We're looking for a treasure Here's a problem my sister-in-law just sent me and she doesn't find the answer. It's to help her daughter.
We have a map of an island. On this island there's a palm tree P, a house M and a big rock R. The rock is 8 meters far from the palm tree, and 5 meters far from the house. The house is 4 meters far from the palm tree. And we know that:


*

*the treasure  is < 6 meters from the palm tree

*the treasure  is > 4 meters from the house

*the treasure  is < 5 meters from the rock


On the island below put points M and P using the previous clues.
Color the zone where we have to dig to be sure to find the treasure.

Could you explain the basic principles of how to solve that problem/draw what's asked (the picture is not the actual one)?
 A: My answer is not much different than what @Nick, who was quicker, already posted. But here it is, there is one detail at the end that is a bit different. 
See the picture below. Start with a circle centered at the rock, R, with radius $8$, I have labeled this circle 8R, meaning "8 from the rock". Pick a point P on that circle, and call it the palm. The house is on the circle 4P and also on 5R, these two circles intersect at two points, just pick one of them to be the house H. The treasure is inside <6P, outside >4H,and inside <5R, so it looks like a triangular area on the picture. 
But, given we do not know which direction is North, etc, and which direction the palm is from the rock, etc, we have only determined how far away from the rock the treasure could be. If the palm was at a different direction from the rock, then that triangular area would end up in a different direction too, although certainly same distance from the rock. So it could be anywhere in the area sprinkled with blue on the picture, outside the green circle and inside the <5R circle. 

A: I would say that the basic principle behind this question is the definition of a circle. A circle is the set of all points in a plane equidistant from a given point - the center. So, to set up the Rock, Palm, and House given that

We have a map of an island. On this island there's a palm tree P, a house M and a big rock R. The rock is 8 meters far from the palm tree, and 5 meters far from the house. The house is 4 meters far from the palm tree.

Basic Information
Basic principle: Knowing the distances between three objects, we can construct a triangle.

*

*Place the rock. Now draw a circle of radius 8 meters. All the points on the circle are 8 meters from the rock and thus candidates for the position of the palm tree. Pick a random point on the circle, and let's let that be the palm tree.

*Draw a circle of radius 5 with the rock as its center - all the points on this circle are candidates for the palm

*With the house at the center draw a circle with radius 4 - all the points on this circle are candidates for the palm

*The intersection points between the circles in step 2 and 3 must be the position of the palm which preserves all the clues up to this point.


Identifying the Treasure
Basic Principle: Again, using circles to identify all points in a plane at a given distance we can identify the region in which the treasure must lie.


*

*the treasure is < 6 meters from the palm tree

*the treasure is > 4 meters from the house

*the treasure is < 5 meters from the rock



*

*Construct three circles centered at the palm tree (radius 6), the house (radius 4), and the rock (radius 5).

*The treasure must lie inside the region bounded by the circles centered at the palm tree and the rock and also outside the region bounded by the circle centered at the house.

The Solution

On the island below put points M and P using the previous clues.
Color the zone where we have to dig to be sure to find the treasure.


A: First of all, to place the palm tree, house and big rock, we notice that any three points form a triangle or a line in the plane. Let's assume it is a triangle, with side lengths as given: 8, 5 and 4 meters respectively. Draw a triangle with these measurements (in centimeters).
The second step is to look at the distances of the treasure from the objects. In general, when an object is $<x$ meters from a point, it is within a circle of radius $x$ meters. So draw a circle of radius $6$ around the palm tree, an circle of radius $5$ around the rock and a circle of radius $4$ around the house. Your treasure is in the intersection of the two first circles, but outside the third circle.
A: The palm tree, the rock, and the house are three vertices of a triangle whose side lengths are $4$, $5$, and $8$:

The treasure must be within the circle of radius $6$ centered at $P$, outside the circle of radius $4$ centered at $H$, and inside the circle of radius $5$ centered at $R$. You should draw these circles and find the desired region, which should end up looking a bit like the Starfleet emblem from Star Trek.
