MathDB

11.4

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(3)

3 spheres touch plane and line - VI Soros Olympiad 1999-00 Round 1 11.4

Source:

5/21/2024
Let the line LL be perpendicular to the plane PP. Three spheres touch each other in pairs so that each sphere touches the plane PP and the line LL. The radius of the larger sphere is 11. Find the minimum radius of the smallest sphere.
geometry3D geometrysphere
p^{2k}+q^{2n}=r^2 (VI Soros Olympiad 1990-00 R2 11.4)

Source:

5/28/2024
For prime numbers pp and qq, natural numbers nn, kk, rr, the equality p2k+q2n=r2p^{2k}+q^{2n}=r^2 holds. Prove that the number rr is prime.
number theoryDiophantine equationdiophantine
circumcenter of (BCD) lies on (ABD), AB=AC

Source: VI Soros Olympiad 1990-00 R3 11.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
Given isosceles triangle ABCABC (AB=ACAB = AC). A straight line ℓ\ell is drawn through its vertex BB at a right angle with ABAB . On the straight line ACAC, an arbitrary point DD is taken, different from the vertices, and a straight line is drawn through it at a right angle with ACAC, intersecting ℓ\ell at the point FF. Prove that the center of the circle circumscribed around the triangle BCDBCD lies on the circumscribed circle of triangle ABDABD.
geometryConcyclic