AMO 2024 Junior College (Grade 11–12)

Question 1

How many real solutions does $x^9+2x^7+3x^5+9x^3=0$ have?

1. 0
2. 1
3. 7
4. 9

Question 2

If $x=\tan34^\circ$, express $\tan17^\circ$ in terms of x.

1. $\frac{2x}{1-x^2}$
2. $\frac{1\pm\sqrt{1+x^2}}x$
3. $\frac{-1+\sqrt{1+x^2}}x$
4. $x/2$

Question 3

In triangle ACD, E lies on AD, B lies on AC, and $F=BD\cap CE$. Given $AE=2$, $AD=8$, $AB=4$, $AC=7$, find $EF:FC$.

1. 8:7
2. 7:8
3. 9:8
4. 1:1

Question 4

Let P be a polynomial satisfying $P(x^3-1)=x^6+4x^3+5$. Find $P(x^3+1)$.

1. $x^6+8x^3+17$
2. $x^6+8x^3+13$
3. $x^6+4x^3+17$
4. $x^6+4x^3+13$

Question 5

A positive integer d is less than 50. Dividing 344 by d leaves remainder 3; dividing 715 by d leaves remainder 2. Which statement is true?

1. $10\le d\le19$
2. $0<d\le29$
3. $0<d\le39$
4. $40\le d\le49$

Question 6

Triangle ABC has centroid G. The line through G parallel to AB meets BC at D and AC at E. If the area of ABC is 900, find the area of DEC.

1. 600
2. 500
3. 400
4. 320

Question 7

A bent coin has probability $1/3$ of heads on each independent toss. What is the probability that the third head occurs on toss six?

1. $1/2$
2. $16/81$
3. $3/121$
4. $80/729$

Question 8

Twin primes differ by two. Find the sum of the largest pair of three-digit twin primes.

1. 1980
2. 1764
3. 1784
4. 1946

Question 9

The average of the arithmetic and geometric means of two nonnegative numbers is 2.25. Find the average of their square roots. The nonnegative domain makes the real square roots explicit.

1. $\sqrt3$
2. $\sqrt2$
3. 1.5
4. 3

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20