Greece National Olympiad 2010

Bagi yang sudah kelaperan pingin ngerjain soal. Ini aku kasih problem dari Olimpiade matematika Yunani 2010. Soal aku terjemahkan sesuai pemahamanku.

Problem 1. Carilah solusi integer dari persamaan Diophantine berikut :

$x^4-6x^2+1=7\cdot2^y$

Problem 2. Jika $x,y$ adalah bilangan real positif dan $x+y=2a$ . Buktikan bahwa :

$x^3y^3(x^2+y^2)^2\leq 4a^{10}$

Kapan kesamaan terjadi?

Solusi

Sory yang problem 3 dan 4 tidak aku terjemahkan, aku kasih apa adanya dalam bahasa inggris. Takut nerjemahkannya salah karena aku tidak suka ma geometry ( sama sekali gelap ). Klo yang 4 sih combin tapi pake unsur di geometry jadinya gelap lagi.

Problem 3. A triangle $ABC$ is inscribed in a circle $C(O,R)$ and has incenter $I$ . Lines $AI, BI, CI$ meet the circumcircle $(O)$ of triangle $ABC$ at points $D, E, F$ respectively. The circles with diameter $ID, IE, IF$ meet the sides $BC, CA, AB$ at pairs of points $(A_1, A_2), (B_1, B_2), (C_1, C_2)$ respectively. Prove that the six points $A_1, A_2, B_1, B_2, C_1, C_2$ are concyclic.

Problem 4. On the plane are given $k+n$ distinct lines, where $k\geq 2$ is integer and $n$ is integer as well. Any three of these lines do not pass through the same point . Among these lines exactly $k$ are parallel and all the other $n$ lines intersect each other. All $k+n$ lines define on the plane a partition of triangular , polygonic or not bounded regions. Two regions are colled different, if the have not common points or if they have common points only on their boundary. A regions is called ''good'' if it contained in a zone between two parallel lines . If in a such given configuration the minimum number of ''good'' region is 176 and the maximum number of these regions is 221 , find $k$ and $n$ .