Find the number of 10-tuples (a1; a2;....., a10) of integers such that

1.	Find the number of 10-tuples (a1; a2;....., a10) of integers such that \|a1\| ≤ 1 and a21 + a22 + a23+ ...... + a210 - a1a2 - a2a3 - a3a4 -.... a9a10 - a10a1 = 2.
Answer» Let a₁₁= a₁. Multiplying the given equation by 2 we get (a₁- a₂₎2 + (a₂ - a₃)2 + .... (a₁₀- a₁)² = 4. Note that if a_i - a_i+1 = ±2 for some i = 1; .....10, then a_j - a_j+1= 0 for all j ≠ i which contradicts the equality ∑¹⁰_i=1(a_i - a_i+1) = 0. Therefore a_i- a_i+1 = 1 for exactly two values of i in {1, 2....10}, a_i- a_i+1= -1 for two other values of i and a_i- a_i+1 = 0 for all other values of i. There are (10, 2) x (8, 2) = 45 x 28 possible ways of choosing these values. Note that a₁ = -1, 0 or 1, so in total there are 3 x 45 x 28 possible integer solutions to the given equation.

Find the number of 10-tuples (a1; a2;....., a10) of integers such that |a1| ≤ 1 and a21 + a22 + a23+ ...... + a210 - a1a2 - a2a3 - a3a4 -.... a9a10 - a10a1 = 2.

Answer»

Let a₁₁= a₁. Multiplying the given equation by 2 we get

(a₁- a₂₎2 + (a₂ - a₃)2 + .... (a₁₀- a₁)² = 4.

Note that if a_i - a_i+1 = ±2 for some i = 1; .....10, then a_j - a_j+1= 0 for all j ≠ i which contradicts the equality ∑¹⁰_i=1(a_i - a_i+1) = 0. Therefore a_i- a_i+1 = 1 for exactly two values of i in {1, 2....10}, a_i- a_i+1= -1 for two other values of i and a_i- a_i+1 = 0 for all other values of i. There are (10, 2) x (8, 2) = 45 x 28 possible ways of choosing these values. Note that a₁ = -1, 0 or 1, so in total there are 3 x 45 x 28 possible integer solutions to the given equation.

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