Olympiad (Olimpiada de Mayo) 2019 L2 P5

Can $Ana$ complete the second row so that $Beto$ does not achieve his goal?

BQ = 2QC, CR = 2RA, &lt; PRQ = 90^o , equal angles wanted

Determine the maximum number of distinct numbers the Gus list can contain.

Number Theory

divide a square paper into 3 parts, by 2 straight cuts, create an obtuse

BQ = 2QC, CR = 2RA, < PRQ = 90^o , equal angles wanted