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Prove these two expression

Source: 2019 Jozsef Wildt International Math Competition-W. 26

May 18, 2020
Summation

Problem Statement

Let n∈Nn \in \mathbb{N}, n≥2n \geq 2, a1,a2,⋯ ,an∈Ra_1, a_2, \cdots , a_n \in \mathbb{R} and an=max{a1,a2,⋯ ,an}a_n = max \{a_1, a_2,\cdots , a_n\}
[*]If tkt_k, tk′∈Rt'_k \in \mathbb{R}, k∈{1,2,⋯ ,n}k \in \{1, 2,\cdots , n\} , tk≤tk′t_k \leq t'_k, for any k∈{1,2,⋯ ,n−1}k \in \{1, 2, \cdots, n - 1\} and ∑k=1ntk=∑k=1ntk′\sum \limits_{k=1}^nt_k=\sum \limits_{k=1}^nt'_kProve that ∑k=1ntkak≥∑k=1ntk′ak\sum \limits_{k=1}^nt_ka_k\geq \sum \limits_{k=1}^nt'_ka_k [*] If bkb_k, ck∈R+∗c_k \in \mathbb{R}^*_+, k∈{1,2,⋯ ,n}k \in \{1, 2,\cdots , n\} , bk≤ckb_k \leq c_k for any k∈{1,2,⋯ ,k−1}k \in \{1, 2,\cdots, k - 1\} and b1b2⋯bn=c1c2⋯cnb_1b_2\cdots b_n=c_1c_2\cdots c_nProve that ∏k=1nbkak≥∏k=1nckak\prod \limits_{k=1}^n b_k^{a_k}\geq \prod \limits_{k=1}^nc_k^{a_k}
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