MathDB

3

Part of 2009 May Olympiad

Problems(2)

sum game 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13

Source: XV May Olympiad (Olimpiada de Mayo) 2009 L2 P3

9/19/2022
In the following sum: 1+2+3+4+5+61 + 2 + 3 + 4 + 5 + 6, if we remove the first two “+” signs, we obtain the new sum 123+4+5+6=138123 + 4 + 5 + 6 = 138. By removing three “++” signs, we can obtain 1+23+456=4801 + 23 + 456 = 480. Let us now consider the sum 1+2+3+4+5+6+7+8+9+10+11+12+131 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13, in which some “++” signs are to be removed. What are the three smallest multiples of 100100 that we can get in this way?
algebra
26 cards with 1 number each , two of 1-13

Source: XV May Olympiad (Olimpiada de Mayo) 2009 L1 P3

9/22/2022
There are 2626 cards and each one has a number written on it. There are two with 11, two with 22, two with 33, and so on up to two with 1212 and two with 1313. You have to distribute the 2626 cards in piles so that the following two conditions are met: ∙\bullet If two cards have the same number they are in the same pile. ∙\bullet No pile contains a card whose number is equal to the sum of the numbers of two cards in that same pile. Determine what is the minimum number of stacks to make. Give an example with the distribution of the cards for that number of stacks and justify why it is impossible to have fewer stacks.
Clarification: Two squares are neighbors if they have a common side.
combinatorics