MathDB

3

Part of 2012 Indonesia TST

Problems(8)

Equal area only if orthocenter-connecting is perpendicular

Source: 2012 Indonesia Round 2 TST 1 Problem 3

2/26/2012
Given a convex quadrilateral ABCDABCD, let PP and QQ be points on BCBC and CDCD respectively such that ∠BAP=∠DAQ\angle BAP = \angle DAQ. Prove that the triangles ABPABP and ADQADQ have the same area if the line connecting their orthocenters is perpendicular to ACAC.
geometrygeometry proposed
Sum of 3 elements are all different

Source: 2012 Indonesia Round 2 TST 4 Problem 3

3/18/2012
Let SS be a subset of {1,2,3,4,5,6,7,8,9,10}\{1,2,3,4,5,6,7,8,9,10\}. If SS has the property that the sums of three elements of SS are all different, find the maximum number of elements of SS.
combinatorics proposedcombinatorics
Geometric mean of fractions is larger

Source: 2012 Indonesia Round 2 TST 2 Problem 3

3/4/2012
Let a1,a2,…,an,b1,b2,…,bna_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n be positive reals such that a1+b1=a2+b2=…+an+bna_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n and a1a2…anb1b2…bnn≥n.\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n. Prove that a1a2…anb1b2…bnn≥a1+a2+…+anb1+b2+…+bn.\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.
inequalities unsolvedinequalities
Incircle and excircle of triangles of a cyclic quadrilateral

Source: 2012 Indonesia Round 2 TST 3 Problem 3

3/18/2012
Given a cyclic quadrilateral ABCDABCD with the circumcenter OO, with BCBC and ADAD not parallel. Let PP be the intersection of ACAC and BDBD. Let EE be the intersection of the rays ABAB and DCDC. Let II be the incenter of EBCEBC and the incircle of EBCEBC touches BCBC at T1T_1. Let JJ be the excenter of EADEAD that touches ADAD and the excircle of EADEAD that touches ADAD touches ADAD at T2T_2. Let QQ be the intersection between IT1IT_1 and JT2JT_2. Prove that O,P,QO,P,Q are collinear.
geometrycircumcircleincentercyclic quadrilateralprojective geometrygeometry unsolved
Lines from vertices to some point are perpendicular

Source: 2012 Indonesia Round 2.5 TST 1 Problem 3

5/10/2012
The incircle of a triangle ABCABC is tangent to the sides AB,ACAB,AC at M,NM,N respectively. Suppose PP is the intersection between MNMN and the bisector of ∠ABC\angle ABC. Prove that BPBP and CPCP are perpendicular.
geometrygeometry proposedAngle Chasingcomplex numberscyclic quadrilateral
Four Simson lines intersect; prove it's a rectangle

Source: 2012 Indonesia Round 2.5 TST 3 Problem 3

5/21/2012
Suppose l(M,XYZ)l(M, XYZ) is a Simson line of the triangle XYZXYZ that passes through MM.
Suppose ABCDEFABCDEF is a cyclic hexagon such that l(A,BDF),l(B,ACE),l(D,ABF),l(E,ABC)l(A, BDF), l(B, ACE), l(D, ABF), l(E, ABC) intersect at a single point. Prove that CDEFCDEF is a rectangle.
Should the first sentence read: Suppose l(M,XYZ)l(M, XYZ) is a Simson line of the triangle XYZXYZ with respect to MM. ? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.
geometryrectanglegeometry unsolved
QM of distances

Source: 2012 Indonesia Round 2.5 TST 2 Problem 3

5/21/2012
The cross of a convex nn-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a 3×43 \times 4 rectangle is 32+32+42+42+52+526=536\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}.
Suppose SS is a dodecagon (1212-gon) inscribed in a unit circle. Find the greatest possible cross of SS.
quadraticsgeometryrectanglevectorgeometry unsolved
For every pair, there exists a point that makes 60 degrees

Source: 2012 Indonesia Round 2.5 TST 4 Problem 3

5/31/2012
Let P1P2…PnP_1P_2\ldots P_n be an nn-gon such that for all i,j∈{1,2,…,n}i,j \in \{1,2,\ldots,n\} where i≠ji \neq j, there exists k≠i,jk \neq i,j such that ∠PiPkPj=60∘\angle P_iP_kP_j = 60^\circ. Prove that n=3n=3.
geometry unsolvedgeometry