Problem C. 1334. (January 2016)
C. 1334. Given that the cyclic quadrilateral \(\displaystyle ABCD\) is either a trapezium (where \(\displaystyle AB\parallel CD\)) or its side \(\displaystyle AB\) is a diameter of the circumscribed circle, then the equations
\(\displaystyle BD\cdot x+AD\cdot \sqrt{1-x^2} = AC;\)
\(\displaystyle AC\cdot x+BC\cdot \sqrt{1-x^2} = BD\)
are held for some \(\displaystyle x\) of \(\displaystyle 0<x<1\).
(5 pont)
Deadline expired on February 10, 2016.
Statistics:
25 students sent a solution. 5 points: Csorba Benjámin, Gera Dóra, Horváth András János, Kocsis Júlia, Komoróczy Ádám, Kormányos Hanna Rebeka, Kósa Szilárd, Matusek Márton, Mészáros 01 Viktória, Nagy 911 Viktória, Szajkó Gréta, Tatai Mihály, Tóth Adrián. 4 points: Csapó Márton, Kasó Ferenc, Sudár Ákos, Szücs Patrícia, Tar Viktor. 3 points: 1 student. 2 points: 3 students. 0 point: 3 students.
Problems in Mathematics of KöMaL, January 2016