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比賽時間:2018年4月2日(9:30 pm)至4月9日(10:00 pm)

分數 Marks

最後更新:2018年4月5日 8:21 pm

排名 名字 Q1 Q2 Q3 Q4 Q5 Q6 總分 得分時間
1 SkydustMario 1 1 1 1 1 1 6 2018年4月5日 2:15 pm
2 APerfectIceCube 1 1 1 0 1 1 5 2018年4月4日 8:10 am
3 MM4031 1 0 0 1 1 1 4 2018年4月3日 6:25 pm
4 addisonshiu 1 1 1 0 - - 3 2018年4月3日 8:43 pm
N/A CookieMan018 0 0 - - - - 0 N/A
N/A happyeefun 0 0 0 - - - 0 N/A
N/A Harry_HK - 0 0 0 0 0 0 N/A
N/A Impu1s3 0 0 0 0 0 0 0 N/A
N/A TaiKok 0 0 0 0 0 0 0 N/A

"-" 代表參賽者未有嘗試作答該題。

比賽規則 Rules And Regulations

  1. 比賽共設6題。每題1分,答錯不扣分。
    This competition comprises 6 questions. Each question carries 1 mark. No marks will be deducted for incorrect answers.
  2. 所有答案均須填在答題表格上的指定位置。
    All answers must be written in the specified areas of the Answer Form.
  3. 所有題目的答案均為數字,無須提供單位或計算步驟。所有答案必須約至最簡。除非題目特別指明,所有答案必須使用其真確值。
    The answers to every question are numbers. No units or steps are required. All answers must be in the simplest form. Unless otherwise specified, all answers must be exact.
  4. 參賽者可以使用計算機。
    Calculators are allowed in this competition.
  5. 參賽者可以多次提交答案,惟必須使用同一個名字。每次提交可以有題目留空。參賽者可以透過多次提交以更改答案。
    Contestants can submit their answers more than once, but the name used in each submission must be the same. It is not required to fill in all the blanks in each submission. Contestants can change their answers by another submission.
  6. 參賽者雖然可以多次提交答案,但不得濫用此權利。若有猜答案的嫌疑,該參賽者會被取消資格。
    Though contestants can submit their answers more than once, this must not be abused. If a contestant is suspected to be guessing the answer, he/she will be disqualified.

作答方式 Submission of Answers

答題表格 Answer Form


表達式輸入方式 Input of Expressions

算子/常數
Operator/Constant

輸入方式
Input Method

例子
Example

分數
Fraction

a/b $ \frac{1}{2} $
1/2

次方

Power

x^n $ 2^3 $
2^3

平方根

Square Root

sqrt(x) $ \sqrt{2} $
sqrt(2)

圓周率

Pi

$ \pi =3.14159\ldots $

pi N/A
表達式

Expression

輸入方式
Input Method
$ \frac{2\pi}{5} $ 2pi/5
$ 3\pi+4 $ 3pi+4
$ \sqrt{2}-1 $ sqrt(2)-1
$ \frac{3-6\sqrt{2}}{4} $ (3-6sqrt(2))/4


題目 Questions

第一題 Question 1

阿美有R$100,阿邦有R$240,卡爾有R$440。他們想透過群組基金令到三人持有等同數量的Robux。當某個金額的Robux被捐到一個群組時,只有70%會落入群組基金。但是,群組基金可以全額轉移到任何用戶。他們最少要向群組捐多少Robux才能達到目標?

Amy, Bob and Carl have 100, 240 and 440 Robux, respectively. They want to transfer Robux to each other using group funds so that they can have the same amount of Robux. When a certain amount of Robux is being donated to a group, only 70% of the original amount is transferred to the group funds. The group funds can then be transferred to any user in full amount. What is the minimum amount of Robux needed to be donated to the group?


