MathDB

1

2013 numberd from 1-2013

2013

cards that have written, each one, a different integer number, from

1

to

2013

; all the cards face down (you can't see what number they are). It is allowed to select any set of cards and ask if the average of the numbers written on those cards is integer. The answer will be true. a) Find all the numbers that can be determined with certainty by several of these questions. b) We want to divide the cards into groups such that the content of each group is known even though the individual value of each card in the group is not known. (For example, finding a group of

3

cards that contains

1, 2

, and

3

, without knowing what number each card has.) What is the maximum number of groups that can be obtained?

3

1

2013-digit numbers wanted. sum of pairwise product of digits = 1 mod 4

Find how many are the numbers of

2013

digits

d_1d_2…d_{2013}

with odd digits

d_1,d_2,…,d_{2013}

such that the sum of

1809

terms

d_1 \cdot d_2+d_2\cdot d_3+…+d_{1809}\cdot d_{1810}

has remainder

1

when divided by

4

and the sum of

203

terms

d_{1810}\cdot d_{1811}+d_{1811}\cdot d_{1812}+…+d_{2012}\cdot d_{2013}

has remainder

1

when dividing by

4

.

5

1

last numbers on board from 1-2013

Given several nonnegative integers (repetitions allowed), the allowed operation is to choose a positive integer

a

and replace each number

b

greater than or equal to

a

by

b-a

(the numbers

a

, if any, are replaced by

0

). Initially, the integers from

1

are written on the blackboard until

2013

inclusive. After a few operations the numbers on the board have a sum equal to

10

. Determine what the numbers that remained on the board could be. Find all the possibilities.

2

1

computational, convex ABCD,<A=<C, B bisector passes midpoint of CD, CD=3AB

In a convex quadrilateral

ABCD

the angles

\angle A

and

\angle C

are equal and the bisector of

\angle B

passes through the midpoint of the side

CD

. If it is known that

CD = 3AD

, calculate

\frac{AB}{BC}

.

6

1

Pretty numbers

A positive integer

n

is called pretty if there exists two divisors

d_1,d_2

of

n

(1\leq d_1,d_2\leq n)

such that

d_2-d_1=d

for each divisor

d

of

n

(where

1<d<n

). Find the smallest pretty number larger than

401

that is a multiple of

401

.

4

1

Inequality with 3 variables

Let

x\geq 5, y\geq 6, z\geq 7

such that

x^2+y^2+z^2\geq 125

. Find the minimum value of

x+y+z

.

2013 Argentina National Olympiad

Subcontests

2013 numberd from 1-2013

2013-digit numbers wanted. sum of pairwise product of digits = 1 mod 4

last numbers on board from 1-2013

computational, convex ABCD,<A=<C, B bisector passes midpoint of CD, CD=3AB

Pretty numbers

Inequality with 3 variables

2013 Argentina National Olympiad

Subcontests

2013 numberd from 1-2013

2013-digit numbers wanted. sum of pairwise product of digits = 1 mod 4

last numbers on board from 1-2013

computational, convex ABCD,&lt;A=&lt;C, B bisector passes midpoint of CD, CD=3AB

Pretty numbers

Inequality with 3 variables

computational, convex ABCD,<A=<C, B bisector passes midpoint of CD, CD=3AB