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AMO 2022 – Junior College (Grade 11–12)

Question 1

What is the simplified value of $\frac1{\sqrt4+\sqrt5}+\frac1{\sqrt5+\sqrt6}+\frac1{\sqrt6+\sqrt7}+\cdots+\frac1{\sqrt{624}+\sqrt{625}}$?

Question illustration
  1. 8
  2. 11
  3. 20
  4. 23
  5. 35

Question 2

Triangle $ABC$ is inscribed in a circle with center $O$. Given $AC=4\sqrt3$ inches and $\angle CBA=60^\circ$, find the radius of the circle.

Question illustration
  1. 3 inches
  2. 4 inches
  3. 6 inches
  4. 8 inches
  5. $2\sqrt3$ inches

Question 3

What is the remainder when $9x^9+8x^8+7x^7+6x^6+5x^5+4x^4+3x^3+2x^2+1$ is divided by $x+1$?

Question illustration
  1. 15
  2. 25
  3. 45
  4. -3
  5. -15

Question 4

A large bag contains pennies (1 cent), nickels (5 cents), and dimes (10 cents), with each type equally likely to be selected. If you draw three coins, what is the probability of obtaining 25 cents? Clarification: use independent draws, each with probability $1/3$ for each type, as intended by the source.

Question illustration
  1. $\frac13$
  2. $\frac25$
  3. $\frac16$
  4. $\frac19$
  5. $\frac29$

Question 5

Janet makes necklaces with exactly 11 beads, each red or green. Necklaces related by rotation count as the same, but flipping is not allowed. How many distinct necklaces can she make?

Question illustration
  1. 120
  2. 128
  3. 188
  4. 216
  5. 318

Question 6

Three circles have centers $A,C,E$ and radius 1. Line $HQ$ is tangent to the circle centered at $A$; $AF$ meets $HC$ at $P$. Point $G$ lies beyond $Q$ on line $HQ$, and $PG=3$. Find $GM$. The figure specifies that $A,C,E$ lie on one another's circles, $F$ is the second intersection of the circles centered at $C,E$, and $M,A,C$ are collinear with $M$ on the circle centered at $A$.

Question illustration
  1. $\sqrt{\frac\pi3-5\sqrt3}$
  2. $\frac\pi2$
  3. $\sqrt{\frac{10}3+2\sqrt3}$
  4. $\sqrt{\frac{40}3-2\sqrt3}$
  5. $\sqrt{\frac{20}3-2\sqrt5}$

Question 7

Two people sit at random on an 18-foot bench. What is the probability that they sit at least 6 feet apart? Model their positions as independent, uniformly distributed points on the bench.

Question illustration
  1. $\frac13$
  2. $\frac35$
  3. $\frac38$
  4. $\frac49$
  5. $\frac59$

Question 8

For constants $A,B$, the identity $\frac A{2x-3}+\frac B{5x+3}=\frac{51x-3}{10x^2-9x-9}$ holds whenever the denominators are nonzero. Find $A+B$.

Question illustration
  1. 10
  2. 15
  3. 18
  4. 21
  5. 25

Question 9

In the $xz$-plane, segment $AB$ joins $A=(0,0,4)$ to $B=(3,0,0)$. Rotating $AB$ about the $z$-axis sweeps a cone. Which equation describes its surface? Clarification: restrict the selected equation to $0\le z\le4$, because a segment sweeps only a finite surface.

Question illustration
  1. $16x^2+16y^2-9z^2+72z-144=0$
  2. $4x^2+16y^2+z^2+16-25=0$
  3. $3x^2+4y^2-5z^2+25z-1=0$
  4. $x^2+y^2-9z^2+25=0$
  5. $x^2+y^2+z^2-1=0$