第二題 Question 2

在一個遊戲中,有一位老手及若干位新手。新手不知友誼為何物,所以他們會向遊戲中其他人員發送交友邀請,而且會接受所有交友邀請。老手明白交友的真正意義,所以他真正喜歡那個人才會接受那個人的交友邀請。最終,總共有95個交友邀請被接受。求遊戲中的人數。

On a server, there are a number of new players and one veteran. Every new player is unaware of the true meaning of being friends and thus sends friend requests to all other players on the server. They would also accept all friend requests sent to them. The veteran treasures friendship and will only accept friend requests from the people he likes. In the end, 95 friend requests are being accepted. Find the number of players on the server.


第三題 Question 3

在Jailbreak,遊戲時間的一天可以分成六個四小時時段。遊戲中任何一個時段都有機會降雨,但是在一個時段期間最多只會出現一次降雨。已知每個時段都有10%機會降雨,求遊戲時間中一天出現最少兩次降雨的概率。

In Jailbreak, an in-game day can be divided into six 4-hour periods. There can be precipitation at any time of a period. However, there can only be precipitation every period. Suppose there is a 10% chance of precipitation in every period. Find the probability of encountering at least 2 precipitations in an in-game day.


第四題 Question 4

假設NLB11A打算為MTRB K76 City編寫一個程式來隨機產生車牌號碼。所產生的車牌號碼必須符合以下各項條件:

  1. 車牌號碼由兩個不同的英文字母和一個四位數字組成,該四位數字的千位數不是0。
  2. 四位數字中必須包含最少兩個在0至9之間不同的數字。
  3. 為了避免引起部份人士的遐想,四位數字中任何兩個相鄰的數位不得為「69」,但是「96」可以接受。例如「1692」和「6905」不符合此條件,但「9613」和「6009」符合此條件。

求符合各項條件的車牌號碼之數量。

Suppose NLB11A is planning to use a script to generate license plate numbers at MTRB K76 City. The license plate number must satisfy the following conditions.

  1. It comprises of 2 different letters and a 4-digit number that does not start with a zero.
  2. There must be at least 2 unique numbers from 0 to 9 in the 4-digit number.
  3. To avoid making some people go crazy, any two neighbouring digits in the 4-digit number must not be “69”. However, “96” is acceptable. For example, “1692” and “6905” are not acceptable, but “9613” and “6009” are acceptable.

Find the number of possible license plate numbers.


第五題 Question 5

勞博士之星綫共有6個站,由勞博士鐵路公司營運,現時只有「每站停車」列車。某一天,勞博士鐵路公司終於推出「快速」列車,只停其中4個站,不停靖明站和茲輋士打灣站。同時,勞博士之星綫亦變得受歡迎。某君想了解乘客的目的地,於是隨機訪問了150位乘客。調查中,有24位男性乘客仍然只能乘搭「每站停車」列車到達目的地,可以乘搭「每站停車」或「快速」列車的乘客與只能乘搭「每站停車」列車的乘客的比例為11:4 。已知男性乘客與女性乘客的比例為16:9,求不是前往靖明站或茲輋士打灣站的女性乘客數量。

The Roblox Star Line has 6 stations. It is operated by The Ryan Railroad. Currently, there is only “all stations” train (stops at every station). Finally, one day, The Ryan Railroad launched the “express” train into service. The “express” train only stops at 4 stations, skipping Ching Ming and Chichester Bay stations. At the same time, The Roblox Star Line has become popular. A person wanted to know about the destinations of passengers, so he randomly interviewed 150 passengers. From the interviews, he found that 24 male passengers can still only travel to their destinations by taking the “all stations”. The ratio of passengers who can take either “all stations” train or “express” train to passengers who can only take “all stations” train is 11:4. Given that the ratio of male passengers to female passengers is 16:9. Find the number of female passengers who are NOT heading to Ching Ming station or Chichester Bay station.


