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AMO 2023 – Secondary 2 (Grade 8)

Question 1

$M,N$ are positive integers satisfying $M^2+16^2=N^2+2023^2$. What is $M+N$?

Clarification: the equation alone does not determine a unique sum. Select the only listed value that can be $M+N$.

Question illustration
  1. 2022
  2. 2024
  3. 2035
  4. 2039

Question 2

Two real numbers $X,Y$ are randomly chosen with $0<X<5$ and $2<Y<7$. What is the probability that $3<X+Y<8$? Interpret the choices as independent and uniform over these intervals.

Question illustration

Supplementary diagram of the given data:
Question illustration
  1. 60%
  2. 66%
  3. 68%
  4. 72%

Question 3

A circle of radius 5 units is centered at $(2,3)$ in the coordinate plane. It meets the $x$-axis at $B,C$. What is $BC$?

Question illustration

Supplementary diagram of the given data:
Question illustration
  1. 5 units
  2. 6 units
  3. 7 units
  4. 8 units

Question 4

Given $2023=7\times17^2$ and $2024=2^3\times11\times23$, how many distinct positive factors does $2023\times2024$ have?

Question illustration
  1. 24
  2. 72
  3. 96
  4. 100

Question 5

$ABCDEFGH$ is a unit cube with $AB=1$. A square pyramid $ABCDG$ is constructed inside it. What is its surface area? The printed choices use “units”; for area these mean square units.

Question illustration
  1. $1+\sqrt5$ units
  2. $4-\sqrt2$ units
  3. $2+\sqrt2$ units
  4. $2+\sqrt6$ units

Question 6

For nonnegative integers, $f(n)=\begin{cases}3&n=0,\\f(n-1)+4&n>0.\end{cases}$ Express $f(n)$ in terms of $n$.

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

Question 7

If the dot pattern continues, how many more dots are there at step 1000 than at step 999?

Question illustration
  1. 2023
  2. 2024
  3. 3001
  4. 3011

Question 8

If $(60-x)(40-x)=80$, find $(60-x)^2+(40-x)^2$.

Question illustration
  1. 100
  2. 560
  3. 750
  4. 1000

Question 9

In the figure, $AB=AC$, $D$ lies on $AC$, and $BD=BC=AD$. Find $\angle BAC=x$ in degrees.

Question illustration
  1. $24^\circ$
  2. $30^\circ$
  3. $36^\circ$
  4. $45^\circ$