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== Problems == | == Problems == | ||
| − | 1. Find all positive integer solutions <math> x, y, z</math> of the equation <math> 3^x \plus{} 4^y \equal{} 5^z.</math> | + | 1. Find all positive integer solutions <math> x, y, z</math> of the equation <math> 3^x \plus{} 4^y \equal{} 5^z.</math> ''(IMO Shortlist 1991)'' |
| + | |||
| + | 2. Find the number of integers <math>n</math> such that <cmath>1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.</cmath> ''(Harvard-MIT Math Tournament)'' | ||
| + | |||
| + | 3. Compute <cmath>\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}}.</cmath> ''(Harvard-MIT Math Tournament)'' | ||
| + | |||
| + | 4. | ||
| + | Let <math>x,y,z</math> be positive real numbers such that <math> xy+yz+zx\geq3 </math>. Prove that<math> \frac{x}{\sqrt{4x+5y}}+\frac{y}{\sqrt{4y+5z}}+\frac{z}{\sqrt{4z+5x}}\geq1 </math> | ||
| + | |||
| + | == Online Math Circle == | ||
| + | |||
| + | Go to the OMC or Online Math Circle at: | ||
| + | |||
| + | newyorkmathcircle.weebly.com | ||
| + | |||
| + | == Q&A == | ||
| + | Edit the article here: | ||
| + | |||
| + | ---- | ||
| + | |||
| + | Ex: | ||
| + | Q: PI IS TASTY | ||
| + | A: Yes it is | ||
| + | |||
| + | ---- | ||
Latest revision as of 16:18, 22 February 2014
The home of DL2000
Problems
1. Find all positive integer solutions 
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 
such that ![\[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\]](http://latex.artofproblemsolving.com/1/2/a/12ae846f4e21c9fe5289c70c4737f00890817773.png)
(Harvard-MIT Math Tournament)
3. Compute ![\[\sum_{a_1=0}^\infty\sum_{a_2=0}^\infty\cdots\sum_{a_7=0}^\infty\dfrac{a_1+a_2+\cdots+a_7}{3^{a_1+a_2+\cdots+a_7}}.\]](http://latex.artofproblemsolving.com/9/3/3/933e7f1dfd766f8b0e55bb5bb9d3672ff92c0509.png)
(Harvard-MIT Math Tournament)
4.
Let 
be positive real numbers such that 
. Prove that
Online Math Circle
Go to the OMC or Online Math Circle at:
newyorkmathcircle.weebly.com
Q&A
Edit the article here:
Ex: Q: PI IS TASTY A: Yes it is
