AMO 2025 Secondary 3 (Grade 9)

Question 1

Find the positive integer $n$ satisfying $1^3+2^3+3^3+\cdots+n^3=2025$.

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$25$
$15$
$10$
$9$

Positive integers $A,B$ satisfy $\frac A{x+2}+\frac B{x-3}=\frac{45x-10}{x^2-x-6}$. Find $A+B$. Clarification: interpret the equality as an identity for every $x\ne-2,3$; the source does not explicitly state this.

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$60$
$50$
$49$
$45$

Question 3

Let $A=(1,5)$ and $B=(7,10)$. Choose $C$ on $y=x$ to minimize $AC+CB$. What is the $x$-coordinate of $C$?

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$\frac{45}{8}$
$\frac{43}{7}$
$\frac{5\pi}{6}$
$\frac{25}{8}$

Question 4

Ten identical apples are distributed among five distinct people, each receiving at least one. How many distributions are possible?

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$145$
$126$
$120$
$108$

Question 5

In right triangle $ABC$, $\angle ACB=90^\circ$, $BC=AB/2$, and $F$ is the midpoint of $AB$. Find $\angle BFC$.

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$45^\circ$
$60^\circ$
$65^\circ$
$75^\circ$

Question 6

A cube of edge length 12 inches is cut by planes $AFC$ and $DEG$, as shown. Find the volume of the drum-like middle piece, in cubic inches.

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