Mathematical and Physical Journal
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Problem B. 4692. (February 2015)

B. 4692. The sides of an acute-angled triangle are \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\), the opposite angles are \(\displaystyle \alpha\), \(\displaystyle \beta\), \(\displaystyle \gamma\), and the lengths of the corresponding altitudes are \(\displaystyle m_a\), \(\displaystyle m_b\), \(\displaystyle m_c\), respectively. Prove that \(\displaystyle \frac{m_a}{a} + \frac{m_b}{b} + \frac{m_c}{c} \ge 2\cos \alpha \cos\beta \cos \gamma \left(\frac{1}{\sin 2\alpha} + \frac{1}{\sin 2\beta} + \frac{1}{\sin 2\gamma}\right) + \sqrt{3}\,\).

Suggested by K. Williams, Szeged

(5 pont)

Deadline expired on March 10, 2015.


Statistics:

52 students sent a solution.
5 points:Andi Gabriel Brojbeanu, Andó Angelika, Baran Zsuzsanna, Bereczki Zoltán, Bodolai Előd, Cseh Kristóf, Csépai András, Döbröntei Dávid Bence, Fekete Panna, Gál Hanna, Gáspár Attila, Gema Szabolcs, Hansel Soma, Katona Dániel, Kerekes Anna, Kovács 246 Benedek, Kovács 972 Márton, Lajkó Kálmán, Leitereg Miklós, Nagy Dávid Paszkál, Nagy-György Pál, Németh 123 Balázs, Öreg Botond, Páli Petra, Pap Tibor, Polgár Márton, Porupsánszki István, Sal Kristóf, Schrettner Bálint, Schwarcz Tamás, Szabó 157 Dániel, Szajbély Zsigmond, Szebellédi Márton, Szemerédi Levente, Tompa Tamás Lajos, Vághy Mihály, Vágó Ákos, Váli Benedek, Varga-Umbrich Eszter, Vu Mai Phuong, Wiandt Péter, Williams Kada.
4 points:Gál Boglárka, Geng Máté, Kasó Ferenc, Nagy Odett, Nagy-György Zoltán, Stein Ármin.
3 points:1 student.
1 point:3 students.

Problems in Mathematics of KöMaL, February 2015