158 个页面

## 比賽時間 Duration

2018年5月21日 (8:30pm) 至 5月30日 (10:00pm)
21st May 2018 (8:30pm) to 30th May 2018 (10:00pm)

N/A

## 比賽規則 Rules And Regulations

1. 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.
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.
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.

### 表達式輸入方式 Input of Expressions

Operator/Constant

Input Method

Example

Fraction

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

Power

x^n $2^3$
2^3

Square Root

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

Pi

$\pi =3.14159\ldots$

pi N/A

Expression

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

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

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

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

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

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.

Number of new players

Probability
0 41%
1 32%
2 16%
3 11%

### 第六題 Question 6

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.

## 乙部 Part B

### 第七題 Question 7

Find the remainder when the binary number 110110...1101102 (3333 digits) is divided by 1110.

### 第八題 Question 8

$F_n$ 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 $$b 為實數，使得 {{F}_{2013}}a+{{F}_{2015}}b={{F}_{2017}}$$ {{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

$\phi$ 為直線 $y=\left( \sin \theta \right)x$$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

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

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

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.

End of questions