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== Problems == | == Problems == | ||
| − | 1. Find all positive integer solutions <math> x, y, z</math> of the equation <math> 3^x | + | 1. Find all positive integer solutions <math>x, y, z</math> of the equation <math>3^x + 4^y = 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)'' | 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)'' | 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)'' | ||
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| + | 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 == | == Online Math Circle == | ||
Latest revision as of 12:31, 30 July 2025
The home of DL2000
Problems
1. Find all positive integer solutions 
of the equation 
(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
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newyorkmathcircle.weebly.com
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