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1 | \noindent 1. What is two-fifths of the recurring decimal $0.2\dot{5}$?
\begin{itemize}
\item[A] $0.1$
\item[B] $0.0\dot{1}$
\item[C] $0.0\dot{1}\dot{0}$
\item[D] $0.\dot{1}\dot{0}$
\item[E] $0.1\dot{0}$
\end{itemize} | E |
2 | \noindent 2. A twip is a very short unit of length, derived from imperial units, and is equal to approximately $0.000018$ metres. A league is a long unit of length which is equal to approximately $4800$ metres. Roughly how many twips are there in a league?
\begin{itemize}
\item[A] $270,000,000$
\item[B] $27,00... | A |
3 | \noindent 3. Two standard dice are placed on a table, with one on top of the other, so that only nine of the faces of the dice may be seen. The touching faces have the same number on them. The sum of the numbers on the visible faces is 33. What is the number on the touching faces?
\begin{itemize}
\item[A] 1
\i... | B |
4 | \noindent 4. The sizes of the three angles in a triangle, in degrees, are $x$, $7x$ and $x^2$. What is the size of the largest angle?
\begin{itemize}
\item[A] $10^{\circ}$
\item[B] $18^{\circ}$
\item[C] $100^{\circ}$
\item[D] $120^{\circ}$
\item[E] $121^{\circ}$
\end{itemize} | C |
5 | \noindent 5. When $4^5 \times 5^4$ is correctly calculated, how many digits are there in the answer?
\begin{itemize}
\item[A] 4
\item[B] 6
\item[C] 10
\item[D] 16
\item[E] 20
\end{itemize} | B |
6 | \noindent 6. One face of a solid polyhedron is an octagon. What is the smallest possible number of edges the solid could have?
\begin{itemize}
\item[A] 9
\item[B] 10
\item[C] 12
\item[D] 16
\item[E] 24
\end{itemize} | D |
7 | \noindent 7. Which is the largest prime factor of $3^8 - 1$?
\begin{itemize}
\item[A] 41
\item[B] 37
\item[C] 31
\item[D] 29
\item[E] 23
\end{itemize} | A |
8 | \noindent 8. In the following expressions, $x$ is non-zero. When one of these expressions is removed, the mean of the remaining four is $11x$. Which expression is removed?
\begin{itemize}
\item[A] $4x$
\item[B] $8x$
\item[C] $12x$
\item[D] $16x$
\item[E] $20x$
\end{itemize} | D |
9 | \noindent 9. A palindromic number is one where the digits read the same forwards as backwards, such as 123 321. What is the hundreds digit of the largest six-digit palindromic number that is divisible by 18?
\begin{itemize}
\item[A] 9
\item[B] 7
\item[C] 5
\item[D] 3
\item[E] 1
\end{itemize} | E |
10 | \noindent 10. The prime factorization of 2024 is $2^3 \times 11 \times 23$. How many two-digit numbers are factors of 2024?
\begin{itemize}
\item[A] 2
\item[B] 4
\item[C] 6
\item[D] 7
\item[E] 8
\end{itemize} | D |
11 | \noindent 11. Which one of the following expressions is a square number for each positive integer $n$?
\begin{itemize}
\item[A] $n+1$
\item[B] $n(n+1)+1$
\item[C] $n(n+1)(n+2)+1$
\item[D] $n(n+1)(n+2)(n+3)+1$
\item[E] $n(n+1)(n+2)(n+3)(n+4)+1$
\end{itemize} | D |
12 | \noindent 12. $p, q, r$ and $s$ are two-digit primes which between them use all the non-zero digits except 5. What is the value of $p+q+r+s$?
\begin{itemize}
\item[A] 220
\item[B] 210
\item[C] 200
\item[D] 190
\item[E] more information needed
\end{itemize} | A |
13 | \noindent 13. The diagram shows a partially completed number pyramid. When correctly completed, the number on any brick above the bottom row should be the sum of the two numbers on the two bricks on which it rests. What number should appear on the brick marked `$z$'?
