MathDB

P1

Part of 2019 India IMO Training Camp

Problems(4)

Line through incenter tangent to a circle

Source: Indian TST D1 P1

7/17/2019
In an acute angled triangle ABCABC with AB<ACAB < AC, let II denote the incenter and MM the midpoint of side BCBC. The line through AA perpendicular to AIAI intersects the tangent from MM to the incircle (different from line BCBC) at a point PP> Show that AIAI is tangent to the circumcircle of triangle MIPMIP.
Proposed by Tejaswi Navilarekallu
geometryincenter
$f(n)$ divides $f(2^n) -2^{f(n)}$

Source: Indian TST 4 P1

7/17/2019
Determine all non-constant monic polynomials f(x)f(x) with integer coefficients for which there exists a natural number MM such that for all n≥Mn \geq M, f(n)f(n) divides f(2n)−2f(n)f(2^n) - 2^{f(n)} Proposed by Anant Mudgal
number theorypolynomial
Integer inequality

Source: Indian TST 2019 Practice Test 1 P1

7/17/2019
Let a1,a2,…,ama_1,a_2,\ldots, a_m be a set of mm distinct positive even numbers and b1,b2,…,bnb_1,b_2,\ldots,b_n be a set of nn distinct positive odd numbers such that a1+a2+⋯+am+b1+b2+⋯+bn=2019a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019 Prove that 5m+12n≤581.5m+12n\le 581.
inequalitiesindiaTST2019number theory
Easy Geometry

Source: Indian TST 2019 Practice Test 2 P1

7/17/2019
Let the points OO and HH be the circumcenter and orthocenter of an acute angled triangle ABC.ABC. Let DD be the midpoint of BC.BC. Let EE be the point on the angle bisector of ∠BAC\angle BAC such that AE⊥HE.AE\perp HE. Let FF be the point such that AEHFAEHF is a rectangle. Prove that D,E,FD,E,F are collinear.
geometrycircumcircleangle bisectorrectangleradical axis