MathDB

3

Part of 2014 India Regional Mathematical Olympiad

Problems(5)

Prove that BO perpendicular to AC if DE parallel to AC

Source: Mumbai Region RMO 2014 Problem 3

12/7/2014
Let ABCABC be an acute-angled triangle in which ∠ABC\angle ABC is the largest angle. Let OO be its circumcentre. The perpendicular bisectors of BCBC and ABAB meet ACAC at XX and YY respectively. The internal angle bisectors of ∠AXB\angle AXB and ∠BYC\angle BYC meet ABAB and BCBC at DD and EE respectively. Prove that BOBO is perpendicular to ACAC if DEDE is parallel to ACAC.
trigonometrygeometryangle bisector
Indian RMO 2014 P3 (WB)

Source:

12/7/2014
Suppose for some positive integers rr and ss, 2r2^r is obtained by permuting the digits of 2s2^s in decimal expansion. Prove that r=sr=s.
find positive integers (x,y) so that (2x + 7y) divides (7x + 2y)

Source: CRMO 2014 region 2 p3

9/29/2018
Find all pairs of (x,y)(x, y) of positive integers such that 2x+7y2x + 7y divides 7x+2y7x + 2y.
number theoryDivisibilitypositive integers
Indian RMO 2014 P3

Source:

1/1/2015
Prove that for any natural number n<2310n < 2310 , n(2310−n)n(2310-n) is not divisible by 23102310.
Indian RMO P3

Source:

1/1/2015
let m,nm,n be natural number with m>nm>n . find all such pairs of (m,n)(m,n) such that
gcd(n+1,m+1)=gcd(n+2,m+2)=..........=gcd(m,2m−n)=1gcd(n+1,m+1)=gcd(n+2,m+2) =..........=gcd(m, 2m-n) = 1
number theorygreatest common divisornumber theory unsolved