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Problem 2

Part of 1994 Korea National Olympiad

Problems(2)

csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12

Source: Korean Mathematical Olympiad 1994, Final Round P2 FKMO

7/21/2018
Let α,β,γ \alpha,\beta,\gamma be the angles of a triangle. Prove that csc2α2+csc2β2+csc2γ2≥12csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12 and find the conditions for equality.
Trigonometric inequalitytrigonometryInequalitygeometryalgebrainequalities
S_1 = {1}, S_k=(S_{k-1} \oplus \{k\}) \cup \{2k-1\}

Source: Korean Mathematical Olympiad 1994, Final Round P5 FKMO

7/21/2018
Given a set S⊂NS \subset N and a positive integer n, let S⊕{n}={s+n/s∈S}S\oplus \{n\} = \{s+n / s \in S\}. The sequence SkS_k of sets is defined inductively as follows: S1=1S_1 = {1}, Sk=(Sk−1⊕{k})∪{2k−1}S_k=(S_{k-1} \oplus \{k\}) \cup \{2k-1\} for k=2,3,4,...k = 2,3,4, ... (a) Determine N−∪k=1∞SkN - \cup _{k=1}^{\infty} S_k. (b) Find all nn for which 1994∈Sn1994 \in S_n.
Integer sequenceSubsetsetnumber theory