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Stage 2 · Mixed

10 problems · · mathsolympiadprep.com

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1. Sophie has written three tests. Her marks were $73\%$, $82\%$, and $85\%$. She still has two tests to write. All tests are equally weighted. Her goal is an average of $80\%$ or higher. With which of the following pairs of marks on the remaining tests will Sophie not reach her goal: $79\%$ and $82\%$, $70\%$ and $91\%$, $76\%$ and $86\%$, $73\%$ and $83\%$, $61\%$ and $99\%$?

   CEMC Pascal

   Algebra Solution and answer checking →

2. Two identical smaller cubes are stacked next to a larger cube. Each of the two smaller cubes has a volume of 8. What is the volume of the larger cube?

   CEMC Cayley

   Geometry Solution and answer checking →

3. For what value of $k$ is the line through the points $(3,2k+1)$ and $(8,4k-5)$ parallel to the $x$-axis?

   CEMC Cayley

   Algebra Solution and answer checking →

4. In the diagram, what is the value of $x$?

   

   1. A$65$
   2. B$75$
   3. C$85$
   4. D$95$
   5. E$105$

   CEMC Pascal 2014

   Geometry Solution and answer checking →

5. The current calendar year, $2025$, is a perfect square. In $n$ years from 2025, the calendar year will
   again be a perfect square. The smallest possible value of $n$ is

   1. A$2025$
   2. B$100$
   3. C$9$
   4. D$46$
   5. E$91$

   CEMC Fermat 2025

   Number theory Solution and answer checking →

6. Which of the following is equal to $9^{4}$?

   CEMC Cayley

   Algebra Solution and answer checking →

7. The entire exterior of a solid $6 \times 6 \times 3$ rectangular prism is painted. Then, the prism is cut into $1 \times 1 \times 1$ cubes. How many of these cubes have no painted faces?

   CEMC Pascal

   Geometry Solution and answer checking →

8. If $2 \times 2 \times 3 \times 3 \times 5 \times 6 = 5\times 6 \times n \times n$, then $n$ could equal

   1. A$2$
   2. B$3$
   3. C$4$
   4. D$5$
   5. E$6$

   CEMC Pascal 2014

   Number theory Solution and answer checking →

9. Alicia starts a sequence with $m=3$. What is the fifth term of her sequence following the algorithm: Step 1: Alicia writes down the number $m$ as the first term. Step 2: If $m$ is even, Alicia sets $n=\frac{1}{2} m$. If $m$ is odd, Alicia sets $n=m+1$. Step 3: Alicia writes down the number $m+n+1$ as the next term. Step 4: Alicia sets $m$ equal to the value of the term that she just wrote down in Step 3. Step 5: Alicia repeats Steps 2, 3, 4 until she has five terms, at which point she stops.

   CEMC Pascal

   Algebra Solution and answer checking →

10. The minute hand on a clock points at the 12. After rotating $120^{\circ}$ clockwise, which number will it point at?

    CEMC Cayley

    Geometry Solution and answer checking →

Answer key — Stage 2 · Mixed

Worked solutions for every problem are on the site, one page per problem.

1. $73\% and 83\%$ open
2. $64$ open
3. $3$ open
4. $C$ open
5. $E$ open
6. $3^{8}$ open
7. $16$ open
8. $E$ open
9. $43$ open
10. $4$ open

Problems belong to the competitions that set them. Each question on this sheet names the competition it came from, and the licences the material is used under are listed below. Free to copy for classroom use.

Apache-2.0 · CC-BY-NC-4.0