Subcontests
(5)50 stacks of coins that have 1,2,3,…,50 coins respectively
On the table there are 50 stacks of coins that have 1,2,3,…,50 coins respectively. Ana and Beto play the following game in turns:
First, Ana chooses one of the 50 piles on the table, and Beto decides if that pile is for Ana or for him.
Then, Beto chooses one of the 49 remaining piles on the table, and Ana decides if that pile is for her or for Beto.
They continue playing alternately in this way until one of the players has 25 batteries.
When that happens, the other player takes all the remaining stacks on the table and whoever has the most coins wins.
Determine which of the two players has a winning strategy. 1000 numbered cards placed in 100 numbered boxes, removing digits
There are 100 boxes that were labeled with the numbers 00, 01, 02,…, 99 . The numbers 000, 001, 002, …, 999 were written on a thousand cards, one on each card. Placing a card in a box is permitted if the box number can be obtained by removing one of the digits from the card number. For example, it is allowed to place card 037 in box 07, but it is not allowed to place the card 156 in box 65.Can it happen that after placing all the cards in the boxes, there will be exactly 50 empty boxes?
If the answer is yes, indicate how the cards are placed in the boxes; If the answer is no, explain why it is impossible junior area chasing, 3 equal segments, 2 equilaterals (2023 May Olympiad L1 p3)
On a straight line ℓ there are four points, A, B, C and D in that order, such that AB=BC=CD. A point E is chosen outside the straight line so that when drawing the segments EB and EC, an equilateral triangle EBC is formed . Segments EA and ED are drawn, and a point F is chosen so that when drawing the segments FA and FE, an equilateral triangle FAE is formed outside the triangle EAD. Finally, the lines EB and FA are drawn , which intersect at the point G. If the area of triangle EBD is 10, calculate the area of triangle EFG. deleting multiple or divisor in pairs from 2-50 on a blackboard
The 49 numbers 2,3,4,...,49,50 are written on the blackboard . An allowed operation consists of choosing two different numbers a and b of the blackboard such that a is a multiple of b and delete exactly one of the two. María performs a sequence of permitted operations until she observes that it is no longer possible to perform any more. Determine the minimum number of numbers that can remain on the board at that moment. min, max (a+c+e) if a+b+c+d+e=1002 pos. integers
Let a,b,c,d, and e be positive integers such that a≤b≤c≤d≤e and that a+b+c+d+e=1002.
a) Determine the largest possible value of a+c+e.
b) Determine the lowest possible value of a+c+e. 4 kids and their mothers, hidden chocolate eggs game
At Easter Day, 4 children and their mothers participated in a game in which they had to find hidden chocolate eggs. Augustine found 4 eggs, Bruno found 6, Carlos found 9 and Daniel found 12. Mrs. Gómez found the same number of eggs as her son, Mrs. Junco found twice as many eggs as her son, Mrs. Messi found three times as many eggs as her son, and Mrs. Núñez found five times as many eggs as her son. At the end of the day, they put all the eggs in boxes, with 18 eggs in each box, and only one egg was left over. Determine who the mother of each child is.