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Some positive integers \(n\) have the property that the sum \([n+{\rm reverse}(n)]\) consists entirely of odd (decimal) digits. For instance, \(36+63=99\) and \(409+904=1313\). We will call such numbers reversible; so \(36, \; 63, \; 409\), and \(904\) are reversible. Leading zeroes are not allowed in either \(n\) or \({\rm reverse}(n)\). There are \(120\) reversible numbers below on-thousand. H..

In laser physics, a "white cell" is a mirror system that acts as a delay line for the laser beam. The beam enters the cell, bounces around on the mirrors, and eventually works its way back out. The specific white cell we will be considering is an ellipse with the equation \(4^x+y^2=100\). The section corresponding to \(-0.01 \le x \le 0.01\) at the top is missing, allowing the light to enter and..

Let \(\rm ABC\) be a triangle with all interior angles being less than \(120\) degrees. Let \(\rm X\) be any point inside the triangle and let \({\rm XA}=p,\; {\rm XC}=q\), and \({\rm XB}=r\). Fermat challenged Torricelli to find the position of \(\rm X\) such that \(p+q+r\) was minimized. Torricelli was able to prove that if equilateral triangles \(\rm AOB, \; BNC\) and \(\rm AMC\) are construc..