Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
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including olympiad champions, researchers, and professionals.
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 7 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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What percentage of the square's area is colored blue?
by
Jason Dyer
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15 can be written as the difference between two positive perfect squares: \[15 = 16-1= 4^2 - 1^2.\] Which of the following numbers cannot be written as the difference of two positive perfect squares?
by
Arjen Vreugdenhil
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A factory manufactures wooden squares whose side length should be one meter. However, the side length of the squares is not always \(\SI{1}{\meter}\) but varies uniformly between \(\SI{1}{\meter}\) and \(\SI{1.1}{\meter}.\) Note that the height and the breadth of each square piece remain equal to each other.
Two inspectors, Alice and Bob, want to calculate the average area of a square.
- Alice claims that the average area is \(\SI{1.1025}{\meter\squared}\) because the average side length is \(\SI{1.05}{\meter}\) and \(1.05 \times 1.05 = 1.1025.\)
- Bob claims that the average area is \(\SI{1.1050}{\meter\squared}\) because the the area is between \((\SI{1}{\meter} \times \SI{1}{\meter})\) and \((\SI{1.1}{\meter} \times \SI{1.1}{\meter}),\) and \(\frac{1 \times 1 + 1.1 \times 1.1}{2} = 1.105\).
Who is correct?
by
Agnishom Chattopadhyay
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An accelerometer is made using tubular sections, as shown below. The accelerometer is placed in a rocket that travels in the \(\vec{x}\) direction, and the liquid levels are steady at \(\SI{20}{\centi\meter}\) and \(\SI{10}{\centi\meter}.\)
Find the magnitude of the rocket’s acceleration (in \(\text{m/s}^2\)). Submit your answer to 3 decimal places.
\(\)
Details and Assumptions:
- The diameter of the tube is much smaller than the heights of the liquid in the two arms.
- \(g = 10 \text{ m/s}^{2}.\)
by
Vignesh Rao
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A ladder is placed on a stairway (as shown). What is the length of the shorter leg?
\(\)
Details and Assumptions:
- Each step has the same horizontal run of \(15 \text{ cm}\) and vertical run of \(10 \text{ cm}.\)
- Each leg of the ladder rests on the horizontal midpoint of a step.
- The angles that the legs make with the steps are congruent.
by
Hosam Hajjir
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