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4

Part of 2018 Vietnam Team Selection Test

Problems(1)

2018 VNTST Problem 4

Source: 2018 Vietnam Team Selection Test

3/31/2018
Let a∈[12, 32]a\in\left[ \tfrac{1}{2},\ \tfrac{3}{2}\right] be a real number. Sequences (un), (vn)(u_n),\ (v_n) are defined as follows: un=32n+1⋅(−1)⌊2n+1a⌋, vn=32n+1⋅(−1)n+⌊2n+1a⌋.u_n=\frac{3}{2^{n+1}}\cdot (-1)^{\lfloor2^{n+1}a\rfloor},\ v_n=\frac{3}{2^{n+1}}\cdot (-1)^{n+\lfloor 2^{n+1}a\rfloor}.
a. Prove that (u0+u1+⋯+u2018)2+(v0+v1+⋯+v2018)2≤72a2−48a+10+242019.{{({{u}_{0}}+{{u}_{1}}+\cdots +{{u}_{2018}})}^{2}}+{{({{v}_{0}}+{{v}_{1}}+\cdots +{{v}_{2018}})}^{2}}\le 72{{a}^{2}}-48a+10+\frac{2}{{{4}^{2019}}}. b. Find all values of aa in the equality case.
algebraSequenceinequalities