MathDB

3

Part of 1998 Iran MO (2nd round)

Problems(2)

Good n-numbers - Iran NMO 1998 (Second Round) Problem3

Source:

10/4/2010
Let nn be a positive integer. We call (a1,a2,⋯ ,an)(a_1,a_2,\cdots,a_n) a good n−n-tuple if ∑i=1nai=2n\sum_{i=1}^{n}{a_i}=2n and there doesn't exist a set of aia_is such that the sum of them is equal to nn. Find all good n−n-tuple. (For instance, (1,1,4)(1,1,4) is a good 3−3-tuple, but (1,2,1,2,4)(1,2,1,2,4) is not a good 5−5-tuple.)
combinatorics proposedcombinatorics
f(A,B) - Iran NMO 1998 (Second Round) Problem6

Source:

10/4/2010
If A=(a1,⋯ ,an)A=(a_1,\cdots,a_n) , B=(b1,⋯ ,bn)B=(b_1,\cdots,b_n) be 22 n−n-tuple that ai,bi=0 or 1a_i,b_i=0 \ or \ 1 for i=1,2,⋯ ,ni=1,2,\cdots,n, we define f(A,B)f(A,B) the number of 1≤i≤n1\leq i\leq n that ai≠bia_i\ne b_i. For instance, if A=(0,1,1)A=(0,1,1) , B=(1,1,0)B=(1,1,0), then f(A,B)=2f(A,B)=2. Now, let A=(a1,⋯ ,an)A=(a_1,\cdots,a_n) , B=(b1,⋯ ,bn)B=(b_1,\cdots,b_n) , C=(c1,⋯ ,cn)C=(c_1,\cdots,c_n) be 3 n−n-tuple, such that for i=1,2,⋯ ,ni=1,2,\cdots,n, ai,bi,ci=0 or 1a_i,b_i,c_i=0 \ or \ 1 and f(A,B)=f(A,C)=f(B,C)=df(A,B)=f(A,C)=f(B,C)=d. a)a) Prove that dd is even. b)b) Prove that there exists a n−n-tuple D=(d1,⋯ ,dn)D=(d_1,\cdots,d_n) that di=0 or 1d_i=0 \ or \ 1 for i=1,2,⋯ ,ni=1,2,\cdots,n, such that f(A,D)=f(B,D)=f(C,D)=d2f(A,D)=f(B,D)=f(C,D)=\frac{d}{2}.
combinatorics proposedcombinatorics