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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$\sqrt{21}$

   Italian Mathematical Olympiad - February Round

   Geometry Solution and answer checking →

2. Compute the smallest integer $n > 72$ that has the same set of prime divisors as $72$.

   HMMT November 2024

   Number theory Solution and answer checking →

3. How many positive integers $n \leq 2009$ have the property that $\left\lfloor\log_{2}(n)\right\rfloor$ is odd?

   Annual Harvard-MIT Mathematics Tournament

   Algebra Solution and answer checking →

4. A polygon is regular if all its sides and angles are the same. Find the measure of each angle in a regular dodecagon (12-sided polygon).

   Berkeley Math Circle: Monthly Contest 5

   Geometry Solution and answer checking →

5. Suppose $f(x)$ is an odd function with domain $\mathbb{R}$. If $f(1) = 2$, $f(2) = 3$, then the value of $f(f(-1))$ is ______.

   China Mathematical Competition

   Algebra Solution and answer checking →

6. How many two-digit prime numbers have the property that both digits are also primes?

   HMMT November 2014

   Number theory Solution and answer checking →

7. The area of a trapezoid is three times that of an equilateral triangle. If the heights of the trapezoid and the triangle are both equal to $8 \sqrt{3}$, what is the length of the median of the trapezoid?

   Tenth Philippine Mathematical Olympiad

   Geometry Solution and answer checking →

8. How many diagonals does a regular undecagon (11-sided polygon) have?

   Annual Harvard-MIT November Tournament

   Combinatorics Solution and answer checking →

9. The product of three integers is $60$. What is the least possible positive sum of the three integers?

   1. A2
   2. B3
   3. C5
   4. D6
   5. E13

   AMC 10 A

   Algebra Solution and answer checking →

10. How many multiples of $7$ between $10^{6}$ and $10^{9}$ are perfect squares?

    United States competition problems

    Number theory 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. $96$ open
3. $682$ open
4. $150°$ open
5. $-3$ open
6. $4$ open
7. $24$ open
8. $44$ open
9. $B$ open
10. $4375$ 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.

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