第一屆（個人賽中三組第 19 題、中四組第 20 題）
董先生參加某國家的總統選舉，得票率（準確至小數點後一個位）為 66.6%。問董先生最少得到多少票？
本題是另一道看下去不難，卻很有深度的題目。表面上它只是一道百分率和近似值的問題，而要找到「66.6%」的例子亦很容易：666/1000 和 333/500 就是最簡單的情況。可是如何找出董先生的得票的最小值呢？
如果大家對數字有一定的敏感度的話，應該會發現 66.6% 和三分之二很接近，但 2/3 四捨五入至小數點後一位的話卻是 66.7%，跟題目不符。因此，董先生的得票應該「比三分之二少一點點」。這基本上也是大會題解背後的思路，大家不妨試試。
這道題是罕有沒有參賽者答對的題目之一。當然，從比賽的角度看這並非好事。然而題目事後卻引起了廣泛討論，有數學老師更想出了一些另類的解法，當中甚至跟表面看來毫不相干的 Pick's formula 扯上關係。這實在是數學其中一個最可愛的地方。
Pick's Formula？願聞其詳。
回覆刪除Andy：
回覆刪除你不記得了嗎？那是當年荃官的聯校活動上孔 sir 提出的，當時你也在場啊。
不過他確實的做法我也不太記得清楚，我一時間也找不到那一張筆記。他大概是在座標平面上畫了 y=0.6655x 和 y=0.6665x 兩條直線，然後用 Pick's formula 數格點的。
哈，記起來了。
回覆刪除你不提起這情景，我也想不起這件事！
Let n be the number of votes for Mr Tung, N be the total number of votes
回覆刪除according to the question,
6655/10000 <= n/N < 6665/10000
2000n / 1331 >= N > 2000n / 1333
n + 669n/1331 >= N > n + 667n / 1333
N bounded by [n+ 667n/1333, n +669n/1331)
Now, find min. n such that the integral part of LHS and RHS are diff.
consider only 667n/1333 and 669n/1331
The remainder of 667n/1333
for odd n (2m+1) = 667 + m --(1)
for even n (2m) = m --------(2)
The remainder of 669n/1331
for odd n (2m+1) = 669 + 7m ---(3)
for even n (2m) = 7m ---------(4)
where m = 0, 1, 2, ...
clearly, (3) is growing faster with respect to n and lead to diff. integral part for LHS and RHS
Thus,
669 + 7m > 1331
m > 662 /7 = 94.57
therefore, min m = 95, i.e. min n = 2*95+1=191
(N = 287)
pop:
回覆刪除Your solution is very nice, and it does not require the observation that 66.6% is close to two-thirds. Would you mind letting your solution be included in a future version of the PCIMC solutions?
Kahoo, it's my pleasure.
回覆刪除As i can only get the numerical solution from the net, that s why i work out my own.
So would you please let me know where i can download the offical detail solution?
pop:
回覆刪除Solution booklets are published every year. This year a combined version of the first five years' solutions will be published. We are therefore trying to include a variety of nice alternative solutions. To express gratitude for providing your alternative solution, we will give you a complimentary copy of the combined version which will be published by the end of the year. Would you mind sending an e-mail to pcimc.sol@gmail.com leaving your details?