MathDB

3

Part of 2001 India IMO Training Camp

Problems(5)

Geometric inequality!

Source: India TST 2001 Day 1 Problem 3

1/31/2015
In a triangle ABCABC with incircle ω\omega and incenter II , the segments AIAI , BIBI , CICI cut ω\omega at DD , EE , FF , respectively. Rays AIAI , BIBI , CICI meet the sides BCBC , CACA , ABAB at LL , MM , NN respectively. Prove that: AL+BM+CN≤3(AD+BE+CF)AL+BM+CN \leq 3(AD+BE+CF) When does equality occur?
inequalitiesgeometryincentergeometry unsolved
Counting pairs of subsets!

Source: India TST 2001 Day 2 Problem 3

1/31/2015
Find the number of all unordered pairs {A,B}\{A,B \} of subsets of an 88-element set, such that A∩B≠∅A\cap B \neq \emptyset and ∣A∣≠∣B∣\left |A \right | \neq \left |B \right |.
combinatorics proposedcombinatorics
A constant independent of n!

Source: India TST 2001 Day 3 Problem 3

1/31/2015
Points B=B1,B2,⋯ ,Bn,Bn+1=CB = B_1 , B_2, \cdots , B_n , B_{n+1} = C are chosen on side BCBC of a triangle ABCABC in that order. Let rjr_j be the inradius of triangle ABjBj+1AB_jB_{j+1} for j=1,⋯ ,nj = 1, \cdots, n , and rr be the inradius of △ABC\triangle ABC. Show that there is a constant λ\lambda independent of nn such that : (λ−r1)(λ−r2)⋯(λ−rn)=λn−1(λ−r)(\lambda -r_1)(\lambda -r_2)\cdots (\lambda -r_n) =\lambda^{n-1}(\lambda -r)
geometryinradiusgeometry proposed
Coloring of a grid!

Source: India TST 2001 Day 4 problem 3

1/31/2015
Each vertex of an m×nm\times n grid is colored blue, green or red in such a way that all the boundary vertices are red. We say that a unit square of the grid is properly colored if: (i)(i) all the three colors occur at the vertices of the square, and (ii)(ii) one side of the square has the endpoints of the same color. Show that the number of properly colored squares is even.
combinatorics unsolvedcombinatorics
Aproximation of a polynomial !

Source: India TST 2001 Day 5 Problem 3

1/31/2015
Let P(x)P(x) be a polynomial of degree nn with real coefficients and let a≥3a\geq 3. Prove that max⁡0≤j≤n+1∣aj−P(j)∣≥1\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1
algebrapolynomialinequalitiesinductionalgebra unsolved