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Trunk numbers

Source: Danube Mathematical Competition,Juniors #1

October 31, 2015

number theory

Problem Statement

Consider a positive integer

n=\overline{a_1a_2...a_k},k\ge 2

.A trunk of

n

is a number of the form

\overline{a_1a_2...a_t},1\le t\le k-1

.(For example,the number

23

is a trunk of

2351

.) By

T(n)

we denote the sum of all trunk of

n

and let

S(n)=a_1+a_2+...+a_k

.Prove that

n=S(n)+9\cdot T(n)

.

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