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AMO 2023 Secondary 3 (Grade 9)

Question 1

How many solutions does $(x!)^2+24=25(x!)$ have? Use the usual factorial for nonnegative integers $x$, including $0!=1$.

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

Question 2

$\alpha$ and $\beta$ are angles with $\sin\alpha=3/5$ and $\sin\beta=12/13$. If $A/B$ is the smallest possible positive value of $\sin(\alpha+\beta)$, written in lowest terms with $B>0$, find $A+B$.

Question illustration
  1. 75
  2. 84
  3. 92
  4. 98

Question 3

What is the sum of all real-number solutions of $x^{\sqrt{x}}=(\sqrt{x})^x$?

Question illustration
  1. 5
  2. 8
  3. 11
  4. 15

Question 4

$E$ lies outside the circle with center $A$. Two secants meet the circle at $C,D$ and $F,G$, as shown. If $CD=12$ cm, $DE=8$ cm and $EF=25$ cm, find $EG$ in centimeters.

Question illustration
  1. 3.6 cm
  2. 4.5 cm
  3. 5.6 cm
  4. 6.4 cm

Question 5

You have one two-dollar bill, one one-dollar bill, one fifty-cent piece, one quarter (25 cents), one dime (10 cents), one nickel (5 cents), and one penny (1 cent). How many different sums can you make, excluding zero dollars?

Question illustration
  1. 63
  2. 120
  3. 127
  4. 128

Question 6

If $\dfrac{43x+5}{12x^2+17x-40}=\dfrac{A}{4x-5}+\dfrac{B}{3x+8}$ is an identity for all $x$ where the fractions are defined, find $AB$.

Question illustration
  1. 20
  2. 35
  3. 40
  4. 45

Question 7

Let $A=2022/2023$ and $B=2023/2024$. What is the greatest common divisor of the numerator and denominator of the greater fraction?

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

Question 8

Which expression equals $\sqrt{2+\sqrt3}+\sqrt{2-\sqrt3}$?

Question illustration
  1. $\sqrt2$
  2. $\sqrt5$
  3. $\sqrt6$
  4. $\sqrt7$

Question 9

A number is written as $53AB$ in base $12$. Which is its base-$11$ representation? Here the digit symbols $A$ and $B$ stand for $10$ and $11$.

Question illustration
  1. 5AA3
  2. 6A07
  3. 5909
  4. 7A05