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AMO 2025 Junior College (Grade 11–12)

Question 1

For $f(x)=x^2-3x-4$, find the $x$-intercept of the tangent at $x=2$.

Question illustration
  1. $12$
  2. $8$
  3. $6$
  4. $2$

Question 2

If $\log_x y+\log_y x=6$, find $(\log_x y)^2+(\log_y x)^2$.

Question illustration
  1. $64$
  2. $35$
  3. $34$
  4. $32$

Question 3

If $\frac{x+1}{y+1}=2$, find $\frac{x+5}{y+3}$. Clarification: the requested expression is defined only for $y\ne-3$; the given equation also requires $y\ne-1$.

Question illustration
  1. $1$
  2. $2$
  3. $4$
  4. $6$

Question 4

Find the product of all real and complex roots of $f(x)=(2x+3)(3x-2)(5x+1)-24$.

Question illustration
  1. $1$
  2. $2$
  3. $8$
  4. $12$

Question 5

A rectangular prism has a $2\times2$ square base and vertical-face diagonals of length 3. The angle $\alpha$ is between the two adjacent-face diagonals meeting at a vertex, as shown. Find $\tan\alpha$.

Question illustration
  1. $\frac{\sqrt{14}}2$
  2. $\frac{\sqrt{14}}5$
  3. $\frac{2\sqrt{14}}5$
  4. $\frac{2\sqrt{14}}3$

Question 6

Find the product of the irrational solutions of $(x^2+2x-20)(x^2+2x-18)=15$.

Question illustration
  1. $15$
  2. $20$
  3. $-15$
  4. $-23$

Question 7

Which shape is represented in polar coordinates by $r(\theta)=\frac{17}{6-6\cos\theta}$?

Question illustration
  1. $\text{Line}$
  2. $\text{Parabola}$
  3. $\text{Ellipse}$
  4. $\text{Hyperbola}$

Question 8

Evaluate $\sum_{k=1}^{2024}\frac1{\sqrt{k}+\sqrt{k+1}}$.

Question illustration
  1. $2026$
  2. $2025$
  3. $45$
  4. $44$

Question 9

Let $x=\frac1{2+\frac1{2+\frac1{2+\cdots}}}$. Which expression equals $x+2$?

Question illustration
  1. $\frac1x$
  2. $\frac1{x^2}$
  3. $x^2$
  4. $\frac2x$