MathDB

2020 May Olympiad

Part of May Olympiad

Subcontests

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Let $ABC$ be a right triangle, right at $B$

Let ABCABC be a right triangle, right at BB, and let MM be the midpoint of the side BCBC. Let PP be the point in bisector of the angle ∠BAC \angle BAC such that PMPM is perpendicular to BC(PBC (P is outside the triangle ABCABC). Determine the triangle area ABCABC if PM=1PM = 1 and MC=5MC = 5.

Maria has a $6 \times 5$ board with some shaded squares

Maria has a 6×56 \times 5 board with some shaded squares, as in the figure. She writes, in some order, the digits 1,2,3,41, 2, 3, 4 and 55 in the first row and then completes the board as follows: look at the number written in the shaded box and write the number that occupies the position indicated by the box shaded as the last number in the next row, and repeat the other numbers in the first four squares, following the same order as in the previous row. For example, if you wrote 2,3,4,1,52, 3, 4, 1, 5 in the first row, then since 44 is in the shaded box, the number that occupies the fourth place (1)(1) is written in the last box of the second row and completes it with the remaining numbers in the order in which. They were. She remains: 2,3,4,5,12, 3, 4, 5, 1. Then, to complete the third row, as in the shaded box is 33, the number located in the third place (4)(4) writes it in the last box and gets 2,3,5,1,42, 3, 5, 1, 4. Following in the same way, he gets the board of the figure. Show a way to locate the numbers in the first row to get the numbers 2,4,5,1,32, 4, 5, 1, 3 in the last row.
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There is a box with 2020 stones

There is a box with 2020 stones. Ana and Beto alternately play removing stones from the box and starting with Ana. Each player in turn must remove a positive number of stones that is capicua. Whoever leaves the box empty wins. Determine which of the two has a strategy winner and explain what that strategy is. Note:Note: A positive integer is capicua if it can be read equally from right to right. left and left to right. For example, 3, 22, 484 and 2002 are capicuas.

ant moving 1N, 2E, 3S, 3W, 5N, 6E, ...,41N (2020 May Olympiad L1 P3)

A clueless ant makes the following route: starting at point A A goes 1 1 cm north, then 2 2 cm east, then 3 3 cm south, then 4 4 cm west, immediately 5 5 cm north, continues 6 6 cm east, and so on, finally 41 41 cm north and ends in point B B . Calculate the distance between A A and B B (in a straight line).
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