1

Part of 2004 France Team Selection Test

Problems(2)

Partition into two sets with equal product

Source: French TST 2004 pb.1

n

is a positive integer, let

A = \{n,n+1,...,n+17 \}

. Does there exist some values of

n

for which we can divide

A

into two disjoints subsets

B

C

such that the product of the elements of

B

is equal to the product of the elements of

C

quadraticsmodular arithmeticnumber theorynumber theory solved

A trivial inequality

Source: French TST 2004, pb.4

n

be a positive integer, and

a_1,...,a_n, b_1,..., b_n

2n

positive real numbers such that

a_1 + ... + a_n = b_1 + ... + b_n = 1

. Find the minimal value of

\frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}

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