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1.	Show that from a set of 11 square integers one can select six numbers a2, b2, c2, d2, e2, f2 such thata2 + b2 + c2 ≡ d2 + e2 + f2 (mod 12)
Answer» If I² ≡ x(mod 12) then x can take 4-values 0, 1, 4, 9. Now let set L₁ contains all number which are congruent mod 12. similarly, L₂, L₃, L₄ are defined Now if out of 11-given square integers if C-1 6 or more numbers belong to either of L_i, i = 1, 2, 3, 4 then we are through select 6 out of these numbers and these are required a², b²,.......,f². C-2 If 5 or 4 are selected from a lot say from Li then at least 2. must come from one of lot out of remaining 3 say L_j. Now select 4-from L_i and 2 from L_j and set a², b², d², e² from first 4 and c², f² from second set. C-3 Now if maximum 3 are selected from one set out of '4' then selection must be 3, 3, 3, 2 Now say L_i, L_j, L_k three are selected where (i ≠ j ≠ k) then select 2-from L_iand set then as a² and d². Similarly select 2 from L_j and set then as b², e² and select 2-from L_k and set then as c², f². Clearly from this, given condition is satisfied. Note that if α is maximum numbers of elements selected from a set out of four sets then α ≥ 3.

Show that from a set of 11 square integers one can select six numbers a2, b2, c2, d2, e2, f2 such thata2 + b2 + c2 ≡ d2 + e2 + f2 (mod 12)

Answer»

If I² ≡ x(mod 12) then x can take 4-values 0, 1, 4, 9.

Now let set L₁ contains all number which are congruent mod 12.

similarly, L₂, L₃, L₄ are defined

Now if out of 11-given square integers if

C-1

6 or more numbers belong to either of L_i, i = 1, 2, 3, 4 then we are through select 6 out of these numbers and these are required a², b²,.......,f².

C-2

If 5 or 4 are selected from a lot say from Li then at least 2. must come from one of lot out of remaining 3 say L_j. Now select 4-from L_i and 2 from L_j and set a², b², d², e² from first 4 and c², f² from second set.

C-3

Now if maximum 3 are selected from one set out of '4' then selection must be 3, 3, 3, 2 Now say L_i, L_j, L_k three are selected

where (i ≠ j ≠ k) then select 2-from L_iand set then as a² and d². Similarly select 2 from L_j and set then as b², e² and select 2-from L_k and set then as c², f². Clearly from this, given condition is satisfied.

Note that if α is maximum numbers of elements selected from a set out of four sets then α ≥ 3.

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