\begin{center} \includegraphics[width=0.3\textwidth... | C |
14 | \noindent 14. P, Q, R, S and T are the digits 1, 2, 3, 4 and 5 in some order. `PRT' and `QRS' are both three-digit primes. Which digit is R?
\begin{itemize}
\item[A] 1
\item[B] 2
\item[C] 3
\item[D] 4
\item[E] 5
\end{itemize} | B |
15 | \noindent 15. The diagram shows two squares, JKLM and NKPO. The length of NL is 10 cm. The shaded region has area $62 \text{ cm}^2$. What is the length of KN in cm?
\begin{center} \includegraphics[width=0.3\textwidth]{SMC-2024-Paper_page3_image2.png} \end{center}
\begin{itemize}
\item[A] 3
\item[B] $\sqrt{18}$... | C |
16 | \noindent 16. A set of cupboards containing eight identical blue doors is arranged in a 2 by 4 grid as shown. A fussy decorator wishes to paint three of the doors red such that at least one door in each row is painted red and at least two of the four corners are painted red. How many ways are there to do this?
\begin{... | B |
17 | \noindent 17. A bag contains four balls each of which is coloured either red or white. If one ball is drawn at random from the bag but not replaced and then a second ball is drawn at random, the probability that both balls are red is $\frac{1}{2}$. What is the probability that both balls are white?
\begin{itemize}
... | E |
18 | \noindent 18. The diagram shows two concentric circles divided by radial lines into 14 pieces of equal area. The radius of the smaller circle is 1. What is the length, $x$, of an outer radial line?
\begin{center} \includegraphics[width=0.3\textwidth]{SMC-2024-Paper_page3_image3.png} \end{center}
\begin{itemize}
\i... | C |
19 | \noindent 19. Five friends are dealt two cards each from a set of twelve cards. The cards are numbered 1 to 12 inclusive. In turn, the friends declare the sum of the values of their two cards. Paolo scores 4, Quinn scores 11, Romy scores 16, Stephen scores 19 and Thomas scores 20. Which of the following statements is t... | C |
20 | \noindent 20. Let $x$ and $y$ be positive integers such that $\frac{1}{x}+\frac{1}{y}=\frac{1}{20}$. What is the maximum possible value of $y$?
\begin{itemize}
\item[A] 40
\item[B] 60
\item[C] 240
\item[D] 420
\item[E] 480
\end{itemize} | D |
21 | \noindent 21. The crossnumber is to be filled with eight of the digits 1 to 9, which are each used once. Which digit is not used?
\begin{center} \includegraphics[width=0.2\textwidth]{SMC-2024-Paper_page4_image1.png} \end{center}
\textbf{Across}
\begin{itemize}
\item[1.] A multiple of 9
\item[3.] A square
\en... | B |
22 | \noindent 22. As shown in the diagram, triangle FGH is divided into four smaller triangles which have areas 4, 8, 12 and 8 respectively. What is the area of triangle IKH?
\begin{center} \includegraphics[width=0.4\textwidth]{SMC-2024-Paper_page4_image2.png} \end{center}
\begin{itemize}
\item[A] 4
\item[B] 5
... | A |
23 | \noindent 23. The plane can be tiled using the `hat tile' shown here. This tile can be subdivided into eight congruent kites. The area of the hat tile is $8\sqrt{3}$. What is the perimeter of the hat tile?
\begin{center} \includegraphics[width=0.3\textwidth]{SMC-2024-Paper_page4_image3.png} \end{center}
\begin{itemize... | E |
24 | \noindent 24. A function f satisfies the equation $f(x)+f\left(\frac{1}{1-x}\right)=24x$ for all real values of $x$ except $x=0$ and $x=1$. What is the value of $f(3)$?
\begin{itemize}
\item[A] 40
\item[B] 42
\item[C] 45
\item[D] 48
\item[E] 50
\end{itemize} | E |
25 | \noindent 25. Three semicircles, each of area 24, overlap as shown in the diagram. The centres of the arcs are X, Y and Z and $\angle ZXY = 30^{\circ}$. What is the total area of the shaded regions?
\begin{center} \includegraphics[width=0.4\textwidth]{SMC-2024-Paper_page4_image4.png} \end{center}
\begin{itemize}
\... | A |
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