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A. 619. There are given four rays, $\displaystyle a$, $\displaystyle b$, $\displaystyle c$ and $\displaystyle d$ in space, starting from the same point, laying in a plane $\displaystyle \varPi$. For an arbitrary acute angle $\displaystyle \varphi$, rotate $\displaystyle \varPi$ by angle $\displaystyle \varphi$ in positive direction around each of the four rays; denote the rotated planes by $\displaystyle A_\varphi$, $\displaystyle B_\varphi$, $\displaystyle \varGamma_\varphi$ and $\displaystyle \varDelta_\varphi$, respectively. Let $\displaystyle \varSigma_\varphi$ be the plane through the intersection line of $\displaystyle A_\varphi$ and $\displaystyle B_\varphi$, and the intersection line of $\displaystyle \varGamma_\varphi$ and $\displaystyle \varDelta_\varphi$. Show that the planes $\displaystyle \varSigma_\varphi$ share a common line.

(5 points)

Deadline expired on 10 June 2014.

Statistics on problem A. 619.
 4 students sent a solution. 5 points: Ágoston Péter, Fehér Zsombor, Williams Kada. 3 points: 1 student.

• Problems in Mathematics of KöMaL, May 2014

•  Támogatóink: Morgan Stanley