Subcontests
(5)In a sports competition
In a sports competition in which several tests are carried out, only the three athletes A,B,C. In each event, the winner receives x points, the second receives y points, and the third receives z points. There are no ties, and the numbers x,y,z are distinct positive integers with x greater than y, and y greater than z.
At the end of the competition it turns out that A has accumulated 20 points, B has accumulated 10 points and C has accumulated 9 points. We know that athlete A was second in the 100-meter event. Determine which of the three athletes he was second in the jumping event. On the blackboard are written the $400$ integers $1, 2, 3, \cdots , 399, 400$
On the blackboard are written the 400 integers 1,2,3,⋯,399,400. Luis erases 100 of these numbers, then Martin erases another 100. Martin wins if the sum of the 200 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 101 numbers and Martín deletes 99?
In each case, explain how the player with the winning strategy can ensure victory. Given a board of $3 \times 3$
Given a board of 3×3 you want to write the numbers 1,2,3,4,5,6,7,8 and a number in their boxes positive integer M, not necessarily different from the above. The goal is that the sum of the three numbers in each row be the same
a) Find all the values of M for which this is possible.
b) For which of the values of M found in a) is it possible to arrange the numbers so that no only the three rows add the same but also the three columns add the same?