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$A^2 + B^2$, $AB$, 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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