Skip to content / Đến nội dung

ThichHocToan.com

1958 AMC 12

Question 1

The value of $\left[2-3(2-3)^{-1}\right]^{-1}$ is:

$5$
$-5$
$\dfrac15$
$-\dfrac15$
$\dfrac53$

Question 2

If $\dfrac1x-\dfrac1y=\dfrac1z,$ then $z$ equals:

$y-x$
$x-y$
$\dfrac{y-x}{xy}$
$\dfrac{xy}{y-x}$
$\dfrac{xy}{x-y}$

Question 3

Of the following expressions, the one equal to $\dfrac{a^{-1}b^{-1}}{a^{-3}-b^{-3}}$ is:

$\dfrac{a^2b^2}{b^2-a^2}$
$\dfrac{a^2b^2}{b^3-a^3}$
$\dfrac{ab}{b^3-a^3}$
$\dfrac{a^3-b^3}{ab}$
$\dfrac{a^2b^2}{a-b}$

Question 4

In the expression $\dfrac{x+1}{x-1},$ each $x$ is replaced by $\dfrac{x+1}{x-1}.$ The resulting expression, evaluated for $x=\dfrac12,$ equals:

$3$
$-3$
$1$
$-1$
none of these

Question 5

The expression $2+\sqrt2+\dfrac1{2+\sqrt2}+\dfrac1{\sqrt2-2}$ equals:

$2$
$2-\sqrt2$
$2+\sqrt2$
$2\sqrt2$
$\dfrac{\sqrt2}{2}$

Question 6

The arithmetic mean between $\dfrac{x+a}{x}$ and $\dfrac{x-a}{x},$ when $x\ne0,$ is:

$2,$ if $a\ne0$
$1$
$1,$ only if $a=0$
$\dfrac ax$
$x$

Question 7

A straight line joins the points $(-1,1)$ and $(3,9).$ Its $x$-intercept is:

$-\dfrac32$
$-\dfrac23$
$\dfrac25$
$2$
$3$

Question 8

Which of these four numbers, $\sqrt{\pi^2},$ $\sqrt[3]{0.8},$ $\sqrt[4]{0.00016},$ $\sqrt[3]{-1}\cdot\sqrt{(0.09)^{-1}},$ is (are) rational?

none
all
the first and fourth
only the fourth
only the first

Question 9

A value of $x$ satisfying the equation $x^2+b^2=(a-x)^2$ is:

$\dfrac{b^2+a^2}{2a}$
$\dfrac{b^2-a^2}{2a}$
$\dfrac{a^2-b^2}{2a}$
$\dfrac{a-b}{2}$
$\dfrac{a^2-b^2}{2}$

Question 10

For what real values of $k,$ other than $k=0,$ does the equation $x^2+kx+k^2=0$ have real roots?

$k<0$
$k>0$
$k\ge1$
all values of $k$
no values of $k$

Question 11

The number of roots satisfying the equation $\sqrt{5-x}=x\sqrt{5-x}$ is:

unlimited
$3$
$2$
$1$
$0$

Question 12

If $P=\dfrac{s}{(1+k)^n},$ then $n$ equals:

$\dfrac{\log(\frac{s}{P})}{\log(1+k)}$
$\log\dfrac{s}{P(1+k)}$
$\log\dfrac{s-P}{1+k}$
$\log\dfrac{s}{P}+\log(1+k)$
$\dfrac{\log s}{\log(P(1+k))}$

Question 13

The sum of two numbers is $10;$ their product is $20.$ The sum of their reciprocals is:

$\dfrac1{10}$
$\dfrac12$
$1$
$2$
$4$

Question 14

At a dance party a group of boys and girls exchange dances as follows: one boy dances with $5$ girls, a second boy dances with $6$ girls, and so on, the last boy dancing with all the girls. If $b$ represents the number of boys and $g$ the number of girls, then:

$b=g$
$b=\dfrac g5$
$b=g-4$
$b=g-5$
It is impossible to determine a relation between $b$ and
$g$ without knowing $b+g$

Question 15

A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:

$1080$
$900$
$720$
$540$
$360$

Question 16

The area of a circle inscribed in a regular hexagon is $100\pi.$ The area of the hexagon is:

$600$
$300$
$200\sqrt2$
$200\sqrt3$
$120\sqrt5$

Question 17

If $x$ is positive and $\log x\ge\log2+\dfrac12\log x,$ then:

$x$ has no minimum or maximum value
the maximum value of $x$ is $1$
the minimum value of $x$ is $1$
the maximum value of $x$ is $4$
the minimum value of $x$ is $4$

Question 18

The area of a circle is doubled when its radius $r$ is increased by $n.$ Then $r$ equals:

$n(\sqrt2+1)$
$n(\sqrt2-1)$
$n$
$n(2-\sqrt2)$
$\dfrac{n\pi}{\sqrt2+1}$

Question 19

The sides of a right triangle are $a$ and $b$ and the hypotenuse is $c.$ A perpendicular from the vertex divides $c$ into segments $r$ and $s,$ adjacent respectively to $a$ and $b.$ If $a:b=1:3,$ then the ratio of $r$ to $s$ is:

$1:3$
$1:9$
$1:10$
$3:10$
$1:\sqrt{10}$

Question 20

If $4^x-4^{x-1}=24,$ then $(2x)^x$ equals:

$5\sqrt5$
$\sqrt5$
$25\sqrt5$
$125$
$25$