MathDB

11.9

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(2)

cut pyramid ABCD into 8 equal and similar pyramids

Source: IV Soros Olympiad 1997-98 Corr. 11.9 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/1/2024
Cut pyramid ABCDABCD into 88 equal and similar pyramids, if:
a) AB=BC=CDAB = BC = CD, ∠ABC=∠BCD=90o\angle ABC =\angle BCD = 90^o, dihedral angle at edge BCBC is right
b) all plane angles at vertex BB are right and AB=BC=BD2AB = BC = BD\sqrt2.
Note. Whether there are other types of triangular pyramids that can be cut into any number similar to the original pyramids (their number is not necessarily 88 and the pyramids are not necessarily equal to each other) is currently unknown
combinatoricsgeometry3D geometrycombinatorial geometrypyramid
0< a <=b <= c, a+b+ c = 7, abc = 9. (IV Soros Olympiad 1997-98 R3 11.9)

Source:

6/2/2024
The numbers aa, bb and cc satisfy the conditions 0<a≤b≤c   ,   a+b+c=7   ,   abc=9.0 < a \le b \le c\,\,\,,\,\,\, a+b+ c = 7\,\,\,, \,\,\,abc = 9. Within what limits can each of the numbers aa, bb and cc vary?
algebrainequalities