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Part of 1996 Vietnam National Olympiad

Problems(2)

VietNam MO 1996

Source:

a+b+c+d \geq \frac{2}{3}(ab+bc+ca+ad+ac+bd)

a;b;c;d

are positive real numbers satisfying

2(ab+bc+cd+da+ac+bd)+abc+bcd+cda+dab=16

algebrapolynomialcalculusderivativeinequalities unsolvedinequalities

Vietnam 1996 Combinatoric

Source: Most difficult problem for me

Let be given integers k and n such that 1<=k<=n. Find the number of ordered k-tuples (a_1,a_2,...,a_n), where a_1, a_2, ..., a_k are different and in the set {1,2,...,n}, satisfying 1) There exist s, t such that 1<=sa_t. 2) There exists s such that 1<=s<=k and a_s is not congruent to s mod 2. P.S. This is the only problem from VMO 1996 I cannot find a solution or I cannot solve. But I'm not good at all in combinatoric...

LaTeXinequalitiescombinatorics unsolvedcombinatorics