Indian National Math Olympiad 2014 (INMO 2014)


1In a triangle ABC, let D be the point on the segment BC such that AB+BD=AC+CD. Suppose that the points B, C and the centroids of triangles ABD and ACD lie on a circle. Prove that AB=AC.
2Let n be a natural number. Prove that,
leftlfloor frac{n}{1} rightrfloor+ leftlfloor frac{n}{2} rightrfloor + cdots + leftlfloor frac{n}{n} rightrfl...
is even.
3Let a,b be natural numbers with ab>2. Suppose that the sum of their greatest common divisor and least common multiple is divisble by a+b. Prove that the quotient is at most frac{a+b}{4}. When is this quotient exactly equal to frac{a+b}{4}
4Written on a blackboard is the polynomial x^2+x+2014. Calvin and hobbes take turns alternatively(starting with Calvin) in the following game. During his turns alternatively(starting with Calvin) in the following game. During his turn, Calvin should either increase or decrese the coeffecient of x by 1. And during this turn, Hobbes should either increase or decrease the constant coefficient by 1. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.
5In a acute-angled triangle ABC, a point D lies on the segment BC. Let O_1,O_2 denote the circumcentres of triangles ABD and ACD respectively. Prove that the line joining the circumcentre of triangle ABC and the orthocentre of triangle O_1O_2D is parallel to BC.
6Let n>1 be a natural number. Let U={1,2,...,n}, and define ADelta B to be the set of all those elements of U which belong to exactly one of A and B. Show that |mathcal{F}|le 2^{n-1}, where mathcal{F} is a collection of subsets of X such that for any two distinct elements of A,B of mathcal{F} we have |ADelta B|ge 2. Also find all such collections mathcal{F} for which the maximum is attained.

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