1

Part of 2010 Vietnam Team Selection Test

Problems(2)

Viet Nam TST 2010 Pro 1

Source:

n

be a positive integer. Let

T_n

be a set of positive integers such that:

{T_n={ \{11(k+h)+10(n^k+n^h)| (1 \leq k,h \leq 10)}}\}

n

for which there don't exist two distinct positive integers

a, b \in T_n

a\equiv b \pmod{110}

modular arithmeticnumber theory unsolvednumber theory

VN TST 2010 Pro 4

Source:

a,b,c

be positive integers which satisfy the condition:

16(a+b+c)\geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}

\sum_{cyc} \left( \frac{1}{a+b+\sqrt{2a+2c}} \right)^{3}\leq \frac{8}{9}