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Indonesia Juniors 2004 day 1 OSN SMP

Source:

October 30, 2021
algebrageometrycombinatoricsnumber theoryindonesia juniors

Problem Statement

p1. Known points A(−1.2)A (-1.2), B(0,2)B (0,2), C(3,0)C (3,0), and D(3,−1)D (3, -1) as seen in the following picture. Determine the measure of the angle AODAOD . https://cdn.artofproblemsolving.com/attachments/f/2/ca857aaf54c803db34d8d52505ef9a80e7130f.png
p2. Determine all prime numbers p>2p> 2 until pp divides 712−372−5171^2 - 37^2 - 51.
p3. A ball if dropped perpendicular to the ground from a height then it will bounce back perpendicular along the high third again, down back upright and bouncing back a third of its height, and next. If the distance traveled by the ball when it touches the ground the fourth time is equal to 106106 meters. From what height is the ball was dropped?
p4. The beam ABCD.EFGHABCD.EFGH is obtained by pasting two unit cubes ABCD.PQRSABCD.PQRS and PQRS.EFGHPQRS.EFGH. The point K is the midpoint of the edge ABAB, while the point LL is the midpoint of the edge SHSH. What is the length of the line segment KLKL?
p5. How many integer numbers are no greater than 20042004, with remainder 11 when divided by 22, with remainder 22 when divided by 33, with remainder 33 when divided by 44, and with remainder 44 when divided by 55?
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