User:DanielL2000

The home of DL2000

Problems

1. Find all positive integer solutions $x, y, z$ of the equation $3^x \plus{} 4^y \equal{} 5^z.$ (Error compiling LaTeX. Unknown error_msg) (IMO Shortlist 1991)

2. Find the number of integers $n$ such that \[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\] (Harvard-MIT Math Tournament)

3. Compute \[\sum_{a_1=0}^\infty\sum_{a_2=0}^\infty\cdots\sum_{a_7=0}^\infty\dfrac{a_1+a_2+\cdots+a_7}{3^{a_1+a_2+\cdots+a_7}}.\] (Harvard-MIT Math Tournament)

4. Let $x,y,z$ be positive real numbers such that $xy+yz+zx\geq3$. Prove that$\frac{x}{\sqrt{4x+5y}}+\frac{y}{\sqrt{4y+5z}}+\frac{z}{\sqrt{4z+5x}}\geq1$

Online Math Circle

Go to the OMC or Online Math Circle at:

newyorkmathcircle.weebly.com

Q&A

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Ex: Q: PI IS TASTY A: Yes it is