Multiple Choice

1. A mystery number $x$ is multiplied by $8$. Then, $12$ is subtracted from the result. This new value is divided by $4$. Finally, the cube root is taken, yielding a total of $3$. What is the value of $x$?

   $15$

   $14.5$

   $16$

   $9$

2. Three-quarters of a barrel of oil was used on Monday. On Tuesday, half of the *remaining* oil was used. If $12$ litres ($L$) of oil are left in the barrel, applying the Working Backwards technique, how many litres did the barrel originally contain?

   $48$

   $96$

   $72$

   $64$

3. Sarah spent $\frac{1}{5}$ of her initial savings $P$. She then deposited $\$ 100$. The resulting amount was then exactly doubled, leaving her with a total of $\$ 520$. What was the initial amount $P$?

   $\$ 180$

   $\$ 150$

   $\$ 220$

   $\$ 200$

4. Amy, Ben, and Chloe share a box of cards. Amy takes half the cards plus $1$. Ben takes half of the *remaining* cards plus $1$. Chloe takes the final $10$ cards. How many cards were in the original box?

   $46$

   $38$

   $48$

   $42$

5. The process of calculating a result $f(x)$ from an input $x$ is described by the function $f(x) = \frac{3x - 5}{2}$. If the final output is $17$, what was the original input $x$?

   $14$

   $13$

   $11$

   $12.5$

6. A linear sequence is defined by the general term $a_n = 5n - 1$. Applying the pattern recognition technique, what is the sum of the $50^{th}$ term and the $51^{st}$ term of this sequence?

   $505$

   $498$

   $503$

   $500$

7. A geometric arrangement of dots yields the sequence $a_n$, where the $n^{th}$ term is described by the generalization $a_n = \frac{n(n+3)}{2}$. Using pattern recognition and generalization, determine the value of the $10^{th}$ term, $a_{10}$.

   $55$

   $65$

   $60$

   $78$

8. By analyzing the pattern of the units digits of successive powers of $7$ (i.e., $7^1, 7^2, 7^3, \dots$), determine the units digit of the number $7^{2023}$.

   $3$

   $1$

   $9$

   $7$

9. An expanding structural pattern follows the rule $S_n = 3n + 7$, where $S_n$ is the total number of components in step $n$. Using generalization, determine which term number $n$ corresponds to a structure containing exactly $154$ components.

   $50$

   $49$

   $47$

   $48$

10. Given the sequence $1, 2, 3, 5, 8, 13, \dots$, where each term is the sum of the two previous terms, applying pattern recognition allows prediction. What is the $10^{th}$ term of this sequence?

    $85$

    $77$

    $68$

    $89$

11. Applying the methodology of solving a simpler case and generalizing the relationship for the number of diagonals $D_n$ in an $n$-sided polygon ($D_n = \frac{n(n-3)}{2}$), calculate the number of diagonals in a convex polygon with $15$ sides.

    $75$

    $105$

    $90$

    $120$

12. A problem involving complex enumeration can be solved by reducing $n$ to small values (e.g., $n=3, 4, 5$). If $20$ people attend a conference and everyone shakes hands exactly once with every other person, how many total handshakes occur?

    $210$

    $195$

    $190$

    $180$

13. A large cube is constructed from smaller $1\text{ cm}^3$ unit cubes to form a cube of side length $4\text{ cm}$. If the exterior of the large cube is painted, how many of the small unit cubes have exactly $2$ faces painted? (Hint: Use visualization by simplifying to a $3 \times 3 \times 3$ case first).

    $16$

    $32$

    $24$

    $8$

14. A point starts at the origin $(0, 0)$ and moves in a sequence of steps: East $1$, North $2$, West $3$, South $4$, East $5$, North $6$, and so on. By analyzing the pattern generated by a few simple cases, determine the $x$-coordinate of the point after exactly $100$ steps.

    $50$

    $-50$

    $0$

    $-100$

15. A single-elimination tennis tournament begins with $128$ participants. Using a simplified case (e.g., $4$ players) to establish the relationship between players and matches, determine the total number of matches required to declare a single winner.

    $256$

    $120$

    $127$

    $64$

16. A bakery sells large pies for $\$ 15$ and small tarts for $\$ 8$. They sold $20$ total items and generated total revenue of $\$ 237$. By applying refined Guess \& Check, how many large pies were sold?

    $13$

    $11$

    $9$

    $10$

17. The product of three consecutive positive integers is $2730$. By determining logical bounds and systematically testing integers near the cube root of the product, find the sum of these three integers.

    $40$

    $39$

    $45$

    $42$

18. An integer $N$ is subject to two simultaneous conditions: when $N$ is divided by $5$, the remainder is $2$; when $N$ is divided by $7$, the remainder is $3$. Using logical elimination and systematic testing, find the smallest positive value of $N$.

    $22$

    $17$

    $32$

    $38$

19. By setting logical bounds and using systematic algebraic elimination (as refined Guess \& Check), determine John's current age given these two constraints: Two years ago, John was $5$ times older than his son David. In $4$ years, John will be $3$ times older than David.

    $35$

    $30$

    $32$

    $40$

20. A student buys $50$ items (pencils and erasers) for a total cost of $\$ 100$. Pencils cost $\$ 3$ each, and erasers cost $\$ 1$ each. Using logical bounds or algebraic substitution, how many pencils did the student buy?

    $25$

    $15$

    $30$

    $20$