MathDB

Problem 3

Part of 2006 Federal Math Competition of S&M

Problems(3)

largest number divisible by each of its digits (Serbia MO 2006 1st Grade P3)

Source:

4/10/2021
Determine the largest natural number whose all decimal digits are different and which is divisible by each of its digits.
number theory
mutually exclusive sets and coprimality

Source: Serbia MO 2006 2nd Grade P3

4/10/2021
For every natural number aa, consider the set S(a)={an+a+1∣n=2,3,…}S(a)=\{a^n+a+1|n=2,3,\ldots\}. Does there exist an infinite set A⊂NA\subset\mathbb N with the property that for any two distinct elements x,y∈Ax,y\in A, xx and yy are coprime and S(x)∩S(y)=∅S(x)\cap S(y)=\emptyset?
number theory
projection of tetrahedron onto planes; area ratio

Source: Serbia MO 2006 3&4th Grades P3

4/10/2021
Show that for an arbitrary tetrahedron there are two planes such that the ratio of the areas of the projections of the tetrahedron onto the two planes is not less than 2\sqrt2.
geometry3D geometrytetrahedronratioinequalities