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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. Of a triangle with vertices $A, B, C$ we know that $AB = 5$, $BC = 4$ and $AC = AM$, where $M$ is the midpoint of side $BC$. What is the length of side $AC$?

   1. A4
   2. B$\sqrt{17}$
   3. C$3\sqrt{2}$
   4. D$2\sqrt{5}$
   5. E$\sqrt{21}$

   Geometry Solution and answer checking →

2. 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 →

3. 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 →

4. 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 →

5. How many distinct real solutions does the equation $x^{6}+2 x^{5}+2 x^{4}+2 x^{3}+2 x^{2}+2 x+1=0$ have?

   1. A0
   2. B1
   3. C2
   4. D4
   5. E6

   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. Let $x \in \left[-\frac{5\pi}{12}, -\frac{\pi}{3}\right]$. Then the maximum value of

   $$y = \tan\left(x + \frac{2\pi}{3}\right) - \tan\left(x + \frac{\pi}{6}\right) + \cos\left(x + \frac{\pi}{6}\right)$$
   is:

   1. A$\frac{12}{5}\sqrt{2}$
   2. B$\frac{11}{6}\sqrt{2}$
   3. C$\frac{11}{6}\sqrt{3}$
   4. D$\frac{12}{5}\sqrt{3}$

   Algebra Solution and answer checking →

8. 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 →

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. Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, “I can’t tell who has the larger number.” Then Bob says, “I know who has the larger number.” Alice says, “You do? Is your number prime?” Bob replies, “Yes.” Alice says, “In that case, if I multiply your number by $100$ and add my number, the result is a perfect square.” What is the sum of the two numbers drawn from the hat?

    1. A$27$
    2. B$37$
    3. C$47$
    4. D$57$
    5. E$67$

    Combinatorics Solution and answer checking →

Answer key — Stage 4 · Mixed

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

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