Mathematical and Physical Journal
for High Schools
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# 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