HMMT 十一月 2011 · 冲刺赛 · 第 4 题

英文解析

4. [ 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)
   √
   29 29 8412
   or 3 < which is true because = > 3. We can reverse this sequence of squarings because,
   16 16 2562
   √
   at each step, we make sure that both our values are positive after taking the square root. Hence, 54 3
   is the smallest.
   17