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1. Problem Suppose is an integer and all the roots of are integers. Find all possible values of 2. Solution Let’s say a,b,c are integers and roots of We can write this polynomial as Comparing (1) and (2) Using (3) and (5) Let’s say , from (6), since a and b are integer values can…
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Problem Let ABC be triangle in which AB = AC. Suppose the orthocentre of the triangle lies on the incircle. Find the ratio AB/BC Solution Method 1 Proof: Given AB=AC, AD is perpendicular bisector of BC and angle bisector of So, incenter and orthocenter lies on AD Since H (Orthocenter) lies on incircle But and…
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1. Problem The in-circle of triangle ABC touches the sides BC, CA and AB in K, L and M respectively. The line through A and parallel to LK meets MK in P and the line through A and parallel to MK meets LK in Q. Show that the line P Q bisects the sides AB…
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1. Problem Let P be an interior point of a triangle ABC and AP, BP, CP meet the sides BC, CA, AB in D, E, F respectively. Show that 2. Solution Proof: = area of figure Since height from C to base is same for both AFC and BFC (proof in Lemma 1) Since height…
The Mathematical Pursuit 