HMMT 十一月 2011 · 冲刺赛 · 第 4 题

HMMT November 2011 — Guts Round — Problem 4

[ 6 ] Determine which of the following numbers is smallest in value: 54 3 , 144 , 108 6 − 108 2. 17

解析

[ 6 ] Determine which of the following numbers is smallest in value: 54 3 , 144 , 108 6 − 108 2. √ √ √ 144 8 Answer: 54 3 We can first compare 54 3 and 144. Note that 3 < 2 and = > 2. Hence, 54 3 √ √ √ √ 54 3 is less. Now, we wish to compare this to 108 6 − 108 2. This is equivalent to comparing 3 to √ √ √ √ √ √ 2( 6 − 2). We claim that 3 < 2( 6 − 2). To prove this, square both sides to get 3 < 4(8 − 4 3) √ 2 29 29 841 or 3 < which is true because = > 3. We can reverse this sequence of squarings because, 2 16 16 256 √ at each step, we make sure that both our values are positive after taking the square root. Hence, 54 3 is the smallest. 17