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AMO 2024 – Secondary 4 (Grade 10)

Question 1

An equilateral triangle of area $36\sqrt3$ square units is divided into four congruent equilateral triangles and folded into a regular tetrahedron as shown. Find its volume in cubic units.

Question illustration
  1. $18\sqrt2$
  2. $18\sqrt3$
  3. $\frac{9\sqrt{39}}2$
  4. $18\sqrt6$

Question 2

The transformation $T(x,y)=(x/3,y/2)$ maps the ellipse $x^2/9+y^2/4=1$ to a new shape. What is the area of the image?

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

Question 3

Choose three distinct numbers from $\{1,3,4,5,10\}$ as the sides of triangle A with the least possible perimeter. Choose three distinct numbers from $\{3,6,8,10,22\}$ as the sides of triangle B with the greatest possible perimeter. Find the sum of their areas.

Question illustration
  1. $6\sqrt5+10\sqrt3$
  2. 30
  3. 32
  4. $6\sqrt{11}+10\sqrt3$

Question 4

Two windshield wipers each have length $L$ and sweep the upper semicircles shown. Their pivot separation is $O_1O_2=L$. What is the total area swept by at least one wiper?

Question illustration
  1. $\frac{\sqrt3}2\pi L^2$
  2. $(\frac{2\pi}3+\frac{\sqrt3}4)L^2$
  3. $\pi L^2-\frac{L^2\sqrt3}4$
  4. $\frac{2\pi L^2}3$

Question 5

An ellipse is inscribed in a $4$-inch by $6$-inch rectangle as shown. Find the shaded area inside the rectangle and outside the ellipse, in square inches.

Question illustration
  1. $12+2\pi$
  2. $36-8\pi$
  3. $24-6\pi$
  4. $10-2\pi$

Question 6

In the diagram, not drawn to scale, $AD\perp AC$ at A and A, B, C lie in that order on a line. Given $AB=3$, $BC=5$ and $CD=28$, find BD.

Question illustration
  1. $15\sqrt2$
  2. $12\sqrt5$
  3. 27
  4. 29

Question 7

Let $f(x)=x^2-5x-14$. Points $A=(2,f(2))$ and $B=(2+\Delta,f(2+\Delta))$ lie on its graph, where $\Delta\ne0$. As $\Delta$ approaches zero, what value does $R=[f(2+\Delta)-f(2)]/\Delta$ approach?

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

Question 8

The source displays $x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}=3$. Under its formal repeated-tail interpretation, what value of x follows? Clarification: treat the tail as 3 and solve $x^3=3$; this is a formal algebraic interpretation, not a claim that the usual sequence of finite power towers converges to 3.

Question illustration
  1. 3
  2. $1/3$
  3. $\sqrt[3]3$
  4. 1

Question 9

The coordinate axes and $3x+4y=12$ bound a right triangle. Rotate it through $360^\circ$ about its longer leg. Find the volume of the solid.

Question illustration
  1. $16\pi$
  2. $12\pi$
  3. $9\pi$
  4. $6\pi$