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AMO 2025 – Secondary 4 (Grade 10)

Question 1

What is the remainder when $2^{2025}$ is divided by 5?

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

Question 2

The transformation $T(x,y)=(2x,y/4)$ is applied to the ellipse $x^2/16+y^2/9=1$. What is the area of the resulting shape?

Question illustration
  1. $6\pi$
  2. $3\pi/2$
  3. $4\pi$
  4. $5\pi$

Question 3

Three mutually externally tangent circles have centres at the vertices of an equilateral triangle. A large circle encloses them and is internally tangent to all three. A small circle in the central gap is externally tangent to all three. What is the ratio of the large radius to the small radius?

Question illustration
  1. $6+4\sqrt5$
  2. $12-2\sqrt2$
  3. $7+4\sqrt3$
  4. $8+2\sqrt3$

Question 4

In the figure, A is the centre of the small circle, E is the midpoint of chord ST, and A, E, C are collinear in that order. AC is the diameter of the semicircle passing through T. Given $AT=8$ and $AE=2$, find CT.

Question illustration
  1. $12\sqrt6$
  2. $16\sqrt3$
  3. $8\sqrt{15}$
  4. $4\sqrt{29}$

Question 5

An equilateral triangle has side a and an incircle. In each corner region, a smaller circle is tangent to the two adjacent sides and to the incircle. Find the combined area of the three smaller circles.

Question illustration
  1. $3\pi a^2/14$
  2. $\pi a^2/\sqrt3$
  3. $\pi a^2/36$
  4. $3\pi a^2\sqrt3/18$

Question 6

If $x+y=6$ and $xy=5$, find $x^4+y^4$.

Question illustration
  1. 2025
  2. 1920
  3. 626
  4. 512

Question 7

Let $g(x)=2026x^2-2025x+4051$. Points $A=(1,g(1))$ and $B=(1+\delta,g(1+\delta))$ lie on its graph, where $\delta\ne0$. As $\delta$ approaches zero, what does $[g(1+\delta)-g(1)]/\delta$ approach? Source clarification: the printed coordinate of B uses undefined f instead of g, and an extra comma and $\delta$ follow the definition; the graph and displayed ratio specify g as used here.

Question illustration
  1. 2007
  2. 2020
  3. 2027
  4. 2028

Question 8

How many distinct real solutions does $x^{x^2-1011x-2026}=1$ have? Interpret real powers wherever they are defined, including negative bases with integer exponents.

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

Question 9

Rotate $P=(4,7)$ counterclockwise about the origin through $\theta=90^\circ$. Use $x^{\prime}=x\cos\theta-y\sin\theta$, $y^{\prime}=x\sin\theta+y\cos\theta$. Find the new coordinates. Source clarification: the printed matrix has lost its row and column layout; these are the standard equations for the stated counterclockwise rotation.

Question illustration
  1. $(-7,4)$
  2. $(14,8)$
  3. $(8,14)$
  4. $(-4,7)$