MathDB

5

Part of 2008 Postal Coaching

Problems(6)

\Delta = nr^2, area and inradius of a triangle with integer sides

Source: Indian Postal Coaching 2008 set 1 p5

5/25/2020
Prove that there are in finitely many positive integers nn such that Δ=nr2\Delta = nr^2, where Δ\Delta and rr are respectively the area and the inradius of a triangle with integer sides.
geometryinteger sidelengthsareainradius
max no of irreducible fractions a/b in interval (0,1/n)

Source: Indian Postal Coaching 2008 set 2 p5

5/25/2020
Let n∈Nn \in N. Find the maximum number of irreducible fractions a/b (i.e., gcd(a,b)=1gcd(a, b) = 1) which lie in the interval (0,1/n)(0,1/n).
FractioncombinatoricsIrreducible
Covering an n-polygon

Source: Indian postal coaching 2008

12/21/2008
Let A1A2...An A_1A_2...A_n be a convex polygon. Show that there exists an index j j such that the circum-circle of the triangle A_j A_{j \plus{} 1} A_{j \plus{} 2} covers the polygon (here indices are read modulo n).
geometrycircumcircleinductionextremal principlecombinatorics unsolvedcombinatorics
PC tangent wanted , starting with a semicircle

Source: Indian Postal Coaching 2008 set 4 p5

5/25/2020
Let ω\omega be the semicircle on diameter ABAB. A line parallel to ABAB intersects ω\omega at CC and DD so that BB and CC lie on opposite sides of ADAD. The line through CC parallel to ADAD meets ω\omega again in EE. Lines BEBE and CDCD meet in FF and the line through FF parallel to ADAD meets ABAB in PP. Prove that PCPC is tangent to ω\omega.
tangentgeometrysemicircleparallelcircle
ABCD is cyclic if angles are equal, incircles related

Source: Indian Postal Coaching 2008 set 5 p5

5/25/2020
A convex quadrilateral ABCDABCD is given. There rays BABA and CDCD meet in PP, and the rays BCBC and ADAD meet in QQ. Let HH be the projection of DD on PQPQ. Prove that ABCDABCD is cyclic if and only if the angle between the rays beginning at HH and tangent to the incircle of triangle ADPADP is equal to the angle between the rays beginning at HH and tangent to the incircle of triangle CDQCDQ. Also fi nd out whether ABCDABCD is inscribable or circumscribable and justify.
geometryincirclecyclic quadrilateralanglesequal angles
if circumcenters of DEF,ABC coincide then ABC is equilateral

Source: Indian Postal Coaching 2008 set 6 p5

5/25/2020
Consider the triangle ABCABC and the points D∈(BC),E∈(CA),F∈(AB)D \in (BC),E \in (CA), F \in (AB), such that BDDC=CEEA=AFFB\frac{BD}{DC}=\frac{CE}{EA}=\frac{AF}{FB}. Prove that if the circumcenters of triangles DEFDEF and ABCABC coincide, then the triangle ABCABC is equilateral.
CircumcenterEquilateralratiogeometry