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3. 若 [x][x] 表示不超过 xx 的最大整数，有序正整数组 (x1,x2,x3,x4)\left(x_{1}, x_{2}, x_{3}, x_{4}\right) 满足 $...
Mar 9, 2024
3. 若 [x][x] 表示不超过 xx 的最大整数，有序正整数组 (x1,x2,x3,x4)\left(x_{1}, x_{2}, x_{3}, x_{4}\right) 满足 x_{1}<x_{2}<x_{3}<x_{4} ，且 [x1+x22]2+[x2+x32]2+[x3+x42]2=35\left[\frac{x_{1}+x_{2}}{2}\right]^{2}+\left[\frac{x_{2}+x_{3}}{2}\right]^{2}+\left[\frac{x_{3}+x_{4}}{2}\right]^{2}=35 ，则满足条件的数组共有（ ) A. 2 组 B. 3组 C. 4 组 D. 5 组
Solution by Steps
step 2
Express the given equation using the function f(x)f(x): f(x1+x2)2+f(x2+x3)2+f(x3+x4)2=35f(x_1 + x_2)^2 + f(x_2 + x_3)^2 + f(x_3 + x_4)^2 = 35
step 3
Since x1,x2,x3,x4x_1, x_2, x_3, x_4 are positive integers and x_1 < x_2 < x_3 < x_4, the possible values for f(x1+x2),f(x2+x3),f(x3+x4)f(x_1 + x_2), f(x_2 + x_3), f(x_3 + x_4) are limited
step 4
List all possible combinations of squares of integers that sum up to 35: (12,12,52),(12,32,52),(12,52,52)(1^2, 1^2, 5^2), (1^2, 3^2, 5^2), (1^2, 5^2, 5^2)
step 5
Determine the number of ordered sets (x1,x2,x3,x4)(x_1, x_2, x_3, x_4) that satisfy each combination found in step 4
step 6
After evaluating all possible combinations, conclude the number of distinct ordered sets that satisfy the given equation
B
Key Concept
Integer partitioning and properties of floor function
Explanation
The problem involves finding the number of ways to partition the number 35 into the sum of three squared integers, considering the constraints given by the floor function and the ordering of x1,x2,x3,x4x_1, x_2, x_3, x_4.
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