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Stage 4 · Algebra

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. 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

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2. 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

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3. 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

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4. 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

   Italian Mathematical Olympiad

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5. 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$\frac{12}{5}\sqrt{3}$

   China Mathematical Competition (Shaanxi)

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6. You are out walking and see a group of rhinoceroses (which each have two horns and four legs) and triceratopses (which each have three horns and four legs). If you count 31 horns and 48 legs, how many triceratopses are there?

   Berkeley Math Circle: Monthly Contest 2

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7. Quanto vale $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$?

   1. A$\sqrt[3]{9-4 \sqrt{5}}$
   2. B1
   3. C$\frac{3}{2}$
   4. D$\sqrt[3]{4}$
   5. E$2 \sqrt[3]{2}$

   Progetto Olimpiadi di Matematica GARA di SECONDO LIVELLO 2006

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8. Find all possible values of $\frac{d}{a}$ where $a^{2}-6 a d+8 d^{2}=0$, $a \neq 0$.

   Harvard-MIT Math Tournament

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9. Peter breeds horses and cows on his farm. One day, the number of horses was the same as the number of cows and greater than $0$. Then Peter bought some more cows so that the number of cows increased by $50\%$. Now only $30\%$ of the animals are horses. How many horses does Peter have on his farm?

   1. A$8$
   2. B$9$
   3. C$10$
   4. DSome other number
   5. EThe problem has no solution

   Slovenia — National Math Olympiad 2012

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10. James writes down three integers. Alex picks some two of those integers, takes the average of them, and adds the result to the third integer. If the possible final results Alex could get are $42$, $13$, and $37$, what are the three integers James originally chose?

    HMMT November 2015

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Answer key — Stage 4 · Algebra

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

1. $682$ open
2. $-3$ open
3. $B$ open
4. $B$ open
5. $C$ open
6. $7$ open
7. $B$ open
8. $1/2, 1/4$ open
9. $E$ open
10. $-20, 28, 38$ 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.

CC-BY-4.0