MathDB

4

Part of 2012 Junior Balkan Team Selection Tests - Romania

Problems(4)

P, Q,R collinear and U, R, O, V concyclic wanted, cyclic ABCD, circumcenters

Source: 2012 Romania JBMO TST2 P4

5/29/2020
The quadrilateral ABCDABCD is inscribed in a circle centered at OO, and {P}=AC∩BD,{Q}=AB∩CD\{P\} = AC \cap BD, \{Q\} = AB \cap CD. Let RR be the second intersection point of the circumcircles of the triangles ABPABP and CDPCDP. a) Prove that the points P,QP, Q, and RR are collinear. b) If UU and VV are the circumcenters of the triangles ABPABP, and CDPCDP, respectively, prove that the points U,R,O,VU, R, O, V are concyclic.
cyclic quadrilateralgeometryConcycliccollinearCircumcentercircumcircle
100 weights, measuring 1,2, ..., 100 grams,

Source: 2012 Romania JBMO TST 1.4 Tournament of Towns, 1999

6/2/2020
100100 weights, measuring 1,2,...,1001,2, ..., 100 grams, respectively, are placed in the two pans of a scale such that the scale is balanced. Prove that two weights can be removed from each pan such that the equilibrium is not broken.
weightscombinatorics
which numbers have to be removed such that sum of remaining numbers is max?

Source: 2012 Romania JBMO TST 3.4

6/2/2020
Consider the set A={1,2,3,...,2n−1}A = \{1, 2, 3, ..., 2n - 1\}, where n≥2n \ge 2 is a positive integer. We remove from the set AA at least n−1n - 1 elements such that: • if a∈Aa \in A has been removed, and 2a∈A2a \in A, then 2a2a has also been removed, • if a,b∈A(a≠b)a, b \in A (a \ne b) have been removed and a+b∈Aa + b \in A, then a+ba + b has also been removed. Which numbers have to be removed such that the sum of the remaining numbers is maximum?
combinatoricsalgebra
infinitely many positive integers that are not lonely,

Source: 2012 Romania JBMO TST 4.4

6/2/2020
A positive integer is called lonely if the sum of the inverses of its positive divisors (including 11 and itself) is not equal with the some of the inverses of the positive divisors of any other positive integer. a) Show that any prime number is lonely. b) Prove that there are infinitely many numbers that are not lonely