MathDB

3

Part of 2013 India Regional Mathematical Olympiad

Problems(5)

Indian RMO- Paper -3

Source: RMO Problem 3

12/11/2013
In an acute-angled triangle ABCABC with AB<ACAB < AC, the circle ω\omega touches ABAB at BB and passes through CC intersecting ACAC again at DD. Prove that the orthocentre of triangle ABDABD lies on ω\omega if and only if it lies on the perpendicular bisector of BCBC.
geometryperpendicular bisectorgeometry unsolved
Almost a Classic Divisibility Problem

Source: Indian RMO 2013 Paper 1 Problem 3

2/1/2014
Find all primes pp and qq such that pp divides q2−4q^2-4 and qq divides p2−1p^2-1.
number theoryeasynice
Indian RMO - Paper 2

Source: Problem 3

12/11/2013
Consider the expression 20132+20142+20152+⋯+n22013^2+2014^2+2015^2+ \cdots+n^2 Prove that there exists a natural number n>2013n > 2013 for which one can change a suitable number of plus signs to minus signs in the above expression to make the resulting expression equal 99999999
algebra unsolvedalgebra
Indian RMO - Paper -4[3]

Source:

12/12/2013
Given real numbers a,b,c,d,e>1a,b,c,d,e>1. Prove that a2c−1+b2d−1+c2e−1+d2a−1+e2b−1≥20 \frac{a^2}{c-1}+\frac{b^2}{d-1}+\frac{c^2}{e-1}+\frac{d^2}{a-1}+\frac{e^2}{b-1} \ge 20
cyclic inequality
3-good Subsets

Source: Indian RMO 2013 Mumbai Region Problem 3

2/1/2014
A finite non-empty set of integers is called 33-good if the sum of its elements is divisible by 33. Find the number of 33-good subsets of {0,1,2,…,9}\{0,1,2,\ldots,9\}.
trigonometrycombinatorics unsolvedcombinatorics