MathDB

1

Part of 2016 India Regional Mathematical Olympiad

Problems(7)

Equal Circumradius of variable triangles!

Source: RMO Delhi 2016, P1

10/11/2016
Given are two circles ω1,ω2\omega_1,\omega_2 which intersect at points X,YX,Y. Let PP be an arbitrary point on ω1\omega_1. Suppose that the lines PX,PYPX,PY meet ω2\omega_2 again at points A,BA,B respectively. Prove that the circumcircles of all triangles PABPAB have the same radius.
geometrycircumcircle
Perpendicularity with incenter in a right triangle

Source: RMO Mumbai 2016, P1

10/11/2016
Let ABCABC be a right-angled triangle with ∠B=90∘\angle B=90^{\circ}. Let II be the incenter of ABCABC. Draw a line perpendicular to AIAI at II. Let it intersect the line CBCB at DD. Prove that CICI is perpendicular to ADAD and prove that ID=b(b−a)ID=\sqrt{b(b-a)} where BC=aBC=a and CA=bCA=b.
geometryincenter
Minimizing sum given product!

Source: RMO Maharashtra and Goa 2016, P1

10/11/2016
Find distinct positive integers n1<n2<⋯<n7n_1<n_2<\dots<n_7 with the least possible sum, such that their product n1×n2×⋯×n7n_1 \times n_2 \times \dots \times n_7 is divisible by 20162016.
inequalitiesnumber theoryconstruction
Geometry

Source: RMO 2016 Karnataka Region P1.

10/16/2016
Let ABCABC be a triangle and DD be the mid-point of BCBC. Suppose the angle bisector of ∠ADC\angle ADC is tangent to the circumcircle of triangle ABDABD at DD. Prove that ∠A=90∘\angle A=90^{\circ}.
geometry
Right angled triangle: IE=IF

Source: RMO Hyderabad 2016 , P1.

10/11/2016
Let ABCABC be a right angled triangle with ∠B=90∘\angle B=90^{\circ}. Let II be the incentre of triangle ABCABC. Suppose AIAI is extended to meet BCBC at FF . The perpendicular on AIAI at II is extended to meet ACAC at EE . Prove that IE=IFIE = IF.
geometryincenter
RMO 2016 ,Q1

Source: Oct 23,2016

10/25/2016
Let ABCABC be an isosceles triangle with AB=AC.AB=AC. Let Γ \Gamma be its circumcircle and let OO be the centre of Γ \Gamma . let COCO meet Γ \Gamma in D.D . Draw a line parallel to ACAC thrugh D.D. Let it intersect ABAB at E.E. Suppose AE:EB=2:1AE : EB=2:1 .Prove that ABCABC is an equilateral triangle.
geometryindia
2016 Chandigarh RMO the sum of any 1008 integers of 2016 is positive

Source:

8/9/2019
Suppose in a given collection of 20162016 integer, the sum of any 10081008 integers is positive. Show that sum of all 20162016 integers is positive.
Sumnumber theoryalgebra