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Stage 4 · Mixed

10 problems · AMC 12 late, AIME early · mathsolympiadprep.com

Print or save as PDF The answer key prints on its own page at the end.

1. Given the function $f(x) = 2\sin{x}\cos{x} - 2\sin^2{x} + 1$ $($ for $x \in \mathbb{R}$ $)$, in a triangle $ABC$ where the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, $a= \sqrt{3}$, angle $A$ is acute, and $f\left(A+ \frac{\pi}{8}\right) = \frac{\sqrt{2}}{3}$. Determine the maximum area of $\triangle ABC$.

   Algebra Solution and answer checking →

2. Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is

   1. A$\frac{1}{2m+1}$
   2. Bm
   3. C1-m
   4. D$\frac{1}{4m}$
   5. E$\frac{1}{8m^2}$

   Geometry Solution and answer checking →

3. Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $400$ feet or less to the new gate be a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

   Combinatorics Solution and answer checking →

4. The solutions to the equation $(z+6)^8=81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A,B,$ and $C$. What is the least possible area of $\triangle ABC?$

   1. A$\frac{1}{6}\sqrt{6}$
   2. B$\frac{3}{2}\sqrt{2}-\frac{3}{2}$
   3. C$2\sqrt3-3\sqrt2$
   4. D$\frac{1}{2}\sqrt{2}$
   5. E$\sqrt 3-1$

   Geometry Solution and answer checking →

5. If $x+y+z=25$ and $y+z=14$, then $x$ is

   1. A8
   2. B11
   3. C6
   4. D-6
   5. E31

   Algebra Solution and answer checking →

6. How many two-digit squares differ by 1 from a multiple of 10 ?
   A 1
   B 2
   C 3
   D 4
   E 5

   Number theory Solution and answer checking →

7. Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\,$ or $10''\,$ or $19''\,$ to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks?

   Combinatorics Solution and answer checking →

8. The perimeter of an isosceles triangle is $24 \mathrm{~cm}$, and a median divides the perimeter into two parts in the ratio 5:3. The length of the base of this triangle is ( $) \mathrm{cm}$.

   1. A7.5
   2. B12
   3. C4
   4. D12 or 4

   Geometry Solution and answer checking →

9. A natural number $a$ has 107 different divisors, including 1 and $a$. Find the sum and product of these divisors.

   Number theory Solution and answer checking →

10. Some students participate in three competitions, the first place gets 5 points, the second place gets 3 points, the third place gets 1 point, and no points for not placing. Then, how many points must a student get to definitely have more points than another student?

    1. A9
    2. B10
    3. C11
    4. D13
    5. E15

    Combinatorics Solution and answer checking →

Answer key — Stage 4 · Mixed

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

1. $A: \frac{3(\sqrt{3} + \sqrt{2})}{4}$ open
2. $\frac{1}{4m}$ open
3. $52$ open
4. $\textbf{(B)}\frac{3}{2}\sqrt{2}-\frac{3}{2}$ open
5. $11$ open
6. $2$ open
7. $465$ open
8. $D$ open
9. $\Sigma_{}=\frac{\sqrt[106]{}-1}{\sqrt[106]{}-1};\Pi_{}=\sqrt{}$ open
10. $D$ open

Problems belong to the competitions that set them and are reproduced from open datasets under their licences; every problem page names its source. Free to copy for classroom use.