Skip to content / Đến nội dung
ThichHocToan.com

AMO 2024 Secondary 3 (Grade 9)

Question 1

If $A,B$ are the roots of $f(x)=2x^2+3x+4$ and $C,D$ are the roots of $g(x)=4x^2+3x+2$, find $ABCD$. Roots are understood over the complex numbers.

Question illustration
  1. 8
  2. 6
  3. -1
  4. 1

Question 2

Jasmine rearranges the letters of “beekeeper”, always starting with b. The arrangements need not be English words. How many different arrangements can she make?

Question illustration
  1. 336
  2. 264
  3. 156
  4. 142

Question 3

Find $\sqrt{1^3+2^3+3^3+\cdots+100^3}$.

Question illustration
  1. 5000
  2. 5025
  3. 5050
  4. 5075

Question 4

In the figure, $\angle ABC=90^\circ$, $\angle BAC=30^\circ$, and $M$ is the midpoint of $AC$. If $AB=12$, find $BM$.

Question illustration
  1. $\sqrt{35}$
  2. $4\sqrt3$
  3. $3\sqrt5$
  4. $10\sqrt3/2$

Question 5

$B,D$ are fixed distinct points. As $E$ moves, $F$ lies on the extension of $BE$ and $DF\perp BF$ at $F$. What curve is traced by $F$?

Question illustration
  1. Parabola
  2. Hyperbola
  3. Circle
  4. Ellipse

Question 6

If $\cos x+\cos y=3/2$ and $\sin x-\sin y=2/3$, find $\cos(x+y)$.

Question illustration
  1. $4/9$
  2. $1/2$
  3. $25/72$
  4. $3\sqrt3/2$

Question 7

Rotate $A=(8,10)$ counterclockwise through $60^\circ$ about $B=(5,6)$. Find the new coordinates.

Question illustration
  1. $((11-4\sqrt3)/2,(9+3\sqrt2)/3)$
  2. $((13-4\sqrt3)/2,(16+3\sqrt3)/2)$
  3. $((13-4\sqrt3)/3,(17+3\sqrt2)/3)$
  4. $(6\sqrt3,5\sqrt6)$

Question 8

What is the remainder when $3^{2024}$ is divided by $11$?

Question illustration
  1. 1
  2. 2
  3. 4
  4. 5

Question 9

The circle $(x-350)^2+(y+750)^2=625$ is inscribed in a square. Find the square’s area.

Question illustration
  1. 625
  2. 1100
  3. 1250
  4. 2500