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Question

If (1)(2003)+(2)(2002)+(3)(2001)++(2003)(1)=(2003)(334)(x), then the value of [x5] is equal to
([.] represents the greatest integer function)

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Solution

Let S=(1)(2003)+(2)(2002)+(3)(2001)++(2003)(1)
K=12+22+32++(2003)2S+K=1(2004)+2(2004)+3(2004)++2003(2004)S+K=(2004)(1+2+3++2003)S+K=2004×2003×20042(2003)(334)(x)+2003×2004×40076=2004×2003×20042(334)(x)+(334)(4007)=(2004)(1002)(334)(x+4007)=(2004)(1002)x+4007=6012x5=20055[x5]=401

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