Pascal Cayley Fermat竞赛

Pascal Cayley Fermat竞赛考试由25个选择题构成，时间是一个小时。题目比aime简单很多。

其中前十个题偏简单后十五个题有一定难度。

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Scoring: There is no penalty for an incorrect answer.
Each unanswered question is worth 2, to a maximum of 10 unanswered questions.

Part A: Each correct answer is worth $5 .$

$3 \mathrm{~cm}$ The area of the rectangle is $\begin{array}{llll}\text { A) } 2 \mathrm{~cm}^{2} & \text { (B) } 9 \mathrm{~cm}^{2} & \text { (C) } 5 \mathrm{~cm}^{2} & 2 \mathrm{~cm}\end{array}$
$\begin{array}{llll}\text { (A) } 2 \mathrm{~cm}^{2} & \text { (B) } 9 \mathrm{~cm}^{2} & \text { (C) } 5 \mathrm{~cm}^{2} & 2 \mathrm{~cm} \ \text { (D) } 36 \mathrm{~cm}^{2} & \text { (E) } 6 \mathrm{~cm}^{2} & \text { (C) } 35 & \text { (D) } 17\end{array}$ 2. The expression $2+3 \times 5+2$ equals $\begin{array}{lll}\text { (A) } 19 & \text { (B) } 27 & \text { (C) } 60 \text { is }\end{array}$
(E) 32
(A) 10
(B) 15
(C) 20
(D) 12
(E) 18

4. When $x=2021$, the value of $\frac{4 x}{x+2 x}$ is
   (A) $\frac{3}{4}$
   (B) $\frac{4}{3}$
   (C) 2021
   (D) 2
   (E) 6
5. Which of the following integers cannot be written as a product of two integers, each greater than 1 ?
   (A) 6
   (B) 27
   (C) 53
   (D) 39
   (E) 77
6. A square piece of paper has a dot in its top right corner and is lying on a table. The square is folded along its diagonal, then rotated $90^{\circ}$ clockwise about its centre, and then finally unfolded, as shown.
   The resulting figure is $\begin{array}{lll}\text { (A) } & \text { (B) } & \text { (C) } & \square\end{array}$
   (D)
   (E)
7. For which of the following values of $x$ is $x$ greater than $x^{2} ?$ $\begin{array}{lllll}\text { (A) } x=-2 & \text { (B) } x=-\frac{1}{2} & \text { (C) } x=0 & \text { (D) } x=\frac{1}{2} & \text { (E) } x=2\end{array}$
8. The digits in a two-digit positive integer are reversed. The new two-digit integer minus the original integer equals 54 . What is the positive difference between the two – digits of the original integer?
   (A) 5
   (B) 7
   (C) 6
   (D) 8
   (E) 9

Part B: Each correct answer is worth 6 .

11. In the sum shown, $P, Q$ and $R$ represent three different single digits. The value of $P+Q+R$ is
    (A) 13
    (B) 12
    (C) 14
    (E) 4
    (D) 3
    \begin{tabular}{r}
    $P \quad 7 \quad R$ \
    $+\quad 3 \quad 9 \quad R$ \
    \hline$R \quad Q \quad 0$
    \end{tabular}
12. How many of the 20 perfect squares $1^{2}, 2^{2}, 3^{2}, \ldots, 19^{2}, 20^{2}$ are divisible by 9 ?
    (A) 2
    (B) 3
    (C) 4
    (D) 5
    (E) 6
13. In the diagram, each of $\triangle W X Z$ and $\triangle X Y Z$ is an isosceles right-angled triangle. The length of $W X$ is $6 \sqrt{2}$. The perimeter of quadrilateral $W X Y Z$ is closest to
    (A) 18
    (B) 20
    (C) 23
    (E) 29
    (D) 25
14. Natascha cycles 3 times as fast as she runs. She spends 4 hours cycling and 1 hour running. The ratio of the distance that she cycles to the distance that she runs is
    (A) $12: 1$
    (B) $7: 1$
    (C) $4: 3$
    (D) $16: 9$
    (E) $1: 1$
15. Let $a$ and $b$ be positive integers for which $45 a+b=2021$. The minimum possible value of $a+b$ is
    (A) 44
    (B) 82
    (C) 85
    (D) 86
    (E) 130
16. If $n$ is a positive integer, the notation $n !$ (read ” $n$ factorial”) is used to represent the product of the integers from 1 to $n$. That is, $n !=n(n-1)(n-2) \cdots(3)(2)(1)$. For example, $4 !=4(3)(2)(1)=24$ and $1 !=1$. If $a$ and $b$ are positive integers with $b>a$, the ones (units) digit of $b !-a !$ cannot be
    (A) 1
    (B) 3
    (C) 5
    (D) 7
    (E) 9
17. The set $S$ consists of 9 distinct positive integers. The average of the two smallest integers in $S$ is 5 . The average of the two largest integers in $S$ is 22 . What is the greatest possible average of all of the integers of $S$ ?
    (A) 15
    (B) 16
    (C) 17
    (D) 18
    (E) 19
18. In the diagram, $\triangle P Q R$ is an isosceles triangle with
    (A) $\frac{1}{3}$
    is an isosceles triangle with vith diameters $P Q, Q R$ and he areas of these three semi- area of the semi-circle with $(\angle P Q R)$ is
    Pemi-circles with diameters $P Q, Q R$ and drawn. The sum of the areas of these three semi- equal to 5 times the area of the semi-circle with $\begin{array}{ll}\text { (B) } \frac{1}{\sqrt{8}} & \text { (C) } \frac{1}{\sqrt{12}} \ \text { (E) } \frac{1}{\sqrt{10}} & \text { The } \cos (\angle P Q R) \text { is }\end{array}$

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