##### 代数工具: 分级测试

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1.

What 2-digit number has the property that if you subtract 5 times its tens digit from it, the result is the 2-digit number with the digits swapped?

2.

Suppose that the symbol $$x \ @ \ y$$ means:

$\frac{1}{x} + \frac{1}{y}.$

If $2 \ @ \ (3 \ @ \ x) = 4,$

what is $$x?$$

3.

What is the sum of all odd 2-digit numbers?

4.

Starting from a point on the ground, a ball is thrown straight up to a height of $$1$$ meter. It bounces up and down vertically, each time reaching a maximum height that is $$\frac{5}{6}$$ its previous maximum height. How many total meters will the ball travel, if it keeps bouncing forever?

5.

Simplify

$\sqrt{15 + 4 \sqrt{14}}$

and rewrite it in simplest radical form as

$a \sqrt{b} + \sqrt{c},$

where $$a,$$ $$b,$$ and $$c$$ are integers. What is the value of $$a + b + c?$$

6.

Elijah counts $$31$$ creepy-crawlers, all of which are spiders (which have $$8$$ legs) or ants (which have $$6$$) legs. The creepy-crawlers have $$214$$ legs altogether. How many spiders are there?

7.

Find the value of the positive number:

$\sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}}$

8.

A square's area plus its perimeter equals $$221.$$ What is the side length of the square?

9.

A straight line passes through the points

$\left( -10, \frac{1}{3} \right)$

and

$\left( -2, \frac{17}{3} \right).$

What is the $$y$$-intercept of another line perpendicular to this first line which passes through the midpoint between

$\left( -10, \frac{1}{3} \right)$

and

$\left( -2, \frac{17}{3} \right)?$

10.

A triangle has corners $$A(0, 0),$$ $$B(12, 0),$$ and $$C(0, 6).$$ If the midpoint of $$AB$$ is connected to $$C$$ and the midpoint of $$AC$$ is connected to $$B,$$ they intersect at a point with coordinates $$(x, y).$$ What is $$y?$$