第六題 Question 6

Addison是Forever Checkpoint Particle的創作人。假設他加入一個函數,只要輸入新一位通關者的通關時間,就可以重新計算遊戲的平均通關時間。有一天,他與兩位朋友玩該遊戲,該兩位朋友分別以4320秒和4880秒的時間通關。可是,當時遊戲出現問題,需要由Addison以人手方式輸入他們的時間。然而,他不小心地將上述的時間輸入成「1320」和「1880」,導致函數傳回一個錯誤的平均通關時間。已知輸入前和輸入後的平均通關時間分別為5200秒和5104秒,求正確的平均通關時間(以秒表示答案)。

Addison is the creator of Forever Checkpoint Particle. Suppose he added a function that calculates the average completion time of the winners. When a number is being inputted, the function automatically re-calculates the average completion time. One day, he was playing the game with two other friends. His friends completed the game in 4320 seconds and 4880 seconds respectively. However, the game was not functioning. Addison had manually input both completion times through the function, but he accidentally put “1320” and “1880” instead. This caused the function to return an incorrect average time. The average time before and after the input is 5200 seconds and 5104 seconds, respectively. Find the correct average completion time in seconds.

數值答案 Numerical Answers

題目

Question

可接受的數值答案

Acceptable Numerical Answers

1 200
2 15
3 $ \frac{22853}{200000} $ / 0.114265 / 11.2465%
4 5662800
5 38
6 5184


解答(中文版)

第一題

卡爾有最多Robux,所以他會將Robux轉移到另外兩人。
設R$ $ x $ 和R$$ y $ 分別為轉移Robux給阿美和阿邦所需向群組捐出的金額。
則最少要向群組捐 $ x+y $ Robux。

最終
阿美會有 $ 100+0.7x $ Robux。
阿邦會有 $ 240+0.7y $ Robux。
卡爾會有 $ 440-x-y $ Robux。

由於三人最終會持有等同數量的Robux,可得
$ \begin{align} & \left\{ \begin{array}{*{35}{l}} 100+0.7x=240+0.7y \\ 100+0.7x=440-x-y \\ \end{array} \right. \\ & \left\{ \begin{array}{*{35}{l}} x-y=200 \\ 1.7x+y=340 \\ \end{array} \right. \\ \end{align} $
解聯立二元一次方程,得解為
$ \left\{ \begin{array}{*{35}{l}} x=200 \\ y=0 \\ \end{array} \right. $

∴ 最少捐出
$ \begin{align} & =200+0 \\ & =\underline{\underline{200}} \\ \end{align} $


第二題

$ n $ 為新手的數量和 $ k $ 為老手喜歡的人之數量。
$ 0\le k\le n $
現在只考慮新手之間的交友邀請。
將新手一個一個加入遊戲中。當加入一個新手,該新手只能接受已經在遊戲中的新手所發送的交友邀請。
$ n=1 $,所接受的交友邀請 $ =0 $
$ n=2 $,所接受的交友邀請 $ =1 $
$ n=3 $,所接受的交友邀請 $ =1+2 $
$ n=4 $,所接受的交友邀請 $ =1+2+3 $
如此類推。
∴ 對於任何數量的新手 $ n $,他們之間所接受的交友邀請 $ =\frac{n\left( n-1 \right)}{2} $ 。(三角形數)

由此可得 $ \frac{n\left( n-1 \right)}{2}+k=95 $
$ \therefore \frac{n\left( n-1 \right)}{2}\le 95 $
解不等式,得解為
$ \frac{1-\sqrt{761}}{2}\le n\le \frac{1+\sqrt{761}}{2} $
$ (-13.29311\ldots \le n\le 14.29311\ldots) $

注意 $ n $ 為非負整數。
如果 $ n=14 $$ k=95-\frac{14\left( 14-1 \right)}{2}=4 $,並且符合 $ 0\le k\le n $
對於其他$ n $的數值,$ k $ 的數值不符合 $ 0\le k\le n $
$ \therefore n=14 $
∴ 遊戲中的人數(包括老手)
$ \begin{align} & =14+1 \\ & =\underline{\underline{15}} \\ \end{align} $


