MathDB

2021 Thailand TST

Part of Thailand Team Selection Test

Subcontests

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Floor(m sqrt(2))Floor(n sqrt(7))<Floor(mn sqrt(14))

Prove that, for all positive integers mm and nn, we have ⌊m2⌋⋅⌊n7⌋<⌊mn14⌋.\left\lfloor m\sqrt{2} \right\rfloor\cdot\left\lfloor n\sqrt{7} \right\rfloor<\left\lfloor mn\sqrt{14} \right\rfloor.

Integer Sets

Let A\mathcal{A} be the set of all n∈Nn\in\mathbb{N} for which there exist k∈Nk\in\mathbb{N} and a0,a1,…,ak−1∈{1,2,…,9}a_0,a_1,\dots,a_{k-1}\in \{1,2,\dots,9\} such that a0≥a1≥⋯≥ak−1a_0 \geq a_1 \geq \cdots \geq a_{k-1} and n=a0+a1⋅101+⋯+ak−1⋅10k−1n = a_0 +a_1 \cdot 10^1 +\cdots +a_{k-1}\cdot 10^{k-1}. Let B\mathcal{B} be the set of all m∈Nm \in\mathbb{N} for which there exist l∈Nl \in\mathbb{N} and b0,b1,…,bl−1∈{1,2,…,9}b_0,b_1,\dots,b_{l-1} \in \{1,2,\dots,9\} such that b0≤b1≤⋯≤bl−1b_0 \leq b_1 \leq \cdots\leq b_{l-1} and m=b0+b1⋅101+⋯+bl−1⋅10l−1m = b_0 + b_1 \cdot 10^1 + \cdots+ b_{l-1}\cdot 10^{l-1}.
[*] Are there infinitely many n∈An\in \mathcal{A} such that n2−3∈A ?n^2-3\in\mathcal{A} \ ? [*] Are there infinitely many m∈Bm\in \mathcal{B} such that m2−3∈B ?m^2-3\in\mathcal{B} \ ?
Proposed by Pakawut Jiradilok and Wijit Yangjit
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