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Stage 3 · Mixed

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

   Algebra Solution and answer checking →

4. What is the smallest possible value of $n$ if a solid cube is made of white plastic and has dimensions $n \times n \times n$, the six faces of the cube are completely covered with gold paint, the cube is then cut into $n^{3}$ cubes, each of which has dimensions $1 \times 1 \times 1$, and the number of $1 \times 1 \times 1$ cubes with 0 gold faces is strictly greater than the number of $1 \times 1 \times 1$ cubes with exactly 1 gold face?

   CEMC Pascal

   Geometry Solution and answer checking →

5. Aaron has 144 identical cubes, each with edge length 1 cm. He uses all of the cubes to construct a solid rectangular prism, which he places on a flat table. If the perimeter of the base of the prism is 20 cm, what is the sum of all possible heights of the prism?

   CEMC Fermat

   Geometry Solution and answer checking →

6. 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 →

7. For how many integers $a$ with $1 \leq a \leq 10$ is $a^{2014}+a^{2015}$ divisible by 5?

   CEMC Fermat

   Number theory Solution and answer checking →

8. Suppose for a quadrilateral $ABCD$, $\angle DAB = 90^\circ$, $\angle ABC = \angle BCD = 60^\circ$. If $AB = 5$ and $CD = 4$, what is the value of $BC$? Here for a line segment $XY$, its length is also denoted by $XY$.

   

   Japan Junior Mathematical Olympiad

   Geometry Solution and answer checking →

9. For how many pairs $(m, n)$ with $m$ and $n$ integers satisfying $1 \leq m \leq 100$ and $101 \leq n \leq 205$ is $3^{m}+7^{n}$ divisible by 10?

   CEMC Cayley

   Number theory Solution and answer checking →

10. 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 →

Answer key — Stage 3 · Mixed

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

1. $15339$ open
2. $2004$ open
3. $24$ open
4. $9$ open
5. $31$ open
6. $3$ open
7. $4$ open
8. $6$ open
9. $2625$ open
10. $16$ open

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