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AMO 2022 – Secondary 3 (Grade 9)

Question 1

O is the center of the circle. Diameter AC is 24 inches, B lies on the circle, and chord AB equals the radius. Find the length of the minor arc AB in inches.

Question illustration
  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 $\sqrt{8T_n+1}$?

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

Question 3

Let $A=\frac{2021}{2022}$ and $B=\frac{2022}{2023}$. What is the greatest common factor of the numerator and denominator of the greater fraction?

Question illustration
  1. 1
  2. 3
  3. 7
  4. 11
  5. 13

Question 4

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

Question illustration
  1. 16
  2. 5
  3. 17
  4. $\frac15$
  5. 19

Question 5

Two vertical poles AB and CD stand on level ground. AB is 6 feet tall and CD is 4 feet tall. Tight ropes AD and BC intersect at K. How high is K above the ground?

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

Question 6

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

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

Question 7

A circular target has concentric circles C₂ and C₁ of diameters 9 and 3 inches, respectively. Every dart hits a uniformly random point in the larger disk. What is the probability of landing inside the shaded smaller disk C₁?

Question illustration
  1. $\frac13$
  2. $\frac14$
  3. $\frac16$
  4. $\frac19$
  5. $\frac1{\pi^2}$

Question 8

The graphs of real functions f and g are shown. Find $g(f(2))$.

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

Question 9

Two forces $\vec u=\overrightarrow{AF_1}$ and $\vec v=\overrightarrow{AF_2}$ act on an object at A. Using the unit grid shown, find the combined force $\vec u+\vec v$.

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