| Problem 1 In a triangle let be a point on the segment such that Suppose that the points and the centroids of triangles and lie on a circle. Prove that  Solution | |
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| Problem 2 Let be a natural number. Prove that
![Rendered by QuickLaTeX.com \[\left[\frac{n}{1}\right]+\left[\frac{n}{2}\right]+\left[\frac{n}{3}\right]+\cdots\left[\frac{n}{n}\right]+[\sqrt{n}]\]](https://www.cheenta.com/wp-content/ql-cache/quicklatex.com-3f0d705a36a5d76d21670822e7193158_l3.png) is even. (Here denotes the largest integer smaller than or equal to . | |
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| Problem 3 Let be natural numbers with . Suppose that the sum of their greatest common divisor and least common multiple is divisible by . Prove that the quotient is at most . When is this quotient exactly equal to  | |
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| Problem 4 Written on a blackboard is the polynomial . Calvin and Hobbes take turns alternatively (starting with Calvin) in the following game. During his turn, Calvin should either increase or decrease the coefficient of by 1 . And during his turn, Hobbes should either increase or decrease the constant coefficient by 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 strategy. | |
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| Problem 5 In an acute-angled triangle a point lies on the segment Let denote the circumcentres of triangles and respectively. Prove that the line joining the circumcentre of triangle and the orthocentre of triangle is parallel to . | |
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| Problem 6 Let be a natural number and For subsets and of we define to be the set of all those elements of which belong to exactly one of and . Let be a collection of subsets of such that for any two distinct elements and in the set has at least two elements. Show that has at most elements. Find all such collections with elements. |
Can one question be the cutoff for INMO 2014.