\[\boxed{\text{753.}\text{\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]
\[Пусть\ x_{1},\ x_{2},\ x_{3},\ x_{4},\ x_{5} - пять\]
\[\ последовательных\ целых\ \]
\[чисел.\]
\[x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = x_{4}^{2} + x_{5}^{2}\]
\[x_{2} = x_{1} + 1,\ \ x_{3} = x_{1} + 2,\ \ \]
\[x_{4} = x_{1} + 3,\ \ x_{5} = x_{1} + 4.\]
\[Составим\ уравнение:\]
\[x_{1}^{2} + \left( x_{1} + 1 \right)^{2} + \left( x_{1} + 2 \right)^{2} =\]
\[= \left( x_{1} + 3 \right)^{2} + \left( x_{1} + 4 \right)^{2}\]
\[x_{1}^{2} + x_{1}^{2} + 2x_{1} + 1 + x_{1}^{2} + 4x_{1} +\]
\[+ 4 = x_{1}^{2} + 6x + 9 + x_{1}^{2} +\]
\[+ 8x_{1} + 16\]
\[3x_{1}^{2} + 6x_{1} + 5 = 2x_{1}^{2} +\]
\[+ 14x_{1} + 25\]
\[x_{1}^{2} - 8x_{1} - 20 = 0\]
\[D = 64 + 80 = 144\]
\[x_{1,2} = \frac{8 \pm 12}{2} = 10;\ - 2\]
\[то\ есть:\]
\[x_{1} = 10,\ x_{2} = 11,\ x_{3} = 12,\ x_{4} =\]
\[= 13,\ x_{5} = 14\ или\]
\[x_{1} = - 2,\ x_{2} = - 1,\ x_{3} =\]
\[= 0,\ x_{4} = 1,\ x_{5} = 2.\ \]
\[Ответ:10,\ 11,\ 12,\ 13,\ 14\ \ или\ \]
\[\ ( - 2),\ - 1,\ 0,\ 1,\ 2.\]