MathDB

2

Part of 1995 Romania Team Selection Test

Problems(4)

(x + y)(y + z)(z + x) = xyzt

Source: Romanian TST 1995

7/14/2009
Find all positive integers x,y,z,t x,y,z,t such that x,y,z x,y,z are pairwise coprime and (x \plus{} y)(y \plus{} z)(z \plus{} x) \equal{} xyzt.
number theory unsolvednumber theory
Intersection of polygons

Source: Romania TST 1995 Test 2 P2

2/22/2014
Suppose that nn polygons of area s=(n−1)2s = (n - 1)^2 are placed on a polygon of area S=n(n−1)22S = \frac{n(n - 1)^2}{2}. Prove that there exist two of the nn smaller polygons whose intersection has the area at least 11.
geometrygeometry proposed
parallelepipeds are cubes when two sums of volumes are equal

Source: Romania TST 1995 3.2

2/17/2020
A cube is partitioned into finitely many rectangular parallelepipeds with the edges parallel to the edges of the cube. Prove that if the sum of the volumes of the circumspheres of these parallelepipeds equals the volume of the circumscribed sphere of the cube, then all the parallelepipeds are cubes.
parallelepipedcombinatorial geometrycombinatoricsVolumecubepartition
lcm(1,2,...,n)

Source: Romania TST 1995

9/28/2009
For each positive integer n n,define f(n)\equal{}lcm(1,2,...,n). (a)Prove that for every k k there exist k k consecutive positive integers on which f f is constant. (b)Find the maximum possible cardinality of a set of consecutive positive integers on which f f is strictly increasing and find all sets for which this maximum is attained.
number theory unsolvednumber theory