MathDB

Problem 3

Part of 1999 Ukraine National Mathematical Olympiad

Problems(3)

2000-digit square with 1999 fives

Source: Ukraine 1999 Grade 8 P3

5/9/2021
Is there a 20002000-digit number which is a perfect square and 19991999 of whose digits are fives?
number theory
9999999+1999000 is composite

Source: Ukraine 1999 Grade 9 P3

5/10/2021
Show that the number 9999999+19990009999999+1999000 is composite.
number theory
angle of intersection of planes

Source: Ukraine 1999 Grade 11 P3

5/11/2021
All faces of a parallelepiped ABCDA1B1C1D1ABCDA_1B_1C_1D_1 are rhombi, and their angles at AA are all equal to α\alpha. Points M,N,P,QM,N,P,Q are selected on the edges A1B1,DC,BC,A1D1A_1B_1,DC,BC,A_1D_1, respectively, such that A1M=BPA_1M=BP and DN=A1QDN=A_1Q. Find the angle between the intersection lines of the plane A1BDA_1BD with the planes AMNAMN and APQAPQ.
3D geometrygeometry