RMO (Homi Bhabha Science Foundation): some miscellaneous problems to practise

Problem 1:

Let f be an bijective function (one-one and onto) from the set $A=\{ 1,2,3, \ldots, n\}$ to n itself. Show that there is a positive integer $M>1$ such that $f^{M}(i)=i$ for each $i \in A$.

Note: $f^{M}$ denotes the composite function $f \circ f \circ f \circ \ldots \circ f$ repeated M times.

Problem 2:

Show that there exists a convex hexagon in the plane such that : (a) all its interior angles are equal (b) its sides are 1,2, 3, 4, 5, 6 in some order.

Problem 3:

There are ten objects with total weight 20, each of the weights being a positive integer. Given that none of the weights exceed 10, prove that the ten objects can be divided into two groups that balance each other when placed on the two pans of a balance.

Problem 4:

At each of the eight corners of a cube, write +1 or -1 arbitrarily. Then, on each of the six faces of the cube write the product of the numbers written at the four corners of that face. Add all the fourteen numbers so written down. Is it possible to arrange the numbers +1 and -1 at the corners initially so that the final sum is zero?

Problem 5:

Given the seven element set $A= \{a,b,c,d,e,f,g \}$, find a collection T of 3-element subsets of A such that each pair of elements from A occurs exactly in one of the subsets of T.

Solutions will be put up tomorrow. Meanwhile, make a whole-hearted attempt to crack these!

Cheers,

Nalin Pithwa

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