MathDB

6

Part of 2014 India Regional Mathematical Olympiad

Problems(5)

Prove that I1D, I2E, I3F are concurrent

Source: Mumbai Region RMO 2014 Problem 6

12/7/2014
Let D,E,FD,E,F be the points of contact of the incircle of an acute-angled triangle ABCABC with BC,CA,ABBC,CA,AB respectively. Let I1,I2,I3I_1,I_2,I_3 be the incentres of the triangles AFE,BDF,CEDAFE, BDF, CED, respectively. Prove that the lines I1D,I2E,I3FI_1D, I_2E, I_3F are concurrent.
geometryincentertrapezoidangle bisectorgeometry unsolved
Indian RMO 2014 P6

Source:

12/7/2014
Let x1,x2,x3…x2014x_1,x_2,x_3 \ldots x_{2014} be positive real numbers such that ∑j=12014xj=1\sum_{j=1}^{2014} x_j=1. Determine with proof the smallest constant KK such that K∑j=12014xj21−xj≥1K\sum_{j=1}^{2014}\frac{x_j^2}{1-x_j} \ge 1
inequalities
\sum_{j=1}^{n} r_j+ \sum_{k=1}^{n} c_k\ne 0, in a grid nxn, sum and product

Source: CRMO 2014 region 2 p6

9/29/2018
Suppose nn is odd and each square of an n×nn \times n grid is arbitrarily filled with either by 11 or by −1-1. Let rjr_j and ckc_k denote the product of all numbers in jj-th row and kk-th column respectively, 1≤j,k≤n1 \le j, k \le n. Prove that
∑j=1nrj+∑k=1nck≠0\sum_{j=1}^{n} r_j+ \sum_{k=1}^{n} c_k\ne 0
combinatoricsnumber theorygrid
Indian RMO P6

Source:

1/1/2015
For any natural number, let S(n)S(n) denote sum of digits of nn. Find the number of 33 digit numbers for which S(S(n))=2S(S(n)) = 2.
inequalitiesmodular arithmeticnumber theory unsolvednumber theory
placing numbers 1,2,3...., 18 so that 3 / sum on three concurrent segments

Source: CRMO 2014 region 4 p6

9/29/2018
In the adjacent fi gure, can the numbers 1,2,3,4,...,181,2,3, 4,..., 18 be placed, one on each line segment, such that the sum of the numbers on the three line segments meeting at each point is divisible by 33?
combinatoricsnumber theoryDivisibility