Mathematical and Physical Journal
for High Schools
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# Problem C. 1283. (March 2015)

C. 1283. The longer base, $\displaystyle AB$, of trapezium $\displaystyle ABCD$ is not greater than three times the base $\displaystyle CD$. Lines $\displaystyle e$ and $\displaystyle f$ are parallel to the legs $\displaystyle BC$ and $\displaystyle DA$, respectively, and each of them halves the area of the trapezium. Let $\displaystyle P$ and $\displaystyle Q$ denote the intersections of $\displaystyle AB$ with $\displaystyle e$ and $\displaystyle f$, respectively, and let $\displaystyle P'$ and $\displaystyle Q'$ denote their intersections with $\displaystyle DC$.

$\displaystyle a)$ Prove that the intersection $\displaystyle M$ of lines $\displaystyle e$ and $\displaystyle f$ lies on the midline of the trapezium.

$\displaystyle b)$ Given that the quadrilateral $\displaystyle PQ'P'Q$ is a parallelogram, find the ratio of the area of triangle $\displaystyle MPQ$ to the area of the trapezium $\displaystyle ABCD$.

(5 pont)

Deadline expired on April 10, 2015.

### Statistics:

 91 students sent a solution. 5 points: Bindics Boldizsár, Bottlik Judit, Csapó Márton, Cseh Noémi, Egyházi Anna, Fehér Balázs, Fényes Balázs, Fetter László, Glasznova Maja, Horváth Botond, Jakus Balázs István, Klász Viktória, Knoch Júlia, Kocsis Júlia, Kormányos Hanna Rebeka, Kósa Szilárd, Kovács Kristóf, Krisztián Jonatán, Mályusz Attila, Mándoki Sára, Marozsák Tóbiás , Márton Dénes, Matusek Márton, Mészáros 01 Viktória, Mihálykó Péter, Mikulás Zsófia, Németh 729 Gábor, Pap-Takács Mónika, Sándor Gergely, Schrettner Jakab, Sebastian Fodor, Souly Alexandra, Sudár Ákos, Szepesvári Csongor, Szűcs Dorina, Szücs Patrícia, Tatai Mihály, Temesvári Bence, Tóth Tamás, Varga-Umbrich Eszter, Várkonyi Lídia, Vida Máté Gergely, Wei Cong Wu. 4 points: 25 students. 3 points: 7 students. 2 points: 2 students. 1 point: 7 students. 0 point: 7 students.

Problems in Mathematics of KöMaL, March 2015