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Stage 3 · Algebra

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

The answer key prints on its own page at the end.

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 $(160,53)$ .

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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)

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

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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?

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Given that $2x$ is the cube root of $216$ , find the square root of $x+6$ .

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If the value of the fraction $\frac{{x}^{2}-4}{x+2}$ is $0$ , then $x=$ ____.

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Given the function $f(x) = x^2 + ax + 2$ , where $x \in [-5, 5]$ ,
(1) When $a = -1$ , find the intervals of monotonicity for the function $f(x)$ .
(2) If the function $f(x)$ is increasing on $[-5, 5]$ , find the range of values for $a$ .

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$\sin 17^\circ \cdot \cos 43^\circ + \sin 73^\circ \cdot \sin 43^\circ$ equals \_\_\_\_\_\_.

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If $sin(\alpha-\frac{\pi}{6})=\frac{2}{3}$ , then $cos(2\alpha+\frac{2\pi}{3})=$ ____.

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Among the following propositions, the true ones are:
① The negation of "If $x^2+y^2 \neq 0$ , then neither $x$ nor $y$ is zero";
② The converse of "All isosceles triangles are similar";
③ The contrapositive of "If $m > 1$ , then the solution set of the inequality $x^2+2x+m > 0$ is $\mathbb{R}$ ".
1. A①
2. B①③
3. C②③
4. D①②③
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Answer key — Stage 3 · Algebra

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

$-0.29$ open
$58(1-x)^2 = 43$ open
$4x-y=0 \text{ or } x+y-5=0$ open
$2004$ open
$\pm 3$ open
$x=2$ open
$[10, +\infty)$ open
$\frac{\sqrt{3}}{2}$ open
$-\frac{1}{9}$ open
$B$ 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.