MathDB

2

May Olympiad 2013, Level 2, Problem 5

600

cards,

200

of them having written the number

5

,

200

having a

2

, and the other

200

having a

1

, a student wants to create groups of cards such that the sum of the card numbers in each group is

9

. What is the maximum amount of groups that the student may create?

erase lengths on 8x8 board May Olympiad (Olimpiada de Mayo) 2013 L1 P5

An

8\times 8

square is drawn on the board divided into

64

1\times1

squares by lines parallel to the sides. Gustavo erases some segments of length

1

so that every

1\times 1

square he erases

0, 1

or

2

sides. Gustavo states that he erased

6

segments of length

1

from the edge of the

8\times 8

square and that the amount of

1\times 1

squares that have exactly

1

side erased is equal to

5

. Decide if what Gustavo said it may be true.

4

2

May Olympiad 2013, Level 2, Problem 4

Is it possible to write

100

odd numbers on a line such that the sum of any

5

consecutive numbers is a square number and the sum of any

9

consecutive numbers is also a square number?

5 numbers wanted, 10 sums given May Olympiad (Olimpiada de Mayo) 2013 L1 P4

Pablo wrote

5

numbers on one sheet and then wrote the numbers

6,7,8,8,9,9,10,10,11

and

12

on another sheet that he gave Sofia, indicating that those numbers are the possible sums of two of the numbers that he had hidden. Decide if with this information Sofia can determine the five numbers Pablo wrote .

3

2

May Olympiad 2013, Level 2, Problem 3

Many distinct points are marked in the plane. A student draws all the segments determined by those points, and then draws a line r that does not pass through any of the marked points, but cuts exactly

60

drawn segments. How many segments were not cut by r? Give all possibilites.

square paper folding (May Olympiad 2013 L1)

Let

ABCD

be a square of side paper

10

and

P

a point on side

BC

. By folding the paper along the

AP

line, point

B

determines the point

Q

, as seen in the figure. The line

PQ

cuts the side

CD

at

R

. Calculate the perimeter of the triangle

PCR

https://3.bp.blogspot.com/-ZSyCUznwutE/XNY7cz7reQI/AAAAAAAAKLc/XqgQnjm8DQYq6Q7fmCAKJwKt3ihoL8AuQCK4BGAYYCw/s400/may%2B2013%2Bl1.png

2

May Olympiad 2013, Level 2, Problem 2

Construct the midpoint of a segment using an unmarked ruler and a trisector that marks in a segment the two points that divide the segment in three equal parts.

ages problem Elisa and grandpa May Olympiad (Olimpiada de Mayo) 2013 L1 P2

Elisa adds the digits of her year of birth and observes that the result coincides with the last two digits of the year her grandfather was born. Furthermore, the last two digits of the year she was born are precisely the current age of her grandfather. Find the year Elisa was born and the year her grandfather was born.

1

2

May Olympiad 2013, Level 2, Problem 1

Sofia summed all the page numbers from a book starting at

1

and getting

2013

. Pablo saw how she did this and realized Sofia skipped a page. How many pages does the book have, and what page did Sofia skip?

2013 =sum 2 integers, no carry over May Olympiad (Olimpiada de Mayo) 2013 L1 P1

Find the number of ways to write the number

2013

as the sum of two integers greater than or equal to zero so that when adding there is no carry over.
Clarification: In the sum

2008+5=2013

there is carry over from the units to the tens

2013 May Olympiad

Subcontests

May Olympiad 2013, Level 2, Problem 5

erase lengths on 8x8 board May Olympiad (Olimpiada de Mayo) 2013 L1 P5

May Olympiad 2013, Level 2, Problem 4

5 numbers wanted, 10 sums given May Olympiad (Olimpiada de Mayo) 2013 L1 P4

May Olympiad 2013, Level 2, Problem 3

square paper folding (May Olympiad 2013 L1)

May Olympiad 2013, Level 2, Problem 2

ages problem Elisa and grandpa May Olympiad (Olimpiada de Mayo) 2013 L1 P2

May Olympiad 2013, Level 2, Problem 1

2013 =sum 2 integers, no carry over May Olympiad (Olimpiada de Mayo) 2013 L1 P1