第二季Roblox數學比賽 (2018年5月)

比賽時間 Duration[]

2018年5月21日晚上八時三十分至5月30日晚上十時正
21st May 2018 (8:30pm) to 30th May 2018 (10:00pm)

分數 Marks[]

N/A

比賽規則 Rules And Regulations[]

比賽共設兩個部分，每部分設有6題。每題1分，答錯不扣分。
This competition comprises of two parts, each consists of 6 questions. Each question carries 1 mark. No marks will be deducted for incorrect answers.
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.
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.
參賽者可以使用計算機。
Calculators are allowed in this competition.
參賽者可以多次提交答案，惟必須使用同一個名字。每次提交可以有留空題目。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.
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[]

https://docs.google.com/forms/d/e/1FAIpQLScGraaX__p3fwEw8994yPhjDqub1II91Up8iYN2hvwFhOVBpA/viewform?usp=sf_link

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

甲部 Part A[]

第一題 Question 1[]

一輛巴士在一條長3000 stud的橋上行駛。起初，巴士以每秒40 stud的速率行駛；經過一段時間後，巴士立即加速至每秒60 stud。最終，巴士用了64秒過橋。巴士在橋上行駛了多少秒後才加速？
A bus is crossing a bridge with a length of 3000 studs. Initially, the bus moves at a constant speed of 40 stud/sec. After a period of time, the bus increases its speed to 60 stud/sec immediately. It takes 64 seconds for the bus to cross the bridge. How many seconds did the bus drive before increasing its speed?

第二題 Question 2[]

為了慶祝KMB 284 City的成功，製作者JohnnyYeung和NLB11A在梁銘記（製作者：likehkbusman）舉行了聚會，除了上述兩人之外有18人參加。他們在梁銘記包了三圍，每圍有8個座位。GS9019和addisonshiu是參加聚會的其中兩位，然而他們是最後到場的人，其他人早已隨意地選擇了座位。如情況許可，上述兩人會選擇座在同一圍。上述兩人與兩位製作者座在同一圍的概率是多少？
To celebrate the success of KMB 284 City, the creators JohnnyYeung and NLB11A held a gathering at LMK (a restaurant made by likehkbusman) with 18 other players. They had booked 3 tables at LMK, which has 8 seats each. GS9019 and addisonshiu were two of the players. However, they were the last to arrive. The other players had already randomly picked a seat. If possible, the two would sit at the same table. What is the probability that the two would sit at the same table as the two creators?

第三題 Question 3[]

東涌市是一個由Facelift Workshop維新工作室製作的遊戲，製作者為NLB11A、ilkehongkongdennis和JVsevensixtwonine。假設一位神祕人物向工作室捐款，扣除市場費用後，工作室獲得350 Robux。現在將該款項分發予遊戲的三位製作者。每位製作者所分到的數額不得為零，而且必須為10的倍數。共有多少種分發的方式？
Tung Chung City is a place created by the Facelift Workshop. The creators are NLB11A, ilkehongkongdennis and JVsevensixtwonine. Suppose a mysterious person donated to them and the Workshop receives 350 Robux after deducting the marketing fee. Now, the amount is being distributed among the three creators. All the creators must receive a non-zero amount of Robux, and the amount must be a multiple of 10. How many ways of distribution are there?

第四題 Question 4[]

假設勞博士鐵路公司正在進行數學研究，使公司能夠更準確地估計勞博士之星綫任何一日的乘客數量。研究發現乘客數量(P)只受兩個因素影響——列車座位數量(s)及以分鐘計算的平均候車時間(t)。另外，P 一部份與 s 成正比，一部份與 t 成反比。下表顯示研究中所得的數據。鐵路系統進行改善工程後的一天，每架列車的座位數量增加至54，而平均候車時間減低至1分鐘。估計當天的乘客數量。
Suppose the Ryan Railroad was conducting a mathematical analysis to better estimate the number of passengers of the Roblox Star Line on a given day. It was found that the number of passengers (P) depended on two factors only – number of seats on a train (s) and average waiting time in minutes (t). It was found that P partly varies directly as s and partly varies inversely as t. The following table shows the data recorded. One day after the renovation of the railway system, a train had 54 seats and the average waiting time was decreased to 1 minute. Estimate the number of passengers on that day.

P	s	t
480	36	2
640	48	1.5

第五題 Question 5[]

某一日，Bauhinia Group製作了一個駕駛巴士的遊戲。當遊戲首次對外開放時，有27位唯一玩家。為了可以出一條關於該遊戲的無聊數學題，假設所有玩家都是假想的數學個體。由於Roblox的朋友活動欄目幫助宣傳遊戲，所有玩家第一次進入遊戲都會帶來若干位新玩家。下表為新玩家上升的數量與其概率。已知新玩家的數量只能透過上述的方式上升，求最終唯一玩家數量的期望值。
One day, the Bauhinia Group created a new bus driving place. When the game is first released, 24 unique players visited the place. For the sake of this problem, all of the players are assumed to be very mathematical beings. Thanks to the friends’ activity section provided by Roblox, new players can be brought to the game by the players when they first visited the game. The following table shows the number of new players and their probabilities. Given that the number of unique players can only be increased through the above way. Find the expected number of unique players for the game in the end.

