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Stage 3 · Algebra

10 problems · AMC 10/12, early questions · mathsolympiadprep.com

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

1. Let $r = \sqrt{\frac{\sqrt{53}}{2} + \frac{3}{2}}$. There is a unique triple of positive integers $(a, b, c)$ such that $r^{100} = 2r^{98} + 14r^{96} + 11r^{94} - r^{50} + ar^{46} + br^{44} + cr^{40}$. What is the value of $a^{2} + b^{2} + c^{2}$?

   CEMC Fermat

   Algebra Solution and answer checking →

2. At the end of the year 2000, Steve had $\$100$ and Wayne had $\$10000$. At the end of each following year, Steve had twice as much money as he did at the end of the previous year and Wayne had half as much money as he did at the end of the previous year. At the end of which year did Steve have more money than Wayne for the first time?

   CEMC Fermat

   Algebra Solution and answer checking →

3. A set consists of five different odd positive integers, each greater than 2. When these five integers are multiplied together, their product is a five-digit integer of the form $AB0AB$, where $A$ and $B$ are digits with $A \neq 0$ and $A \neq B$. (The hundreds digit of the product is zero.) For example, the integers in the set $\{3,5,7,13,33\}$ have a product of 45045. In total, how many different sets of five different odd positive integers have these properties?

   CEMC Cayley

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4. In the sum shown, each letter represents a different digit with $T \neq 0$ and $W \neq 0$. How many different values of $U$ are possible?

   $$\begin{array}{rrrrr} & W & X & Y & Z \\ + & W & X & Y & Z \\ \hline & W & U & Y & V \end{array}$$

   CEMC Fermat

   Algebra Solution and answer checking →

5. The set $S$ consists of 9 distinct positive integers. The average of the two smallest integers in $S$ is 5. The average of the two largest integers in $S$ is 22. What is the greatest possible average of all of the integers of $S$?

   CEMC Fermat

   Algebra Solution and answer checking →

6. A sequence of numbers $t_{1}, t_{2}, t_{3}, \ldots$ has its terms defined by $t_{n}=\frac{1}{n}-\frac{1}{n+2}$ for every integer $n \geq 1$. What is the largest positive integer $k$ for which the sum of the first $k$ terms is less than 1.499?

   CEMC Fermat

   Algebra Solution and answer checking →

7. For how many positive integers $k$ do the lines with equations $9x+4y=600$ and $kx-4y=24$ intersect at a point whose coordinates are positive integers?

   CEMC Fermat

   Algebra Solution and answer checking →

8. For each positive digit $D$ and positive integer $k$, we use the symbol $D_{(k)}$ to represent the positive integer having exactly $k$ digits, each of which is equal to $D$. For example, $2_{(1)}=2$ and $3_{(4)}=3333$. There are $N$ quadruples $(P, Q, R, k)$ with $P, Q$ and $R$ positive digits, $k$ a positive integer with $k \leq 2018$, and $P_{(2k)}-Q_{(k)}=\left(R_{(k)}\right)^{2}$. What is the sum of the digits of $N$?

   CEMC Fermat

   Algebra Solution and answer checking →

9. There are functions $f(x)$ with the following properties: $f(x)=ax^{2}+bx+c$ for some integers $a, b$ and $c$ with $a>0$, and $f(p)=f(q)=17$ and $f(p+q)=47$ for some prime numbers $p$ and $q$ with $p<q$. For each such function, the value of $f(pq)$ is calculated. The sum of all possible values of $f(pq)$ is $S$. What are the rightmost two digits of $S$?

   CEMC Fermat

   Algebra Solution and answer checking →

10. Sergio recently opened a store. One day, he determined that the average number of items sold per employee to date was 75. The next day, one employee sold 6 items, one employee sold 5 items, and one employee sold 4 items. The remaining employees each sold 3 items. This made the new average number of items sold per employee to date equal to 78.3. How many employees are there at the store?

    1. A$50$
    2. B$5$
    3. C$20$
    4. D$40$
    5. E$30$

    CEMC Fermat 2014

    Algebra Solution and answer checking →

Answer key — Stage 3 · Algebra

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

1. $15339$ open
2. $2004$ open
3. $24$ open
4. $3$ open
5. $16$ open
6. $1998$ open
7. $7$ open
8. $11$ open
9. $71$ open
10. $C$ open

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