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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. Which of the following could not be the units digit [ones digit] of the square of a whole number?

   1. A1
   2. B4
   3. C5
   4. D6
   5. E8

   Number theory Solution and answer checking →

2. The regression equation $\hat{y}=0.85x-82.71$ is used to predict the weight of female college students based on their height, where $x$ is in centimeters (cm) and $\hat{y}$ is in kilograms (kg). Calculate the residual for an individual with the measurements $(160,53)$.

   Algebra Solution and answer checking →

3. An isosceles triangle has one side of length $3$ and another side of length $8$. The perimeter of this triangle is:

   1. A$14$
   2. B$19$
   3. C$11$
   4. D$14$ or $19$

   Geometry Solution and answer checking →

4. There are 6 identical cards, with the numbers "$1921$", "$1994$", "$1935$", "$1949$", "$1978$", and "$1980$" written on them. These cards are placed with their backs facing up, shuffled, and one card is randomly drawn. The probability of drawing a card with an even number on it is ____.

   Combinatorics Solution and answer checking →

5. A certain medicine underwent two price reductions, with the retail price per bottle decreasing from $58 to$43. Given that the percentage reduction for both price reductions is $x$, the equation to be written is ______. (Write down the equation only, do not solve it)

   Algebra Solution and answer checking →

6. For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which $n(A)+n(B)+n(C)=n(A\cup B\cup C)$ and $|A|=|B|=100$, then what is the minimum possible value of $|A\cap B\cap C|$?

   1. A96
   2. B97
   3. C98
   4. D99
   5. E100

   Combinatorics Solution and answer checking →

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

   Number theory Solution and answer checking →

8. The area of the triangle formed by the line $\frac{x}{5} + \frac{y}{2} = 1$ and the coordinate axes is

   1. A2
   2. B5
   3. C7
   4. D10

   Geometry Solution and answer checking →

9. A line $l$ passes through point P(1, 4) and has equal intercepts on both coordinate axes. What is the equation of line $l$?

   Algebra Solution and answer checking →

10. An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is \$20$ or more?
    $

    1. A${{{\frac{1}{4}}}}$
    2. B${{{\frac{2}{5}}}}$
    3. C${{{\frac{3}{7}}}}$
    4. D${{{\frac{1}{2}}}}$
    5. E${{{\frac{2}{3}}}}$

    Combinatorics Solution and answer checking →

Answer key — Stage 3 · Mixed

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

1. $E$ open
2. $-0.29$ open
3. $B$ open
4. $\frac{1}{2}$ open
5. $58(1-x)^2 = 43$ open
6. $97$ open
7. $4$ open
8. $5$ open
9. $4x-y=0 \text{ or } x+y-5=0$ open
10. $\frac{1}{2}$ 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.