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Part of 2014 IMC

Problems(2)

IMC 2014, Problem 6

Source: IMC 2014

For a positive integer

x

n^{\mathrm{th}}

decimal digit by

d_n(x)

d_n(x)\in \{ 0,1, \dots, 9\}

x=\sum_{n=1}^{\infty} d_n(x)10^{n-1}

. Suppose that for some sequence

(a_n)_{n=1}^{\infty}

, there are only finitely many zeros in the sequence

(d_n(a_n))_{n=1}^{\infty}

. Prove that there are infinitely many positive integers that do not occur in the sequence

(a_n)_{n=1}^{\infty}

.
(Proposed by Alexander Bolbot, State University, Novosibirsk)

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IMC 2014, Problem 1

Source: IMC 2014

Determine all pairs

(a, b)

of real numbers for which there exists a unique symmetric

2\times 2

M

with real entries satisfying

\mathrm{trace}(M)=a

\mathrm{det}(M)=b

.
(Proposed by Stephan Wagner, Stellenbosch University)

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