Maths Olympiad Prep

Track / Stage 6 / 289 of 400 #1289 of 1964

Problem 1289

National olympiad, first round
Geometry Difficulty 6.4 Find the answer

During the van ride from the Grand Canyon to the beach, Michael asks his dad about the costs of renewable energy resources. "How much more does it really cost for a family like ours to switch entirely to renewable energy?"

Jerry explains, "Part of that depends on where the family lives. In the Western states, solar energy pays off more than it does where we live in the Southeast. But as technology gets better, costs of producing more photovoltaic power go down, so in just a few years more people will have reasonably inexpensive options for switching to clearner power sources. Even now most families could switch to biomass for between $200\$200 and $1000\$1000 per year. The energy comes from sawdust, switchgrass, and even landfill gas. We pay that premium ourselves, but some families operate on a tighter budget, or don't understand the alternatives yet."

"Ew, landfill gas!" Alexis complains mockingly.

Wanting to save her own energy, Alexis decides to take a nap. She falls asleep and dreams of walking around a 2−D2-\text{D} coordinate grid, looking for a wormhole that she believes will transport her to the beach (bypassing the time spent in the family van). In her dream, Alexis finds herself holding a device that she recognizes as a tricorder\textit{tricorder} from one of the old Star Trek\textit{Star Trek} t.v. series. The tricorder has a button labeled "wormhole" and when Alexis presses the button, a computerized voice from the tricorder announces, "You are at the origin. Distance to the wormhole is 24002400 units. Your wormhole distance allotment is two\textit{two}."'

Unsure as to how to reach, Alexis begins walking forward. As she walks, the tricorder displays at all times her distance from her starting point at the origin. When Alexis is 24002400 units from the origin, she again presses the "wormhole" buttom. The same computerized voice as before begins, "Distance to the origin is 24002400 units. Distance to the wormhole is 38403840 units. Your wormhole distance allotment is two\textit{two}."

Alexis begins to feel disoriented. She wonders what is means that her wormhole distance allotment is two\textit{wormhole distance allotment is two}, and why that number didn't change as she pushed the button. She puts her hat down to mark her position, then wanders aroud a bit. The tricorder shows her two readings as she walks. The first she recognizes as her distance to the origin. The second reading clearly indicates her distance from the point where her hat lies - where she last pressed the button that gave her distance to the wormhole.

Alexis picks up her hat and begins walking around. Eventually Alexis finds herself at a spot 24002400 units from the origin and 38403840 units from where she last pressed the button. Feeling hopeful, Alexis presses the tricorder's wormhole button again. Nothing happens. She presses it again, and again nothing happens. "Oh," she thinks, "my wormhole allotment was two\textit{two}, and I used it up already!"

Despair fills poor Alexis who isn't sure what a wormhole looks like or how she's supposed to find it. Then she takes matters into her own hands. Alexis sits down and scribbles some notes and realizes where the wormhole must be. Alexis gets up and runs straight from her "third position" to the wormhole. As she gets closer, she sees the wormhole, which looks oddly like a huge scoop of icecream. Alexis runs into the wormhole, then wakes up.

How many units did Alexis run from her third position to the wormhole?

The source for this one didn't record the answer, so there is nothing to check what you type against. Work it on paper and mark yourself against the solution below.

Official solution

1. Define Points and Distances:
Let the origin be I I , the second location be P1 P_1 , the third location be P2 P_2 , and the wormhole be W W . We know the following distances:
- IP1=2400 IP_1 = 2400 units
- IP2=2400 IP_2 = 2400 units
- P1W=3840 P_1W = 3840 units

2. Construct the Diagram:
We can visualize the problem by constructing two circles:
- A circle centered at I I with radius 2400 2400 units.
- A circle centered at P1 P_1 with radius 3840 3840 units.

The intersection points of these two circles will give us the possible locations for W W .

3. **Calculate the Distance Between I I and P1 P_1 :**
Since IP1=2400 IP_1 = 2400 units, we place P1 P_1 at coordinates (2400,0) (2400, 0) .

4. Find the Intersection Points:
The intersection points of the two circles can be found using the distance formula and the properties of circles. The equation of the circle centered at I I is:
x2+y2=24002 x^2 + y^2 = 2400^2
The equation of the circle centered at P1 P_1 is:
(x−2400)2+y2=38402 (x - 2400)^2 + y^2 = 3840^2

5. Solve the System of Equations:
Substituting x=2400 x = 2400 into the second equation:
(2400−2400)2+y2=38402  ⟹  y2=38402 (2400 - 2400)^2 + y^2 = 3840^2 \implies y^2 = 3840^2
This gives us two possible values for y y :
y=±3840 y = \pm 3840

6. **Calculate the Distance WP2 WP_2 :**
Since P2 P_2 is also 2400 units from the origin and lies on the circle centered at I I , we need to find the distance from P2 P_2 to W W . Using the Pythagorean Theorem, we can find the distance from I I to the foot of the altitude from I I to WP1 WP_1 :
IH=IP12−P1H2=24002−19202=1440 IH = \sqrt{IP_1^2 - P_1H^2} = \sqrt{2400^2 - 1920^2} = 1440

7. **Calculate WP WP :**
Using the relationship between the areas, we have:
IH⋅WP1=IP1⋅WP  ⟹  WP=IH⋅WP1IP1=1440⋅38402400=2304 IH \cdot WP_1 = IP_1 \cdot WP \implies WP = \frac{IH \cdot WP_1}{IP_1} = \frac{1440 \cdot 3840}{2400} = 2304

8. **Double the Distance for WP2 WP_2 :**
Since WP2=2×WP WP_2 = 2 \times WP :
WP2=2×2304=4608 WP_2 = 2 \times 2304 = 4608

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.