2012年2月27日 星期一

不等式的疑惑

本屆初賽高中組其中一道題為:

Let k be a constant. It is given that for real number x , the minimum value of
x^2 - 5x + k is 2012. Find k.


A solution provided by one of the contestant:

x^2 - 5x + k >= 2012

(x - 5/2)^2 + k >= 2012 + 25/4

So, k >= 2018.25

k = 2019

官方答案的 k 為 2018.

大家看到問題在哪嗎?

提示:

A^2 + k >= C
k >= C - A^2

5 則留言:

Marco_Dick 提到...

這類問題應該在比賽前post讓大家知道規矩嘛……

wahas 提到...
作者已經移除這則留言。
wahas 提到...

Sorry I am someone who just accidentally visit this page but I do not understand why k = 2018.
If k = 2018, then if x = 2.5, x^2-5x+k = (2.5)^2-5(2.5)+2018 =2011.75<2012
Why is k 2018 but not 2019?
Thanks anyway.

goose 提到...

There have been amendments in the regulations of the current Heat Event. If the correct
answer to a question is not an integer between 0 and 9999, one should pick the integer in the above
range which is closest to the correct answer. In case of an answer midway between two such
integers, round up to the larger integer. Read the instructions on the answer sheet in detail.

2018.25 is more near to 2018 instead of 2019
so....2018

芳日 提到...

(x-5/2)^2+k >= 2018.25
k >= [2018.25-(x-5/2)^2] <= 2018.25

so the discrepency