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
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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 6 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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The line segments shown divide parallelogram \(ABCD \) into 8 regions, and the areas of the orange triangles are 48, 21, and 63.
What is the area of the blue triangle?
by
Markus Michelmann
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The weight of this hourglass is monitored constantly.
The last grain of sand falls from the top of the hourglass at \(t_1\), and hits the bottom at \(t_2\).
Which line best represents the readout of the weight on the scale?
by
Laszlo Mihaly
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Is there a positive integer divisible by 2018 which only consists of the digits 3 and 0 (in base 10)?
by
Mustafa Elq
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If one of the two series below is equal to 2, what is the value of the other one?
\[\begin{cases} 1 + x + x^2 + x^3 + \cdots \\\\ x + 2x^2 + 3x^3 + 4x^4 + \cdots \end{cases} \]
by
Pi Han Goh
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Your friend, David Hilbert, runs a casino with space for an infinite number of guests. He installs a slot machine with the following properties:
- Each guest at the casino plays one round at the slot machine after all the guests have arrived.
- If there are \(n\) guests at the casino, then the odds of winning the slot machine are \(\frac{1}{n}.\)
- When a guest wins, they get all of the money in the machine.
To understand how profitable this machine is, he wants to know the probability that no one wins the slot machine.
If David fills up his casino with a number of guests approaching infinity, what does this probability approach?
by
Blan Morrison
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