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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..
Find the smallest \(x+y+z\) with integers \(x>y>z>0\) such that \(x+y\), \(x-y\), \(x+z\), \(x-z\), \(y+z\), \(y-z\) are all perfect squares. \(x>y>z>0\) 을 만족하는 자연수 \(x, \; y\; z\) 에 대해서 \(x+y\), \(x-y\), \(x+z\), \(x-z\), \(y+z\), \(y-z\) 가 모두 완전제곱수가 되는 \(x+y+z\) 의 최솟값을 구하시오. 늘 그렇듯이 처음엔 무식하게 하나하나 완전제곱수인 것을 모두 확인하는 방법을 사용했지만, 아니나 다를까 시간이 너무 오래 걸린다. 그래서 다음의 방법을 생각 !! \[x+y=a^2 \label{a}\tag{1}\] ..