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Solved by 17 users: fetetriste, Norbert, dan, svirg, DAV, zloy_mipt, popo, RAVEman, UdH-WiNGeR, defrager, aioni, FordPerfect, mazahaka, Dest, abortmozga.ru, avg79, regmar.
FordPerfect09 mar 2010C++200.03539 
DAV18 jun 2009C++4400.02557 
DAV04 jun 2009C++4300.01595 
svirg07 nov 2008C++1400.01597 
DAV17 nov 2008C++2400.01626 
DAV17 nov 2008C++1100.01629 
DAV17 nov 2008C++1800.01629 
fetetriste11 dec 2008C++3100.01664 
dan06 nov 2008C++400.01666 
dan06 nov 2008C++500.01666 
dan06 nov 2008C++200.01672 
RAVEman07 mar 2009C++1900.01715 
popo01 feb 2009C++600.01720 
dan06 nov 2008C++900.01733 
dan06 nov 2008C++1100.01750 
fetetriste29 oct 2008C++2700.01782 
fetetriste29 oct 2008C++2600.01815 
fetetriste29 oct 2008C++2900.01815 
fetetriste10 dec 2008C++3000.01815 
UdH-WiNGeR08 mar 2009Java500.15859 
RAVEman07 mar 2009C++1700.01925 
abortmozga.ru23 oct 2010C++19800.01962 
abortmozga.ru23 oct 2010C++19700.01982 
Norbert02 nov 2008C++100.01994 

Flight to Mars

Time limit = 1 second(s)

Memory limit = 64 Mb

In October 1, 203* the Only Superpower launches the first manned expedition to Mars on the Avenger spacecraft. To ensure safe landing on Mars, a robotic space probe orbiting Mars has dropped three beacons marking a safe landing area on the Martian surface.

However, to cut the cost of the expedition, Avenger is not equipped with a parachute. So Avenger just moves at a constant speed until the Martian surface is reached.

The taikonaut commander asks you to determine if Avenger hits the surface inside the safe landing area. This area is the spherical triangle with the apexes at the beacon landing points.


The first input line contains three integer numbers (x0, y0, z0) — the Cartesian coordinates of the Martian center. The next input line contains one integer number R (1 ≤ R ≤ 10 000) — the radius of Mars.

The next three lines contain spherical coordinates of the beacons. Each coordinate is specified by two integer numbers d_i and l_i (-90 ≤ d_i ≤ 90, -180 ≤ l_i ≤ 180), where d_i is the latitude, l_i is the longitude, both measured in degrees. The north latitude is positive, and the south latitude is negative. The west longitude is positive, and the east longitude is negative. The next line contains the Cartesian coordinates (x1,y1,z1) of the Avenger starting point in the circummartian space The next line contains the direction (v_x,v_y,v_z) of the Avenger's movement.

All coordinates do not exceed 10000 by absolute value. The North Pole is towards (0, 0, 1) direction from the Martian center. The Prime Meridian crosses the Equator at (1, 0, 0) direction from the Martian center. The beacon landing points do not belong to one great circle. Also there exists a point in space, visible from all beacon landing points.

Since the spacecraft moves really fast, you may disregard the Martian rotation and orbital movement and think of it as a fixed sphere. Gravitational attraction of Mars is also considered negligible.

Output If Avenger successfully crashes inside or on the boundary of the safe landing area, output «YES», otherwise output «NO». It is guaranteed that if Avenger crashes outside of the safe landing area the distance between the crash point and the border of the safe landing area is at least R10-6.

0 0 0
0 0
90 0
0 90
1 1 1
-1 -1 -1

ACM ICPC 2008-2009, Moscow Subregional Contest, Moscow, October 26, 2008
26 October 2008

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