MathDB

1

Part of 2011 India IMO Training Camp

Problems(3)

Prove that the radius of the circle..

Source: Indian IMO Training camp-2011

5/17/2011
Let ABCABC be a triangle each of whose angles is greater than 30∘30^{\circ}. Suppose a circle centered with PP cuts segments BCBC in T,Q;CAT,Q; CA in K,LK,L and ABAB in M,NM,N such that they are on a circle in counterclockwise direction in that order.Suppose further PQK,PLM,PNTPQK,PLM,PNT are equilateral. Prove that:
a)a) The radius of the circle is 2abca2+b2+c2+43S\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S} where SS is area.
b)a⋅AP=b⋅BP=c⋅PC.b) a\cdot AP=b\cdot BP=c\cdot PC.
geometrytrapezoidtrigonometrycyclic quadrilateralgeometry unsolved
Indian TST Number theory

Source: Selection Test 2

6/20/2011
Find all positive integer nn satisfying the conditions
a)n2=(a+1)3−a3a)n^2=(a+1)^3-a^3
b)2n+119b)2n+119 is a perfect square.
geometry3D geometryAMCAIMEDiophantine equationnumber theory unsolvednumber theory
ABC be an acute-angled

Source: Indian TST Day 4 Problem 1

6/30/2011
Let ABCABC be an acute-angled triangle. Let AD,BE,CFAD,BE,CF be internal bisectors with D,E,FD, E, F on BC,CA,ABBC, CA, AB respectively. Prove that EFBC+FDCA+DEAB≥1+rR\frac{EF}{BC}+\frac{FD}{CA}+\frac{DE}{AB}\geq 1+\frac{r}{R}
inequalitiestrigonometryinequalities unsolved