MathDB

5

Part of 2016 India Regional Mathematical Olympiad

Problems(7)

Sum of four squares

Source: RMO Delhi, 2016 P5

10/11/2016
a.) A 7-tuple (a1,a2,a3,a4,b1,b2,b3)(a_1,a_2,a_3,a_4,b_1,b_2,b_3) of pairwise distinct positive integers with no common factor is called a shy tuple if a12+a22+a32+a42=b12+b22+b32 a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2and for all 1≤i<j≤41 \le i<j \le 4 and 1≤k≤31 \le k \le 3, ai2+aj2≠bk2a_i^2+a_j^2 \not= b_k^2. Prove that there exists infinitely many shy tuples.
b.) Show that 20162016 can be written as a sum of squares of four distinct natural numbers.
number theorySum of Squares
Centroids Coincide

Source: RMO Mumbai 2016, P5

10/11/2016
Let ABCABC be a triangle with centroid GG. Let the circumcircle of triangle AGBAGB intersect the line BCBC in XX different from BB; and the circucircle of triangle AGCAGC intersect the line BCBC in YY different from CC. Prove that GG is the centroid of triangle AXYAXY.
geometrycircumcircle
Lower bound on a product

Source: RMO Maharashtra and Goa 2016, P5

10/11/2016
Let x,y,zx,y,z be non-negative real numbers such that xyz=1xyz=1. Prove that (x3+2y)(y3+2z)(z3+2x)≥27.(x^3+2y)(y^3+2z)(z^3+2x) \ge 27.
inequalitiesalgebra
Geometry

Source: RMO 2016 Karnataka Region P5

10/16/2016
Let ABCABC be a right-angled triangle with ∠B=90∘\angle B=90^{\circ}. Let II be the incentre if ABCABC. Extend AIAI and CICI; let them intersect BCBC in DD and ABAB in EE respectively. Draw a line perpendicular to AIAI at II to meet ACAC in JJ, draw a line perpendicular to CICI at II to meet ACAC at KK. Suppose DJ=EKDJ=EK. Prove that BA=BCBA=BC.
geometry
RMO 2016 ,Q5

Source:

10/25/2016
Let ABCABC be a triangle , ADAD an altitude and AEAE a median . Assume B,D,E,CB,D,E,C lie in that order on the line BCBC . Suppose the incentre of triangle ABEABE lies on ADAD and he incentre of triangle ADCADC lies on AEAE . Find ,with proof ,the angles of triangle ABCABC .
geometry
ABC right angle triangle: FK=BC

Source: RMO Hyderabad 2016 , P5

10/12/2016
Let ABCABC be a right angled triangle with ∠B=90∘\angle B=90^{\circ}. Let ADAD be the bisector of angle AA with DD on BCBC . Let the circumcircle of triangle ACDACD intersect ABAB again at EE; and let the circumcircle of triangle ABDABD intersect ACAC again at FF . Let KK be the reflection of EE in the line BCBC . Prove that FK=BCFK = BC.
geometrycircumcirclegeometric transformationreflection
2016 Chandigarh RMO find K,L on the BC,CD so (ABK)=(AKL)=(ADL), rect.ABCD

Source:

8/9/2019
Given a rectangle ABCDABCD, determine two points KK and LL on the sides BCBC and CDCD such that the triangles ABK,AKLABK, AKL and ADLADL have same area.
geometryrectangleareasarea of a triangle