Track / Stage 5 / 386 of 400 #986 of 1964
Problem 986 AIME late Difficulty 5.8 Prove it
Sis: *Take n ( ⩾ 2 ) n(\geqslant 2) n ( ⩾ 2 ) ( 0 , 1 ) (0,1) ( 0 , 1 ) 1 3 n 3 2 \frac{1}{3} n^{\frac{3}{2}} 3 1 n 2 3
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Official solution Let the n n n a 1 b 1 t 1 ⩽ 1 t ∑ b , > t b i ⩽ B t {\frac{a_{1}}{b_{1}}t} 1 \leqslant \frac{1}{t} \sum_{b,>t} b_{i} \leqslant \frac{B}{t} b 1 a 1 t 1 ⩽ t 1 ∑ b , > t b i ⩽ t B n = ∑ b 1 1 1 ⩽ t 2 + B t .
n=\sum_{b_{1}1} 1 \leqslant t^{2}+\frac{B}{t} .
n = b 1 1 ∑ 1 ⩽ t 2 + t B .
Taking t = B 1 3 t=B^{\frac{1}{3}} t = B 3 1 2 B 2 3 ⩾ n 2 B^{\frac{2}{3}} \geqslant n 2 B 3 2 ⩾ n B ⩾ ( n 2 ) 3 / 2 > 1 3 n 3 / 2 .
B \geqslant\left(\frac{n}{2}\right)^{3 / 2}>\frac{1}{3} n^{3 / 2} .
B ⩾ ( 2 n ) 3/2 > 3 1 n 3/2 .
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Source: NuminaMath-1.5 ,
licensed Apache-2.0 .
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