MathDB

1

Part of 2002 Romania Team Selection Test

Problems(4)

Diagonals BD,CE concurrent with diameter AO in cyclic ABCDE

Source: Romanian TST 2002

2/5/2011
Let ABCDEABCDE be a cyclic pentagon inscribed in a circle of centre OO which has angles ∠B=120∘,∠C=120∘,\angle B=120^{\circ},\angle C=120^{\circ}, ∠D=130∘,∠E=100∘\angle D=130^{\circ},\angle E=100^{\circ}. Show that the diagonals BDBD and CECE meet at a point belonging to the diameter AOAO.
Dinu Șerbănescu
symmetrytrigonometrygeometry proposedgeometry
set finding

Source: Romanian IMO Team Selection Test TST 2002, problem 1

7/4/2005
Find all sets AA and BB that satisfy the following conditions: a) A∪B=ZA \cup B= \mathbb{Z}; b) if x∈Ax \in A then x−1∈Bx-1 \in B; c) if x,y∈Bx,y \in B then x+y∈Ax+y \in A. Laurentiu Panaitopol
inductionnumber theory unsolvednumber theory
x formed by digits of the sequence is rational

Source: Romanian TST 2002

2/5/2011
Let (an)n≥1(a_n)_{n\ge 1} be a sequence of positive integers defined as a1,a2>0a_1,a_2>0 and an+1a_{n+1} is the least prime divisor of an−1+ana_{n-1}+a_{n}, for all n≥2n\ge 2.
Prove that a real number xx whose decimals are digits of the numbers a1,a2,…an,…a_1,a_2,\ldots a_n,\ldots written in order, is a rational number.
Laurentiu Panaitopol
number theory proposednumber theory
Partition of set {1,2,3...4mn} so that every sum is a square

Source: Romanian TST 2002

2/5/2011
Let m,nm,n be positive integers of distinct parities and such that m<n<5mm<n<5m. Show that there exists a partition with two element subsets of the set {1,2,3,…,4mn}\{ 1,2,3,\ldots ,4mn\} such that the sum of numbers in each set is a perfect square.
Dinu Șerbănescu
number theory proposednumber theory