新玩家數量 Number of new players	概率 Probability
0	41%
1	32%
2	16%
3	11%

第六題 Question 6[]

某人想製作一個滾珠遊戲，他打算用圖6.1中以磚頭製成的漏斗，漏斗底部有一個洞讓球穿過。若要製造漏斗，先將兩塊大小相同的磚頭放在同一軸上，然後將它們圍繞 y 軸旋轉。如圖6.2所示，磚頭的斜邊長度和厚度分別為20 stud和2 stud，磚頭的寬足以防止磚頭與磚頭之間出現空隙。磚頭的傾斜度為，以24塊磚頭製成漏斗。該人想直徑為6 stud的球體穿過底部，求兩塊磚頭中心之間的最短距離。
A person wants to create a marble rolling game. Funnels made from parts in Figure 6.1 are planned to be used. A hole exists at the bottom to allow the ball to fall through. To make a funnel, two parts of the same size are put on the same axis with a certain separation. Then, the parts are rotated about the y-axis to form the shape. In Figure 6.2, the parts have a slant length of 20 studs and a thickness of 2 studs. The width is long enough to prevent any gaps between the parts. The inclination of the parts is . The funnel is made from 24 parts. The person wants a sphere with a diameter of 6 studs to pass through the hole in the bottom. Find the minimum distance between the centres of the two parts.

圖6.1 Figure 6.1

File:S2q6b.png

圖6.2 Figure 6.2

乙部 Part B[]

第七題 Question 7[]

求二進制數字 110110...110110₂（共3333位）除以 11₁₀ 的餘數。
Find the remainder when the binary number 110110...110110₂ (3333 digits) is divided by 11₁₀.

第八題 Question 8[]

將

$F_n$ 記為斐波那契數列中的第

$n$ 項，當中

${{F}_{1}}={{F}_{2}}=1$ ；當

$n \ge 3$ ，

${{F}_{n}}={{F}_{n-1}}+{{F}_{n-2}}$ 。

$a$ 和

$b$ 為實數，使得

${{F}_{2013}}a+{{F}_{2015}}b={{F}_{2017}}$ 和

${{F}_{2014}}a+{{F}_{2016}}b={{F}_{2018}}$ 。求

$ab$ 。
Denote

$F_n$ as the

$n$ th term of the Fibonacci sequence, where

${{F}_{1}}={{F}_{2}}=1$ and

${{F}_{n}}={{F}_{n-1}}+{{F}_{n-2}}$ for

$n \ge 3$ .

$a$ and

$b$ are real numbers such that

${{F}_{2013}}a+{{F}_{2015}}b={{F}_{2017}}$ and

${{F}_{2014}}a+{{F}_{2016}}b={{F}_{2018}}$ . Find

$ab$ .

第九題 Question 9[]

設

$\phi$ 為直線

$y=\left( \sin \theta \right)x$ 和

$y=\left( \cos \theta \right)x$ 之間的銳角，當中

$0{}^\circ \le \theta \le 360{}^\circ$ 。求

$\tan \phi$ 的最大值。
Let

$\phi$ be the acute angle between the lines

$y=\left( \sin \theta \right)x$ and

$y=\left( \cos \theta \right)x$ , where

$0{}^\circ \le \theta \le 360{}^\circ$ . Find the maximum value of

$\tan \phi$ .

第十題 Question 10[]

AB 和 BC 是一個正六邊形的其中兩邊，其中心位於第一象限。已知 A 和 B 的座標分別為 (-1,-1) 和 (4,-3)。設

$(x,y)$ 為 C 的座標，求

$x + y$ 。
AB and BC are two edges of a regular hexagon. Its centre lies in the first quadrant. It is known that the coordinates of A and B are (-1,-1) and (4,-3), respectively. Let

$(x,y)$ be the coordinates of C. Find

$x + y$ .

第十一題 Question 11[]

如果一個數字的數位從左到右都不會下降，該數字就是一個「上升數字」，例如1234和5567。任意選擇一個八位數字，求該數字為「上升數字」的概率。
A number is said to be an “ascending number” if all its digits from left to right never decreases. For example, 1234 and 5567 are “ascending numbers”. An 8-digit number is chosen randomly. Find the probability that it is an “ascending number”.

第十二題 Question 12[]

如圖，ABCD 是一個邊長為 1 的正方形，有一個以 AB 作為半徑的四分之一圓在內。繪製一個與四分之一圓、 AD 和 CD 相切的圓形，求該圓形的半徑。
In the figure, ABCD is a square with a side length of 1. A quarter circle with AB as the radius is in it. A circle touching the quarter circle, AD and CD is constructed. Find the radius of the circle.

File:S2q12.png

題目完
End of questions