Problem B. 4712. (April 2015)
B. 4712. What percentage of a pencil gets wasted? Assume that a pencil is a cylinder, infinitely long, and the graphite rod inside is also cylindrical. The axes of the two cylinders coincide. When the pencil is sharpened, its point is a perfect cone with an apex angle of 12 degrees. When we write with the pencil, its axis always encloses a 42-degree angle with the plane of the paper. We keep using the pencil until we can no longer write with it since no matter how we rotate it about its axis, the wood will scratch the paper. Then the pencil is sharpened again to the shape of a 12-degree cone, but never longer, that is, the tip of the pencil never changes during sharpening, it only wears in writing. What percentage of the graphite is wasted by scraping it off with the sharpener? Will someone holding the pencil at a 45-degree angle waste more than that or less? If so, by how much?
Suggested by E. M. Gáspár, Budapest
Deadline expired on May 11, 2015.
61 students sent a solution. 5 points: Adorján Dániel, Andó Angelika, Baran Zsuzsanna, Bereczki Zoltán, Bodolai Előd, Cseh Kristóf, Csépai András, Czirkos Angéla, Döbröntei Dávid Bence, Fekete Panna, Gál Boglárka, Gál Hanna, Gáspár Attila, Glasznova Maja, Gyulai-Nagy Szuzina, Hansel Soma, Katona Dániel, Kerekes Anna, Keresztfalvi Bálint, Khayouti Sára, Kovács 162 Viktória, Lajkó Kálmán, Leitereg Miklós, Nagy Ábel, Nagy Kartal, Nagy-György Pál, Nagy-György Zoltán, Németh 123 Balázs, Olexó Gergely, Porupsánszki István, Sal Kristóf, Schefler Barna, Schrettner Bálint, Schwarcz Tamás, Solymosi Zsófia, Szakács Lili Kata, Szauer Marcell, Szebellédi Márton, Tóth Viktor, Váli Benedek, Williams Kada. 4 points: Bindics Boldizsár, Kuchár Zsolt, Szakály Marcell, Vu Mai Phuong, Wiandt Péter. 3 points: 12 students. 2 points: 3 students.