MathDB

4

Part of 2014 Junior Balkan Team Selection Tests - Romania

Problems(5)

AB x CD = AC x BD if <A + <C = 60^o and AB x CD = BC xAD

Source: 2014 Romania JBMO TST 1.4

5/31/2020
Let ABCDABCD be a quadrilateral with ∠A+∠C=60o\angle A + \angle C = 60^o. If AB⋅CD=BC⋅ADAB \cdot CD = BC \cdot AD, prove that AB⋅CD=AC⋅BDAB \cdot CD = AC \cdot BD.
Leonard Giugiuc
geometryanglesratio
path of a chess knight , n colors, nxn board

Source: 2014 Romania JBMO TST 2.4

6/3/2020
Let n≥6n \ge 6 be an integer. We have at our disposal nn colors. We color each of the unit squares of an n×nn \times n board with one of the nn colors. a) Prove that, for any such coloring, there exists a path of a chess knight from the bottom-left to the upper-right corner, that does not use all the colors. b) Prove that, if we reduce the number of colors to ⌊2n/3⌋+2\lfloor 2n/3 \rfloor + 2, then the statement from a) is true for infinitely many values of nn and it is false also for infinitely many values of nn
floor functionColoringcombinatorics
D lies on the circumcircle of triangle XY Z, incircle, orthocenter related

Source: 2014 Romania JBMO TST 3.4

5/31/2020
In the acute triangle ABCABC, with AB≠BCAB \ne BC, let TT denote the midpoint of the side [AC],A1[AC], A_1 and C1C_1 denote the feet of the altitudes drawn from AA and CC, respectively. Let ZZ be the intersection point of the tangents in AA and CC to the circumcircle of triangle ABC,XABC, X be the intersection point of lines ZAZA and A1C1A_1C_1 and YY be the intersection point of lines ZCZC and A1C1A_1C_1. a) Prove that TT is the incircle of triangle XYZXYZ. b) The circumcircles of triangles ABCABC and A1BC1A_1BC_1 meet again at DD. Prove that the orthocenter HH of triangle ABCABC is on the line TDTD. c) Prove that the point DD lies on the circumcircle of triangle XYZXYZ.
geometrycircumcircleincircleorthocenter
concyclic wanted, intersecting chords related

Source: 2014 Romania JBMO TST 4.4

5/31/2020
In a circle, consider two chords [AB],[CD][AB], [CD] that intersect at EE, lines ACAC and BDBD meet at FF. Let GG be the projection of EE onto ACAC. We denote by M,N,KM,N,K the midpoints of the segment lines [EF],[EA][EF] ,[EA] and [AD][AD], respectively. Prove that the points M,N,K,GM, N,K,G are concyclic.
geometryConcyclicChordsmidpoints
turn all coins tail up after a finite number of moves, equilateral

Source: 2014 Romania JBMO TST 5.4

6/3/2020
On each side of an equilateral triangle of side n≥1n \ge 1 consider n−1n - 1 points that divide the sides into nn equal segments. Through these points draw parallel lines to the sides of the triangles, obtaining a net of equilateral triangles of side length 11. On each of the vertices of the small triangles put a coin head up. A move consists in flipping over three mutually adjacent coins. Find all values of nn for which it is possible to turn all coins tail up after a finite number of moves.
Colombia 1997
Equilateralcombinatorial geometrycombinatorics