2020 Portugal MO

Part of Portugal MO

Subcontests

(4)

1

sum of 1/ab, for primes a,b

Determine the fractions of a fraction of the form

\frac{1}{ab}

a,b

are prime natural numbers such that

0 < a < b \le 200

a + b > 200

1

alternating sums of {1,..,10}

Given a subset of

\{1,2,...,n\}

, we define its alternating sum in the following way: we order the elements of the subset in descending order and, starting with the largest, we alternately add and subtract the successive numbers. For example, the alternating sum of the set

\{1,3,4,6,8\}

8-6+4-3+1 = 4

. Determines the sum of the alternating sums of all subsets of

\{1,2,...,10\}

with an odd number of elements.

1

good positive integers - Portugal OPM 2020 p1

It is said that a positive integer is not GOOD, if there exists a permutation of the integers from 1 to n,

(a_1,a_2,...,a_n)

k + a_k

is a perfect square for all

k

5

is a GOOD number, since the permutation

(3,2,1,5,4)

checks the condition:

1 + 3 = 2^2

2 + 2 = 2^2

3 + 1 = 2^2

4 + 5 = 3^2

5 +4 = 3^2

. Find all GOOD numbers up to

12

1

computational, <C=2A, D \in AC , <ABD = <DBC, AB = 10, CD = 3,

[ABC]

\angle C = 2\angle A

D

is marked on the side

[AC]

\angle ABD = \angle DBC

AB = 10

CD = 3

, what is the length of the side

[BC]