MathDB

1

Part of 1998 Belarus Team Selection Test

Problems(6)

x^3= f( [ x ] ) + g(x)

Source: 1998 Belarus TST 1.1

12/25/2020
Do there exist functions f:R→Rf : R \to R and g:R→Rg : R \to R, gg being periodic, such that x3=f(⌊x⌋)+g(x)x^3= f(\lfloor x \rfloor ) + g(x) for all real xx ?
functionalfunctional equationalgebraperiodic
min no of calls for all 6 gossips to share the news

Source: 1998 Belarus TST 2.1

12/25/2020
Any of 66 gossips has her own news. From time to time one of them makes a telephone call to some other gossip and they discuss fill the news they know. What the minimum number of the calls is necessary so as (for) all of them to know all the news?
combinatorics
(S(n))^3 <n^4 for sum of all different natural divisors of odd natural n>1

Source: 1998 Belarus TST 3.1

12/25/2020
Let S(n)S(n) be the sum of all different natural divisors of odd natural number n>1n> 1 (including nn and 11). Prove that (S(n))3<n4(S(n))^3 <n^4.
inequalitiesnumber theorySumDivisors
PF is perpendicular to AB if ON + OH = BK

Source: 1998 Belarus TST 6.1

6/9/2020
Let OO be a point inside an acute angle with the vertex AA and H,NH, N be the feet of the perpendiculars drawn from OO onto the sides of the angle. Let point BB belong to the bisector of the angle, KK be the foot of the perpendicular from BB onto either side of the angle. Denote by P,FP,F the midpoints of the segments AK,HNAK,HN respectively. Known that ON+OH=BKON + OH = BK, prove that PFPF is perpendicular to ABAB.
Ya. Konstantinovski
geometryperpendicular bisectorangle bisectormidpointsangle
least no to be deleted such that sum of any 2 in {1,2,...,2n-1,2n}is composite

Source: 1998 Belarus TST 7.1

12/25/2020
Let n≥2n\ge 2 be positive integer. Find the least possible number of elements of tile set A={1,2,...,2n−1,2n}A =\{1,2,...,2n-1,2n\} that should be deleted in order to the sum of any two different elements remained be a composite number.
number theorycompositioncombinatorics
locus of the intersection points of PS,RQ, intersecting circles related

Source: 1998 Belarus TST 8.1

6/9/2020
Two circles S1S_1 and S2S_2 intersect at different points P,QP,Q. The arc of S1S_1 lying inside S2S_2 measures 2a2a and the arc of S2S_2 lying inside S1S_1 measures 2b2b. Let TT be any point on S1S_1. Let R,SR,S be another points of intersection of S2S_2 with TPTP and TQTQ respectively. Let a+2b<πa+2b<\pi . Find the locus of the intersection points of PSPS and RQRQ.
S.Shikh
geometryLocuscirclesangles