5
Part of 2016 May Olympiad
Problems(2)
game of areas in right triangle May Olympiad (Olimpiada de Mayo) 2016 L2 P5
Source:
9/25/2021
Rosa and Sara play with a triangle , right at . Rosa begins by marking two interior points of the hypotenuse , then Sara marks an interior point of the hypotenuse different from those of Rosa. Then, from these three points the perpendiculars to the sides and are drawn, forming the following figure.
https://cdn.artofproblemsolving.com/attachments/9/9/c964bbacc4a5960bee170865cc43902410e504.png
Sara wins if the area of the shaded surface is equal to the area of the unshaded surface, in other case wins Rosa. Determine who of the two has a winning strategy.
gamecombinatoricsgeometryareaswinning strategy
On the blackboard are written the $400$ integers $1, 2, 3, \cdots , 399, 400$
Source: May Olympiad 2016 L1 P5
3/14/2021
On the blackboard are written the integers . Luis erases of these numbers, then Martin erases another . Martin wins if the sum of the erased numbers equals the sum of those not deleted; otherwise, he wins Luis. Which of the two has a winning strategy? What if Luis deletes numbers and Martín deletes ?
In each case, explain how the player with the winning strategy can ensure victory.
combinatorics