AMO 2021 - Secondary 3 (Grade 9)

Question 1

$O$ is the center of the circle, $AC$ is a diameter of length 24 inches, and $B$ lies on the circle. Chord $AB$ equals the radius. Find the length of the minor arc $AB$, in inches.

1. Not enough information
2. $\pi$ inches
3. $2\pi$ inches
4. $2.5\pi$ inches
5. $4\pi$ inches

Question 2

Let $T_n=1+2+\cdots+n$, where $n$ is a positive integer. What is $\sqrt{8T_n+1}$?

1. $n-1$
2. $n+1$
3. $2n+1$
4. $2n-1$
5. $3n+1$

Question 3

A circular table has diameter 12 feet. Two people choose seats randomly on its circumference. What is the probability that they are at least 6 feet apart? Clarification: positions are independent and uniform, and distance means straight-line distance.

1. $1/2$
2. $1/3$
3. $2/3$
4. $1/4$
5. $3/4$

Question 4

For real $x$, define $f(x)=2^x-15$ and $g(x)=5x+6$. Find $f^{-1}\left(g\left(\frac15+2\right)\right)$.

1. 16
2. 5
3. 17
4. $1/5$
5. 19

Question 5

Two upright poles $AB$ and $CD$ are 6 feet and 4 feet tall. Tight ropes join the top of each pole to the foot of the other and intersect at $K$. What is the height of $K$ above the level ground?

1. Not enough information
2. 2 feet
3. 2.4 feet
4. 2.5 feet
5. 3.2 feet

Question 6

If the identity $\dfrac{16x-3}{10x^2-13x-30}=\dfrac{A}{2x-5}+\dfrac{B}{5x+6}$ holds wherever defined, find $A+B$.

1. 4
2. 5
3. 6
4. 7
5. 8

Question 7

A dart lands uniformly at random inside the larger circle $C_2$, whose diameter is 9 inches. The concentric shaded circle $C_1$ has diameter 3 inches. What is the probability of landing in $C_1$?

1. $1/3$
2. $1/4$
3. $1/6$
4. $1/9$
5. $1/\pi^2$

Question 8

The figure shows the graphs of real functions $f$ and $g$. Find $g(f(2))$.

1. -1
2. 0
3. 1
4. 2
5. 3

Question 9

Two force vectors $\vec u=\overrightarrow{AF_1}$ and $\vec v=\overrightarrow{AF_2}$ act on an object at $A$. Find the resultant vector $\vec u+\vec v$ from the unit grid.

1. $\langle3,7\rangle$
2. $\langle8,-1\rangle$
3. $\langle\sqrt5,-2\rangle$
4. $\langle6,-4\rangle$
5. $\langle1,8\rangle$

Question 10

Adam and Bob play a best-of-five chess match: the first to win three rounds wins. Adam wins each round with probability 0.6 and Bob with probability 0.4. Find the probability that Bob wins the match, rounded to two decimal places. Assume independent rounds and no draws.

1. 0.21
2. 0.24
3. 0.32
4. 0.41
5. 0.64

Question 11

Find the remainder when $x^5-5x^4+11x^3-4x^2-24x+31$ is divided by $x-2$.

1. $x-1$
2. 0
3. -2
4. 5
5. 7

Question 12

What is the product of all non-real roots of $x^4-16=0$?

1. 4
2. -4
3. $4i$
4. $-4i$
5. 8

Question 13

Solve $\begin{bmatrix}2&-3\\3&2\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}22\\7\end{bmatrix}$.

1. $\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}2\\3\end{bmatrix}$
2. $\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\-4\end{bmatrix}$
3. $\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-5\\4\end{bmatrix}$
4. $\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}1\\0\end{bmatrix}$
5. $\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-22\\-7\end{bmatrix}$

Question 14

Let $S=A+B+C$, where each of $A,B,C$ is randomly chosen from $\{1,3,5,7\}$. Find the probability that $S=11$. Clarification: the choices are independent and uniform, with repetition allowed.

1. $5/32$
2. $27/64$
3. $5/128$
4. $1/8$
5. $3/16$

Question 15

In base ten, $923=9\cdot10^2+2\cdot10+3$. In base five the digits are $\{0,1,2,3,4\}$, so $321_5=3\cdot5^2+2\cdot5+1$. What is the base-five representation of the decimal number 237? Source note: the original list of decimal digits accidentally omits 6; the full list is $\{0,1,2,3,4,5,6,7,8,9\}$.

1. 3212
2. 1422
3. 2214
4. 4122
5. 4321

Question 16

Evaluate $360\displaystyle\lim_{n\to\infty}\left(\sum_{i=1}^{n}\frac1{4^i}\right)$.

Question 17

In triangle $ABC$, $BE$ bisects $\angle ABC$ and meets $AC$ at $D$; $AE\parallel BC$. Given $AB=40$ cm, $BC=60$ cm and $AC=80$ cm, find $AD$ in cm.

Question 18

Adam, Beth, Carla and David are seated randomly around a round table. Let $p$ be the probability that Beth and David sit next to each other. Find $30p$.

Question 19

Let $A=2021/2022$ and $B=2022/2023$. Find the sum of the numerator and denominator of the greater fraction.

Question 20

For a positive integer $n$, let $\phi(n)$ count positive integers less than $n$ that are relatively prime to $n$. Find $\phi(35)$.