INSTRUCTIONS
- Do not open the paper until the invigilator tells you to do so.
- Time allowed: $\mathbf{9 0}$ minutes.
No answers, or personal details, may be entered after the allowed time is over. - The use of blank paper for rough working is allowed; squared paper, calculators and measuring instruments are forbidden.
- Use a B or an HB non-propelling pencil. Mark A, B, C, D, E on the Answer Sheet for each question. Mark only one option, boldly, within the box.
- Your Answer Sheet will be read by a machine. Do not write or doodle on the sheet except to mark your chosen options. The machine will read all markings, including bits of eraser stuck to the page, and interpret the mark in its own way.
- Do not expect to finish the whole paper in the time allowed. The questions in this paper have been arranged in approximate order of difficulty with the harder questions towards the end. You are not expected to complete all the questions during the time. You should bear this in mind when deciding which questions to tackle.
- Scoring rules: All candidates start with 25 marks; 0 marks are awarded for each question left unanswered; 4 marks are awarded for each correct answer; 1 mark is deducted for each incorrect answer (to discourage guessing).
- The questions on this paper are designed to challenge you to think, not to guess. You will gain more marks, and more satisfaction, by doing one question carefully than by guessing lots of answers. This paper is about solving interesting problems, not about lucky guessing.
- To accommodate candidates sitting at other times, please do not discuss the paper on the internet until 08:00 BST on Friday 12 November.
- Cicely had her $21^{\text {s }}$ birthday in 1939 .
When did she have her $100^{\text {th }}$ birthday?
A 2020
B 2019
C 2018
D 2010
E 2008 - The sequence, formed from the sequence of primes by rounding each to the nearest ten, begins 0,0 , $10,10,10,10,20,20,20,30, \ldots$.
When continued, how many terms in this sequence are equal to 40 ?
A 1
B 2
C 3
D 4
E 5 - The diagram shows two congruent regular pentagons and a triangle.
The angles marked $x^{\circ}$ are equal.
What is the value of $x$ ?
A 24
B 30
C 36
D 40
E 45 - The positive integer $k$ is a solution of the equation $(k \div 12) \div(15 \div k)=20$.
What is the sum of the digits of $k$ ?
A 15
B 12
C 9
D 6
E 3 - The sum of four consecutive primes is itself prime.
What is the largest of the four primes?
A 37
B 29
C 19
D 13
E 7 - Three points, $P, Q$ and $R$ are placed on the circumference of a circle with centre $O$. The arc lengths $P Q, Q R$ and $R P$ are in the ratio $1: 2: 3$.
In what ratio are the areas of the sectors $P O Q, Q O R$ and $R O P$ ?
A $1: 1: 1$
B $1: 2: 3$
C $1: \pi: \pi^2$
D $1: 4: 9$
E 1:8:27 - Which of these numbers is the largest?
A $2^{5000}$
B $3^{4000}$
C $4^{3000}$
D $5^{2000}$
E $6^{1000}$
- What is the area of the region inside the quadrilateral $P Q R S$ ?
A $18 \quad$ B 24 E more information needed
C 36
D 48
P - Alison has a set of ten fridge magnets showing the integers from 0 to 9 inclusive.
In how many different ways can she split the set into five pairs so that the sum of each pair is a multiple of 5 ?
A 1 B 2
C 3
D 4
E 5 - In a survey, people were asked to name their favourite fruit pie. The pie chart shows the outcome. The angles shown are exact with no rounding.
What is the smallest number of people who could have been $\begin{array}{cccccc}\text { A } 45 & \text { B } 60 & \text { C } 80 & \text { D } 90 & \text { E } 180\end{array}$ cherry $108^{\circ} / 40^{\circ} 40^{\circ}$ plum
- Alitta claims that if $p$ is an odd prime then $p^2-2$ is also an odd prime.
Which of the following values of $p$ is a counterexample to this claim?
A 3
B 5
C 7
D 9
E 11 - For how many positive integers $N$ is the remainder 6 when 111 is divided by $N$ ?
A 5
B 4
C 3
D 2
E 1 - Which of these is the mean of the other four?
A $\sqrt{2}$
B $\sqrt{18}$
C. $\sqrt{200}$
D $\sqrt{32}$
E $\sqrt{8}$ - What is the smallest number of rectangles, each measuring $2 \mathrm{~cm}$ by $3 \mathrm{~cm}$, which are needed to fit together without overlap to form a rectangle whose sides are in the ratio $5: 4$ ?
A 10
B 15
C 20
D 30
E 60 - Three dice, each showing numbers 1 to 6 , are coloured red, blue and yellow respectively. Each of the dice is rolled once. The total of the numbers rolled is 10 . In how many different ways can this happen?