第三題

$ x $ 為在遊戲時間的一天中降雨的次數。
$ P\left( x=0 \right)={{\left( 1-10% \right)}^{6}}={{\left( \frac{9}{10} \right)}^{6}} $
$ P\left( x=1 \right)=C_{1}^{6}{{\left( 1-10% \right)}^{5}}{{\left( 10% \right)}^{1}}=6\times {{\left( \frac{9}{10} \right)}^{5}}\left( \frac{1}{10} \right) $ (二項分佈)
$ \begin{align} P\left( x\ge 2 \right)&=1-P\left( x=0 \right)-P\left( x=1 \right) \\ & =1-{{\left( \frac{9}{10} \right)}^{6}}-6\times {{\left( \frac{9}{10} \right)}^{5}}\left( \frac{1}{10} \right) \\ & =\frac{{{10}^{6}}-{{9}^{6}}-6\times {{9}^{5}}}{{{10}^{6}}} \\ & =\frac{114265}{1000000} \\ & =\underline{\underline{\frac{22853}{200000}}} \end{align} $

小數:$ 0.114265 $
百分比:$ 11.4265% $



第四題

對於英文字母
排列數量 $ =26\times 25=650 $

對於四位數字
四位數字的數量 $ =9000 $
由一個數字組成的四位數字 $ =\{1111,2222,3333,\ldots ,9999\} $
∴ 由一個數字組成的四位數字的數量 $ =9 $

考慮包含「69」的四位數字。
情況一:69 _ _      數量 $ =10\times 10=100 $
情況二:_ 69 _      數量 $ =9\times 10=90 $
情況三:_ _ 69      數量 $ =9\times 10=90 $
注意「6969」在情況一和三總共被數了兩次。
∴ 包含「69」的四位數字的數量 $ =100+90+90-1=279 $

∴ 可接受的四位數字的數量 $ =9000-9-279=8712 $

∴ 車牌號碼的數量
$ \begin{align} & =650\times 8712 \\ & =\underline{\underline{5662800}} \\ \end{align} $


第五題

考慮下表。

男性 女性
只有「每站停車」 24 a
「每站停車」和「快速」 b c


使用已知的比例,可得
$ \begin{align} a+24&=150\times \frac{4}{4+11} \\ a&=16 \end{align} $
$ \begin{align} b+24&=150\times \frac{16}{16+9} \\ b&=72 \end{align} $

$ \begin{align} \therefore 16+72+c+24&=150 \\ c&=38 \end{align} $

∴ 答案為 $ \underline{\underline{38}} $


第六題

$ n $ 為原本的通關人數。

$ 5200n+1320+1880=5104\left( n+2 \right) $
解方程,得解為 $ n=73 $
∴ 正確平均通關時間
$ \begin{align} & =\frac{5200\times 73+4320+4880}{73+2} \\ & =\underline{\underline{5184}} \\ \end{align} $


Solution (English Version)

Question 1

Carl has the most Robux. Therefore, he will transfer Robux to the others.
Let $ x $ Robux and $ y $ Robux be the amount of donation to the group that would be transferred to Amy and Bob, respectively.
Then, the minimum amount of donation would be $ x+y $ Robux.

In the end,
Amy would have $ 100+0.7x $ Robux.
Bob would have $ 240+0.7y $ Robux.
Carl would have $ 440-x-y $ Robux.

Since they have the same amount of Robux,
$ \begin{align} & \left\{ \begin{array}{*{35}{l}} 100+0.7x=240+0.7y \\ 100+0.7x=440-x-y \\ \end{array} \right. \\ & \left\{ \begin{array}{*{35}{l}} x-y=200 \\ 1.7x+y=340 \\ \end{array} \right. \\ \end{align} $
Solving the equations, we have
$ \left\{ \begin{array}{*{35}{l}} x=200 \\ y=0 \\ \end{array} \right. $

∴ Minimum amount of donation
$ \begin{align} & =200+0 \\ & =\underline{\underline{200}} \\ \end{align} $


Question 2

Let $ n $ be the number of newbies on the server and $ k $ be the number of players the veteran likes.
Then, $ 0\le k\le n $.

