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The Labyrinth of Linear Equations

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In a world where paths are determined by equations, you find yourself at the entrance of a labyrinth. The labyrinth is designed as a giant coordinate plane, with the entrance at the origin (0,0). To navigate through, you must follow a path defined by a series of linear equations, each leading to the next until you find the exit. The path is as follows: Start at the origin. Follow the line defined by the equation y=2x+3 until you intersect with the line defined by x=3. At the intersection, turn and follow the line defined by x=3 upwards until you intersect with the line defined by x+6. Once you reach this second intersection, follow the line x+6 to find the exit. Your task is to calculate the coordinates of the exit of the labyrinth.
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The Enigma of the Lost Treasure: Algebraic Expressions Challenge

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A long time ago, a pirate hid his treasure on an island and left a cryptic note for its location. The note says: "Ye who seeks my treasure, listen well. The treasure lies buried where the sum of thrice the age of the oldest tree on the island and twice the number of coconut trees equals 276. If the island has four times as many coconut trees as the oldest tree is years old, solve for the age of the oldest tree and the number of coconut trees." Solve for: The age of the oldest tree on the island. The number of coconut trees on the island. Your Task: Formulate algebraic expressions based on the given clues and solve for the two unknowns.
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Statistical Showdown: The Mysterious Case of Missing Data

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A group of detectives are investigating a series of mysterious disappearances in the town of Numeralia. They've gathered data on the ages of the 30 missing people, which are as follows: 15, 22, 19, 17, 34, 21, 18, 20, 16, 22, 25, 23, 18, 17, 19, 21, 20, 16, 15, 24, 20, 23, 25, 22, 17, 19, 21, 16, 15, 18. As a budding statistician, your task is to help the detectives by providing the following statistical insights: Calculate the mean, median, and mode of the ages. Determine the range of the data. Comment on what these statistics can tell us about the age group of the missing people. Provide your calculations and insights to assist the detectives in understanding if there's a pattern in the ages of the missing individuals.
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The Parabolic Pathfinder

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A parabolic path is modeled by the quadratic equation � = − 2 � 2 + 12 � − 10 y = − 2 x 2 + 12 x − 10 , where � y represents the height in meters above ground and � x represents the horizontal distance in meters from the starting point. At what horizontal distances � x is the path exactly 6 meters above the ground? What is the maximum height reached by the path and at what horizontal distance does this occur? Provide your answers in terms of meters and round to two decimal places if necessary.
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The Paradox of the Infinite Hotel

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Consider a hotel with an infinite number of rooms, each numbered consecutively from 1 onwards. The hotel is fully occupied, but a new guest arrives. To accommodate this guest, the manager decides to move the guest in Room 1 to Room 2, the guest in Room 2 to Room 3, and so on, shifting each existing guest to the room number that is one greater than their current room number. After this rearrangement, in which room will the new guest be accommodated?
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The Mystery of the Missing Angle

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  In a triangle, one of the angles is twice as large as the smallest angle, while the third angle is 20 degrees less than the sum of the other two angles. Can you determine the measures of all three angles in this mysterious triangle?
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The Secret of the Prime Pyramid

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  An ancient pyramid has a numerical puzzle engraved on its entrance. The puzzle states that there are three prime numbers, each less than 20, that form a unique set. The sum of these three prime numbers is equal to the square of the smallest prime number in the set. What are these three prime numbers? Remember, a prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
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