Excel in math and science
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Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, interactive explorations.
Used and loved by over 6 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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\[S=1^{33} + 2^{33} + 3^{33} + 4^{33} + \cdots + 89^{33}\]
What is the units digit of \(S?\)
by
Prem Chebrolu
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Assume that Earth is a perfect sphere with radius \(R.\) The time \(T\) an object takes to fall to the ground from rest from a height of \(R\) above the ground is given by \[T=\sqrt{\frac Rg}\left(A+\frac {\pi^B}C\right),\] where \(A,B,C\) are positive integers.
Find \(A+B+C.\)
Note: Consider only gravity, and forget about atmospheric effects, air resistance, etc.
by
Brian Lie
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After studying various 3D shapes and finding formulas for their volumes, I challenged my students to invent a new shape. Lindsay created a shape \((\)with height \(2 \text{ cm})\) that is circular at the top \((\)with radius \(1\text{ cm})\) but square at the bottom \((\)with side length \(2\text{ cm}).\) Lindsay created this shape from a paraboloid: sliced four times parallel to the paraboloid's axis and two times perpendicular to the paraboloid's axis. Lindsay's shape is pictured below.
Find the volume of Lindsay's shape in \(\text{cm}^{3},\) which can be written as \(\frac{A}{B}+C\pi\) with \(A,B,C\) integers, \(A\) and \(B\) coprime, and \(B\) positive.
Give the value of \(A+B+C.\)
by
Jeremy Galvagni
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Four nails are randomly fixed on a circular soft board.
Emma takes a red elastic rubber band, stretches it around the nails, and lets go.
What is the probability that the rubber band does not take the shape of a quadrilateral?
Source: Putnam 2006 A-6
by
Digvijay Singh
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There is a special point between Sun and Earth where a spacecraft can be parked, so that it always remains directly between them, at a fixed distance \(x\) from Earth, as Earth rotates around the Sun. A spacecraft can remain at this point without using any thrust.
What is the distance \(x\) of this point from the Earth in gigameters? \(\big(1\text{ gigameter}=10^6\text{ km}\big)\)
Details and Assumptions:
- The masses of the Earth and the Sun have a ratio of \(1: 333,000.\)
- The distance between the Earth and the Sun is \(a \approx 150 \cdot 10^6 \text{ km}.\)
- The radii of the Earth and the Sun are negligibly small.
by
Markus Michelmann
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