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I need to prove this inequality, but I do not have a good background in algebra, if you can guide me:

We have: $$ p_1 + 2p_2 + \ldots +kp_k < q_1 + 2q_2 + \ldots +kq_k+(k+1)q_{k+1}+\ldots+tq_t $$ and $$ p_1+p_2+\ldots+p_k = q_1+q_2+\ldots+q_k+\ldots+q_t $$ and $$ 1 \leq l_1 < l_2 < \ldots < l_k < \ldots < l_t $$ where

  • $p_i \in \mathbb{N^*}$,
  • $q_i \in \mathbb{N^*}$,
  • $l_i\in \mathbb{N^*}$ for all $i$.

Then I need to prove that: $$ l_1p_1+l_2p_2+\ldots +l_kp_k < l_1q_1+l_2q_2+\ldots+l_kq_k+\ldots+l_tq_t $$

If anyone can help me please.

P.S. Sorry, I forget a detail :

$$ p_1 \geq p_2 \geq ... \geq p_k\\ and\\ q_1 \geq q_2 \geq ... \geq q_t $$ Best regards

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    $\begingroup$ This is not true for $k=2$ and $t=3$. Take $p_1=1,p_2=98, q_1=q_2=q_3=33,l_1=1, l_2=3, l_3=4.$ $\endgroup$
    – Toni Mhax
    Commented Sep 22, 2023 at 6:01
  • $\begingroup$ Sorry i forget a detail: $p_1 \geq p_2 \geq ... \geq p_k$ and $q_1 \geq q_2 \geq ... \geq q_t$ $\endgroup$
    – BADJARA Mohamed el Amine
    Commented Sep 22, 2023 at 10:40
  • $\begingroup$ Since you say "I need to prove this inequality", does that mean that you already know that it is true? Is it an exercise? What is the motivation? $\endgroup$
    – Tom De Medts
    Commented Sep 22, 2023 at 11:00
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    $\begingroup$ I assume $\mathbb{N}^*$ means the positive integers. This looks like a question about integer partitions. $\endgroup$
    – Brian Hopkins
    Commented Sep 22, 2023 at 15:17
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    $\begingroup$ This is not true also, $p_1=p_2=50$, $l_1=1, l_2=3, l_3=4$, $q_1=60, q_2=30, q_3=10$ $\endgroup$
    – Toni Mhax
    Commented Sep 22, 2023 at 16:14

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