\( A^2 + B^2\), \(AB\), and \(A + B \) are all integers.
Do both \(A\) and \(B\) have to be integers?
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Six straight, zigzagging lines are drawn inside an \(8\times1\) rectangle. The drawing starts at the top left vertex and ends at the top right vertex.
What is the minimum possible sum of the lengths of these 6 segments?
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Without opening his parachute, a skydiver reaches a fall velocity of \(v_1 = 50\text { m/s}.\) When he does open the parachute, he's braked by additional air resistance. After a while, he finally arrives at the ground.
At what fall velocity \(v_2 \) does he reach the ground?
Details and Assumptions:
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There is a collection of points inside a unit cube that are closer to the center of the cube than to any of the cube’s vertices.
What is the volume of this 3D region?
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\(f(n)\) gives the sum of the cubed digits of some positive integer \(n.\) For example, \(f(123)=1^3+2^3+3^3=36.\)
If we repeatedly apply this process on each previous result, the following two different behaviors may arise:
Let the limit set be the set of all fixed points and limit cycles in the range of \(f(n).\)
Find the sum of all the numbers in the limit set (including the four in pink found above). Note: A coding environment is provided below:
Bonus: Prove that the limit set actually contains finitely many numbers.
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