Combinatorics #1

Given any 99 integers, show that it is possible to choose w,x,y,zw,x,y,z such that w+xyz0w+x-y-z\equiv0(mod(\text{mod} 20)20).

#Combinatorics

Note by Victor Loh
6 years, 11 months ago

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Comments

The condition is the same as:w+xy+z(mod20)w+x\equiv y+z \pmod {20}. Notice that if we can find wy,xzw\equiv y, x\equiv z, then we are done. Hence it suffices to consider the case where there's at most one residue of 2020 such that at most 33 numbers are congruent to it.

In this case, we have at least 77 distinct residues considering each integer in mod 20 . Note that there are at least (72)=21{7\choose 2}=21 ways to choose 22 of the numbers and then take their sum. Using the 2020 mod 20 residues as pigeon holes, there exist a residue rr with at least two sums i.e w+xy+zr(mod20)w+x\equiv y+z\equiv r \pmod {20}(we can't have w=yw=y because that would imply xzx\equiv z which contradicts having distinct residues) and we are done.

Xuming Liang - 6 years, 11 months ago

The easy bash: The 10 possible sets of end digits of the 9 integers are [1,2,3,4,5,6,7,8,9],[0,1,2,3,4,5,6,7,8],[2,3,4,5,6,7,8,9,0],[3,4,5,6,7,8,9,0,1][1,2,3,4,5,6,7,8,9], [0,1,2,3,4,5,6,7,8], [2,3,4,5,6,7,8,9,0] ,[3,4,5,6,7,8,9,0,1] [4,5,6,7,8,9,0,1,2],[5,6,7,8,9,1,2,3,4,],[6,7,8,9,1,2,3,4,5][4,5,6,7,8,9,0,1,2] , [5,6,7,8,9,1,2,3,4,] ,[6,7,8,9,1,2,3,4,5][7,8,9,1,2,3,4,5,6],[8,9,1,2,3,4,5,6,7]and[9,0,1,2,3,4,5,6,7][7,8,9,1,2,3,4,5,6],[8,9,1,2,3,4,5,6,7] and [9,0,1,2,3,4,5,6,7]It suffices to show that 1+928=01+9-2-8=0 and 8+071=08+0-7-1=0 as by Pigeonhole Principle, each of the other 8 sets is just a rearrangement of the first 2. Not very elegant but it works. Since this works for single digit numbers, it works for any other set because each number in that set is basically adding a multiple of 20 to the original 1-digit set. (Not clear, I know but basically, I'm saying that for 10+19121810+19-12-18, it works because it is 1+928+10+1010101+9-2-8+10+10-10-10, or 1+9+2+8+01+9+2+8+0.)

Joshua Ong - 6 years, 11 months ago

one more question, given the series:1,3,5,7,9,11,13,15. now choose 5 digits from here whose sum is equal to 30.(u can take a number twice also)

sarfraz Ahmed - 6 years, 11 months ago

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Odd×5=OddOdd\times5=Odd

Joshua Ong - 6 years, 11 months ago
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