Consider the friend requests among newbies only.
Now, we add newbies to the server one by one. When a newbie is added, it can only accept the requests from the previous newbies.
When $ n=1 $ , friend requests accepted $ =0 $
When $ n=2 $ , friend requests accepted $ =1 $
When $ n=3 $ , friend requests accepted $ =1+2 $
When $ n=4 $ , friend requests accepted $ =1+2+3 $
And so on.
∴ For any number of newbies $ n $, friend requests accepted among them $ =\frac{n\left( n-1 \right)}{2} $ . (Triangular numbers)

So, we have $ \frac{n\left( n-1 \right)}{2}+k=95 $ .
$ \therefore \frac{n\left( n-1 \right)}{2}\le 95 $
Solving the inequality, we have
$ \frac{1-\sqrt{761}}{2}\le n\le \frac{1+\sqrt{761}}{2} $
$ (-13.29311\ldots \le n\le 14.29311\ldots) $

Note that $ n $ is a non-negative integer.
For $ n=14 $ , $ k=95-\frac{14\left( 14-1 \right)}{2}=4 $ , which satisfies $ 0\le k\le n $.
For any other values of $ n $ , the value of $ k $ does not satisfy $ 0\le k\le n $.
$ \therefore n=14 $
∴ Number of players in the server (including the veteran)
$ \begin{align} & =14+1 \\ & =\underline{\underline{15}} \\ \end{align} $


Question 3

Let $ x $ be the number of precipitation in an in-game day.
$ P\left( x=0 \right)={{\left( 1-10% \right)}^{6}}={{\left( \frac{9}{10} \right)}^{6}} $
$ P\left( x=1 \right)=C_{1}^{6}{{\left( 1-10% \right)}^{5}}{{\left( 10% \right)}^{1}}=6\times {{\left( \frac{9}{10} \right)}^{5}}\left( \frac{1}{10} \right) $ (Binomial distribution)
$ \begin{align} P\left( x\ge 2 \right)&=1-P\left( x=0 \right)-P\left( x=1 \right) \\ & =1-{{\left( \frac{9}{10} \right)}^{6}}-6\times {{\left( \frac{9}{10} \right)}^{5}}\left( \frac{1}{10} \right) \\ & =\frac{{{10}^{6}}-{{9}^{6}}-6\times {{9}^{5}}}{{{10}^{6}}} \\ & =\frac{114265}{1000000} \\ & =\underline{\underline{\frac{22853}{200000}}} \end{align} $

Decimal number: $ 0.114265 $
Percentage: $ 11.4265% $



Question 4

For the letters,
Number of permutations $ =26\times 25=650 $

For the four-digit number,
Number of four-digit numbers $ =9000 $
Four-digit numbers that only have one unique number $ =\{1111,2222,3333,\ldots ,9999\} $
∴ Number of four-digit numbers that only have one unique number $ =9 $

Consider the numbers that contain "69".
Case 1: 69 _ _      Number $ =10\times 10=100 $
Case 2: _ 69 _      Number $ =9\times 10=90 $
Case 3: _ _ 69      Number $ =9\times 10=90 $
Note that we have counted "6969" twice in cases 1 and 3.
∴ Number of four-digit numbers that contain "69" $ =100+90+90-1=279 $ .

∴ Number of acceptable four-digit numbers $ =9000-9-279=8712 $

∴ Number of license plate numbers
$ \begin{align} & =650\times 8712 \\ & =\underline{\underline{5662800}} \\ \end{align} $


Question 5

Consider the following table.

Male Female
"All stations" only 24 a
"All stations" and "Express" b c


Using the given ratios,
$ \begin{align} a+24&=150\times \frac{4}{4+11} \\ a&=16 \end{align} $
$ \begin{align} b+24&=150\times \frac{16}{16+9} \\ b&=72 \end{align} $

$ \begin{align} \therefore 16+72+c+24&=150 \\ c&=38 \end{align} $

∴ The answer is $ \underline{\underline{38}} $ .


Question 6

Let $ n $ be the original number of people who completed the game.
Then, we have
$ 5200n+1320+1880=5104\left( n+2 \right) $
Solving the equation, we have $ n=73 $ .
∴ The correct average completion time
$ \begin{align} & =\frac{5200\times 73+4320+4880}{73+2} \\ & =\underline{\underline{5184}} \\ \end{align} $


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Roblox 數學比賽