Thanks to Simon Foundation : a youtube video.
Reference: Nordic Mathematical Contest 1987, R. Todev:
Let f be a function, defined for natural numbers, that is strictly increasing, such that values of the function are also natural numbers and which satisfies the conditions and for all natural numbers m and n. Define the smallest possible value of a.
Since, is a function satisfying the conditions of the problem, the smallest possible a is at most 4. Assume that . It is easy to prove by induction that for all . So, taking into account that f is strictly increasing, we get
as well as .
So, we arrive at . But, this is not possible, since is an integer. So, .
Let us get started on plane geometry, some how or the other, now. Rather than starting with a lecture, let me start in a un-conventional format. A little challenging problem for you. Try to prove the following (Morley’s theorem):
In any arbitrary triangle, draw the trisectors of all the angles. Prove that the triangle formed by the points of intersection of adjacent trisectors (of different angles), taken in pair, is always equilateral.
Important Note: Please do not take look at any solution from the internet or any other source. You can use the basic theorems only presented in the Geometry toolkit, like as in, Problem Primer for the Olympiad, B J Venkatachala and/or in the lecture.
Prove that the equation
has an infinity of solutions in Z, the set of integers.
Solution by Andrei Stefanescu
For any we will consider the numbers , and .
Note that the triplet is a solution of the equation, as
Since we can define this triplet for all k’s, there will be an infinity of solutions.
Several Olympiad problems deal with functions defined on certain sets of points. These problems are interesting in that they combine both geometrical and algebraic ideas.
Let be an integer and a function defined on the set of points in the plane, with the property that for any regular n-gon
Prove that f is the zero function.
In Euclidean geometry, the only motions permissible are rigid motions — translations, rotations, and reflections.
Let A be an arbitrary point. Consider a regular n-gon . Let k be an integer, . A rotation with center A of angle sends the polygon to , where and is the image of for all
From the condition of the statement, we have
Observe that in the sum the number appears n times, therefore,
On the other hand, we have
since the polygons are all regular n-gons. From the two equalities above we deduce , hence, f is the zero function.
Let a and b be distinct positive integers such that is divisible by . Prove that
Let and write and with . Then,
is an integer. N^ote that . Similarly, .
Because , we have
Now, we apply the following lemma: Let a and b be two coprime numbers. If c is an integer such that , then .
Hence, we get, implying that . Therefore,
It follows that . QED.
Note that the key step divides g can also be obtained by clever algebraic manipulations such as .
More Olympiad problems later,
A student has 6 weeks (that is, 42 days) to prepare for her examination and she has decided that during this period she will put in a total of 70 hours towards her preparation for the examination. She decides to study in full hours every day, studying at least one hour each day. We have to prove that no matter how she schedules her studying pattern, she will study for exactly 13 hours during some consecutive days.
If denotes the number of hours she studies on the ith day, then we are given that each is a positive integer and the sum . To get the required succession of days, we must find some m and j such that and . Trying out all the possibilities, is close to impossible as well as dumb. Let denote the partial sum which is the number of hours she studies upto the ith day. Our given data then translates into
and we have to find some such that , that is, (clearly, then . Hence, besides the 42 numbers in
we also look at 42 more numbers in which are also 42 different numbers and the largest among them is . Hence, the are actually among the positive integers from 1 to 83 and hence, by the pigeonhole principle, we see that two numbers in must be equal. As we already saw the numbers in B are all distinct and so are the numbers in . Hence, we must have for some i and j, giving the required succession of days when she studied exactly for 13 hours a day.
More on pigeonhole principle,