A 36
B 30
C 27
D 24
E 21 - A circle of radius $r$ and a right-angled isosceles triangle are drawn such that one of the shorter sides of the triangle is a diameter of the circle.
What is the shaded area?
A $\sqrt{2} r$
C $2 \pi r$
D $\frac{\pi r^2}{4}$
$\mathrm{E}(\sqrt{2}-1) \pi r^2$
- The number 840 can be written as $\frac{p !}{q !}$, where $p$ and $q$ are positive integers less than 10 . What is the value of $p+q$ ?
Note that, $n !=1 \times 2 \times 3 \times \cdots \times(n-1) \times n$.
A 8
B 9
C 10
D 12
E 15 - The diagram shows two overlapping triangles: triangle $F G H$ with interior angles $60^{\circ}, 30^{\circ}$ and $90^{\circ}$ and triangle $E G H$ which is a right-angled isosceles triangle.
What is the ratio of the area of triangle $I F G$ to the area of triangle $I E H^?$ ?
A $1: 1$
B $1: \sqrt{2}$
C $1: \sqrt{3}$
D $1: 2$
E 1:3
- Laura and Dina have a running race. Laura runs at constant speed and Dina runs $n$ times as fast where $n>1$. Laura starts $s \mathrm{~m}$ in front of Dina.
What distance, in metres, does Dina run before she overtakes Laura?
A $\frac{n . s}{n-1} \quad$ B $n s$
C $\frac{s}{n-1}$
D $\frac{n s}{n+1}$
$\mathrm{E} \frac{s}{n}$ - The numbers $m$ and $k$ satisfy the equations $2^m+2^k=p$ and $2^m-2^k=q$.
What is the value of $2^{m+k}$ in terms of $p$ and $q$ ?
A $\frac{p^2-q^2}{4} \quad$ B $\frac{p q}{2}$
C $p+q \quad$ D $\frac{(p-q)^2}{4}$
$\mathrm{E} \frac{p+q}{p-q}$ - A triangle with interior angles $60^{\circ}, 45^{\circ}$ and $75^{\circ}$ is inscribed in a circle of radius 2 .
What is the area of the triangle?
A $2 \sqrt{3}$
B 4
C $6+\sqrt{3}$
D $6 \sqrt{3}$
E $3+\sqrt{3}$ - Let $x$ be a real number. What is the minimum value of $\left(x^2-4 x+3\right)\left(x^2+4 x+3\right)$ ?
$A-16$
B $-9$
C 0
D 9
E 16 - Saba, Rayan and Derin are cooperating to complete a task. They each work at a constant rate independent of whoever else is working on the task. When all three work together, it takes 5 minutes to complete the task. When Saba is working with Derin, the task takes 7 minutes to complete. When Rayan is working with Derin, the task takes 15 minutes to complete.
How many minutes does it take for Derin to complete the task on his own?
A 21
B 28
C 35
D 48
E 105
- Five line segments of length $2,2,2,1$ and 3 connect two corners of a square as shown in the diagram.
What is the shaded area?
A 8
B 9
C 10
D 11
E 12
SMC高级挑战赛考试内容
JMC考试涵盖代数、几何、数论、统计概率四大模块考点。从基本考点分布来讲，和其他各大数学竞赛基本一致。
核心考点：
数论
1. Factor & Multiple 因数与倍数
2. Prime Number 质数与合数
3. Remainder 余数问题
4. Complex Number Theory 复杂数论问题
代数
1. Sequence 数列
2. Polynomial 多项式
3. Equation /Function/Graph 方程与函数
4. Inequality 不等式
计数与概率
1. Rule of Sum and Product 加乘原理
2. Permutation & Combination 排列组合
3. Inclusion-exclusion Principle 容斥原理
4. Logic 逻辑分析
几何
1. Triangle/Polygons 三角形和多边形
2. Similar Triangle, Congruent Triangle 相似三角形，全等三角形
3. Circle 圆
4. Solid geometry 立体几何
除此，SMC又有其独特之处。
循序渐进，逐步提升
UKMT 整个系列赛事，题目设计特别注重知识的延续性。我们以其最爱的数字谜题为例，这也是钱老师最喜欢和同学们分享的一个例子。
初级比赛中，简单考察基础的“线性等和问题” （JMC 2018 Q6）
中级比赛中，将线性求和问题延伸到“圆形等和” （IMC 2016 Q13）
高级比赛中，等和问题提升到等积问题，更巧妙融入了因数讨论（SMC 2016 Q10）
