AMO 2021 - Secondary 4 (Grade 10)

Question 1

A diameter of a circle has endpoints $(1,4)$ and $(1,2)$. Point $P$ is $(-7,9)$. What is the distance from the center of the circle to $P$, in units?

1. $1$
2. $3$
3. $5$
4. $10$
5. $15$

Question 2

A summer class has 130 students. Of these, 40% major in history and the rest in art. Each student is either full-time or part-time. There are 28 part-time history students and 32 full-time art students. A student is selected randomly from the history students. What is the probability that this student is full-time? Choose the closest listed decimal.

1. $0.18$
2. $0.21$
3. $0.35$
4. $0.43$
5. $0.46$

Question 3

The expression $\dfrac{x^2+15x+54}{3x+27}\cdot\dfrac{6x-3}{x^2+x-30}\cdot(x^2-25)$ is well-defined. Which expression is equivalent to it?

1. $x^2+16x+15$
2. $x^2-8x+15$
3. $2x^2-15x+23$
4. $2x^2+9x-5$
5. $3x^2+48x+56$

Question 4

A circular playground contains an inscribed rectangular soccer field measuring 80 m by 60 m, as shown. A player loses keys at a random location in the playground. What is the probability that the keys are outside the field? Use the usual model in which all locations in the playground are equally likely.

1. About 22%
2. About 39%
3. About 77%
4. About 36%
5. About 48%

Question 5

Let $f(x)=3x+2$ and $g(x)=x^{1/3}$. Find $f[g(8)]+g[f(2)]\cdot g^{-1}(4)$, where $g^{-1}$ is the inverse function.

1. $10$
2. $80$
3. $136$
4. $164$
5. $640$

Question 6

A council consists of a president, vice-president, treasurer and secretary. Five girls and ten boys are candidates, all equally likely to be elected. Four different people fill the posts. What is the probability that at least one girl is elected?

1. $1/39$
2. $6/39$
3. $11/39$
4. $11/13$
5. $12/13$

Question 7

If $w=1+\dfrac{1}{1+\dfrac{1}{1+w}}$, what is $|w|$?

1. $2$
2. $\sqrt{2.5}$
3. $\sqrt3$
4. $\sqrt5$
5. $\sqrt6$

Question 8

Three circles of radius 1 have centers $A,C,E$, as shown. Line $HC$ is tangent to the circle centered at $A$, and $AF$ meets $HC$ at $P$. Find $PC$. As indicated by the circle incidences in the figure, each center lies on the other two circles, and $F$ is the lower intersection of the circles centered at $C$ and $E$.

1. $\sqrt5/2$
2. $\sqrt3/3$
3. $\sqrt6/3$
4. $\sqrt{10}/2$
5. $\sqrt5/3$

Question 9

Which listed number satisfies $\dfrac{3x-5}{2}\ge\dfrac{x+1}{4}-2$?

1. $-1.3$
2. $-0.6$
3. $0.2$
4. $0.5$
5. $1.6$

Question 10

A right cylinder has radius 5 inches and height $5\sqrt3$ inches. Points $D,E$ lie on the top rim with $DE=5$ inches. Segment $DF$ is perpendicular to the bottom plane, with $F$ on the bottom rim. Find $\angle DFE=\alpha$, in degrees.

1. $45^\circ$
2. $30^\circ$
3. $75^\circ$
4. $40^\circ$
5. $60^\circ$

Question 11

The lines $7y=kx+14$ and $x+2y=6$ are parallel. Find $k$.

1. $-7/2$
2. $7$
3. $21/2$
4. $14$
5. $35/2$

Question 12

Solve $\log_2(x-3)-\log_2(x^2-6x+9)=2$.

1. $3.25$
2. $4.5$
3. $25$
4. $32$
5. $16$

Question 13

Terry marks points $A,B$ on the same straight ground line as an oak tree, with $A$ nearer the tree and $AB=6$ yards. In the photograph, the angles of elevation labelled 1 and 2 are $60^\circ$ and $45^\circ$, respectively. Ignoring camera distortion and using the level-ground model shown, what is the tree height in yards?

1. $6+5\sqrt2$
2. $15-3\sqrt2$
3. $12+4\sqrt2$
4. $9+3\sqrt3$
5. $8+6\sqrt2$

Question 14

Which function describes the graph shown?

1. $f(x)=x^2-2x+1$
2. $g(x)=2x^4-4x+4$
3. $h(x)=x^3-2x^2-x+2$
4. $i(x)=4x^3-8x^2-4x+32$
5. $j(x)=x^3+18x^2-48x+36$

Question 15

Which expression equals $\dfrac{5\sqrt3-5\sqrt2\,i}{\sqrt3+\sqrt2\,i}$, where $i^2=-1$? The illustration marks the denominator on the complex plane; Re and Im denote the real and imaginary axes.

1. $5+6\sqrt3\,i$
2. $10-\sqrt6\,i$
3. $6+3\sqrt5\,i$
4. $5-\sqrt6\,i$
5. $1-2\sqrt6\,i$

Question 16

Let $f(x)=x^2-4x+5$, with $A=(5,f(5))$ and $B=(5+w,f(5+w))$ on its graph. As $w$ approaches zero through nonzero values, what value does $r=\dfrac{f(5+w)-f(5)}{w}$ approach?

Question 17

The clock currently reads 8:25 am. Let $H:M:S$ be the time when the hour and minute hands overlap between 8 am and 9 am, with $0\le M<60$ and $0\le S<60$. Find $60M+S$. Seconds may be fractional.

Question 18

A regular octahedron lies inside a cube, with its six vertices at the centers of the six cube faces. If $V_c$ and $V_o$ are their volumes, find $V_c/V_o$.

Question 19

The parabola $f(x)=x^2+4x-7$ meets $y=-2x$ at $A,B$. Their midpoint is $M=(m,n)$. Find $|mn|$.

Question 20

A sheet of paper is 0.25 mm thick. Each fold in half doubles its thickness: after one, two and three folds it is 0.50 mm, 1 mm and 2 mm thick. In this idealized thought experiment, what is the least number of folds needed to make it more than 1 km thick? If useful, $\log_2 10\approx3.322$.