MathDB

2018 May Olympiad

Part of May Olympiad

Subcontests

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max last number , 1-16 in 4x4 board May Olympiad (Olimpiada de Mayo) 2018 L2 P2

On a board 4×44\times 4 the numbers from 11 to 1616 are written, one in each box. Andres and Pablo choose four numbers each. Andrés chooses the biggest of each row and Pablo, the biggest of each column. The same number can be chosen by both. Then they are removed from the board all chosen numbers. What is the greatest value that the sum of the numbers can have what are left on the board?

max remainder when 2018 is divided by 1-1000 , 2018 May Olympiad L1 p2

A thousand integer divisions are made: 20182018 is divided by each of the integers from 1 1 to 10001000. Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger?
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computational geometry for 15yo, distance inside a parallelogram

In a parallelogram ABCDABCD, let MM be the point on the BCBC side such that MC=2BMMC = 2BM and let NN be the point of side CDCD such that NC=2DNNC = 2DN. If the distance from point BB to the line AMAM is 33, calculate the distance from point NN to the line AMAM.

7 pos. integers with sum 36 around a circle 2018 May Olympiad L1 p4

Anna must write 77 positive integers, not necessarily distinct, around a circle such that the following conditions are met: ∙\bullet The sum of the seven numbers equals 3636. ∙\bullet If two numbers are neighbours, the difference between the largest and the smallest is equal to 22 or 33. Find the maximum value of the largest of the numbers that Anna can write.
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Liars and knights

The 20182018 inhabitants of a city are divided in two groups: the knights(only speak the truth) and the liars(only speak the lie). The inhabitants sat in a circle and everybody spoke "My two neighbours(in the left and in the right) are liars". After this, one inhabitant got off the circle. The 20172017 inhabitants sat again in a circle(not necessarily in the same order), and everybody spoke "None of my two neighbours(in the left and in the right) is of the same group of myself" Can we determine the group of the inhabitant that got off the city?

computational geometry for 13yo, segment inside a regular 10-gon

Let ABCDEFGHIJABCDEFGHIJ be a regular 1010-sided polygon that has all its vertices in one circle with center OO and radius 55. The diagonals ADAD and BEBE intersect at PP and the diagonals AHAH and BIBI intersect at QQ. Calculate the measure of the segment PQPQ.