Find the range of the function

At a maths contest books are given as prizes to students (each students gets one book). In how many ways can the organisers give these prizes if they have copies of one book and other books each in one copy?

Consider . Show that there exists such that

Laurențiu Panaitopol

Determine the lengths of the edges of a right tetrahedron of volume so that the sum of its edges' lengths is minumum.

Consider a tetrahedron with equilateral. Let be the area of the triangle of sides , and . Show that where is the circumradius and is the volume of the tetrahedron.

Stere Ianuș

Let where .
1) Determine such that for every we have .
2) Show that if is irrational then for every there exists such that
Generalize the problem!

In the plane of traingle we consider a variable point which moves on line towards . Halfway there, it stops and starts moving in a straight line line towards . Halfway there, it stops and starts moving in a straight line towards , and halfway there it stops and starts moving in a straight line towards , and so on. Show that will get as close as we want to the vertices of a fixed triangle with area .

Let be a positive integer not divisible by 3. Prove that there are infinitely many positive integers such that and are divisible by , where is the sum of digits of .

Dorel Miheț

Let triangle be circumscribed about circle , and let be the orthocenter of . The circle touches line at . The tangent to the circle at meets at . Let be the midpoint of , and let the line meet again at . The tangent to parallel to meets the line at . Prove that is tangent to .

JEE preps are adversely impacting my thinking ability.
So I will try and do something that I didn't even do when I was actually preparing for the olympiads
Actually think !
It's trivial I know I know...
But it was fun !

Problem (ISL 2015 C1) :
In Lineland there are towns, arranged along a road running from left to right. Each town has a left bulldozer (put to the left of the town and facing left) and a right bulldozer (put to the right of the town and facing right). The sizes of the bulldozers are distinct. Every time when a left and right bulldozer confront each other, the larger bulldozer pushes the smaller one off the road. On the other hand, bulldozers are quite unprotected at their rears; so, if a bulldozer reaches the rear-end of another one, the first one pushes the second one off the road, regardless of their sizes.

Let and be two towns, with to the right of . We say that town can sweep town away if the right bulldozer of can move over to pushing off all bulldozers it meets. Similarly town can sweep town away if the left bulldozer of can move over to pushing off all bulldozers of all towns on its way.

Prove that there is exactly one town that cannot be swept away by any other one.

Solution :
Let the statement be true for .

Let the towns be labelled from left to right and their left and right bulldozer respectively.

Now we have to prove the statement for towns..
Consider the rightmost town and let some collide with

Then there are two cases :
derails all such . Then obviously is the new winner town !

If some derails . Then obviously since there is no other bulldozer between this point and ,
sweeps
Since there are no bulldozers between and , The first towns live unaffected by the remaining towns. And hence by inducton we are done.

For a given positive integer find, in terms of , the minimum value of for which there is a set of distinct positive integers that has sum greater than but every subset of size has sum at most