MathDB

2007 May Olympiad

Part of May Olympiad

Subcontests

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8 children, all of different heights

Eight children, all of different heights, must form an orderly line from smallest to largest. We will say that the row has exactly one error if there is a child that is immediately behind another taller than it, and everyone else (except the first in line) is immediately behind a shorter one. of how many ways the eight children can line up with exactly one mistake?

6 positive integers one at each side of a cube

Jorge chooses 66 different positive integers and writes one on each face of a cube. He threw his bucket three times. The first time his cube showed the number 55 facing up and also the sum of the numbers on the faces sides was 2020. The second time his cube showed the number 77 facing up and also the sum of the numbers on the faces sides was 1717. The third time his cube showed the number 44 up, plus all the numbers on the side faces. They turned out to be primes. What are the numbers that Jorge chose and how did he distribute them on the faces of the cube? Analyze all odds.
Remember that 11 is not prime.
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Prime numbers

Alex and Bruno play the following game: each one, in your turn, the player writes, exactly one digit, in the right of the last number written. The game finishes if we have a number with 66 digits( distincts ) and Alex starts the game. Bruno wins if the number with 66 digits is a prime number, otherwise Alex wins. Which player has the winning strategy?

one lamp at each square of a 7x7 board

A 7×77\times 7 board has a lamp on each of its 4949 squares, which can be on or off. The allowed operation is to choose 33 consecutive cells of a row or a column that have two lamps neighboring each other on and the other off, and change the state of all three. Namely https://cdn.artofproblemsolving.com/attachments/e/b/28737b19c940ff5e1c98d05533c77069e990f5.png Give a configuration of exactly 88 lit lamps located in the first 44 rows of the board such that, through a succession of permitted operations, a single lamp is lit on the board and that it is located in the last row. Show the sequence of operations used to achieve the goal.
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Angles and segments

In the triangle ABCABC we have ∠A=2∠C\angle A = 2\angle C and 2∠B=∠A+∠C2\angle B = \angle A + \angle C. The angle bisector of ∠C\angle C intersects the segment ABAB in EE, let FF be the midpoint of AEAE, let ADAD be the altitude of the triangle ABCABC. The perpendicular bisector of DFDF intersects ACAC in MM. Prove that AM=CMAM = CM.

paper folding a pentagon (May Olympiad 2007 L1)

You have a paper pentagon, ABCDEABCDE, such that AB=BC=3AB = BC = 3 cm, CD=DE=5CD = DE= 5 cm, EA=4EA = 4 cm, ∠ABC=100o\angle ABC = 100^o ,∠CDE=80o \angle CDE = 80^o. You have to divide the pentagon into four triangles, by three straight cuts, so that with the four triangles assemble a rectangle, without gaps or overlays. (The triangles can be rotated and / or turned around.)