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Excel in math and science
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Advanced Factorization
Advanced factorization is a gateway to algebraic number theory, which mathematicians study in order to solve famous conjectures like Fermat's Last Theorem.
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, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If \(P(n)=\frac{n^3+n^2-2n}{n^2-4}+\frac{n+1}{2+n-n^2}\), what is the value of \(\lfloor P(409) \rfloor + \lfloor P(423) \rfloor\)?
Details and assumptions
The function \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer smaller than or equal to \(x\). For example, \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -5 \rfloor = -5\).
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, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If \(x\) and \(y\) are real numbers such that \[x\neq y, x+y=5, \mbox{ and } xy=2,\] what is the value of \[\frac{x^8-y^8}{433(x-y)}?\]
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, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
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If \(\frac{x^2+y^2+z^2}{xy+yz+zx}=7\) and \(\frac{y+z}{x}+\frac{z+x}{y}+\frac{x+y}{z}=6\), what is the value of \(\frac{(x+y+z)^3}{xyz}\)?
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, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
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Let \(a\) be a positive number such that \[a^2+\frac{1}{a^2}=3.\] If the value of \[a^3+\frac{1}{a^3}\] can be expressed as \(m\sqrt{n},\) where \(n\) is a prime number, what is \(m+n?\)
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, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
\(x, y\) and \(z\) are complex numbers such that
\[ \begin{cases} x + y + z & = 46 \\ (x-y)^2 + (y-z)^2 + (z-x) ^ 2 & = xyz.\\ \end{cases} \]
What is the value of \( \frac{ x^3 + y^3 + z^3} { xyz} \)?