Cyclic expressions, fractions: Pre RMO, PRMO, IITJEE foundation 2019

In order to solve the following tutorial sheet, it helps to solve/understand and then apply the following beautiful cyclic relations or identities:

(Note if these look new to you, then you need to check the truth of all them; if all are v v familiar to you, just go ahead and crack the tutorial sheet below):

Core Identities in Cyclic Expressions:
1) (b-c)+(c-a)+(a-b)=0
2) a(b-c)+b(c-a)+c(a-b)=0
3) a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)=-(a-b)(b-c)(c-a)
4) bc(b-c)+ca(c-a)+ab(a-b)=-(a-b)(b-c)(c-a)
5) a(b^{2}-c^{2})+b(c^{2}-a^{2})+c(a^{2}-b^{2})=(a-b)(b-c)(c-a)

Solve or simplify the following:

1) \frac{a}{(a-b)(a-c)} + \frac{b}{(b-c)(b-a)} + \frac{c}{(c-a)(c-b)}
2) \frac{bc}{(a-b)(a-c)} + \frac{ca}{(b-c)(b-a)} + \frac{ab}{(c-a)(c-b)}
3) \frac{a^{2}}{(a-b)(a-c)} + \frac{b^{2}}{(b-c)(b-a)} + \frac{c^{2}}{(c-a)(c-b)}
4) \frac{a^{3}}{(a-b)(a-c)} + \frac{b^{3}}{(b-c)(b-a)} + \frac{c^{3}}{(c-a)(c-b)}
5) \frac{a(b+c)}{(a-b)(c-a)} + \frac{b(a+c)}{(a-b)(b-c)} + \frac{a(a+b)}{(c-a)(b-c)}
6) \frac{1}{a(a-b)(a-c)} + \frac{1}{b(b-c)(b-a)} + \frac{1}{c(c-a)(c-b)}
7) \frac{bc}{a(a^{2}-b^{2})(a^{2}-c^{2})} + \frac{ca}{b(b^{2}-c^{2})(b^{2}-a^{2})} + \frac{ab}{c(c^{2}-a^{2})(c^{2}-b^{2})}
8) \frac{(x-b)(x-c)}{(a-b)(a-c)} + \frac{(x-c)(x-a)}{(b-c)(b-a)} + \frac{(x-a)(x-b)}{(c-a)(c-b)}
9) \frac{bc(a+d)}{(a-b)(a-c)} + \frac{ca(b+d)}{(b-c)(b-a)} + \frac{ab(c+d)}{(c-a)(c-b)}
10) \frac{1}{(a-b)(a-c)(x-a)} + \frac{1}{(b-c)(b-a)(x-b)} + \frac{1}{(c-a)(c-b)(x-c)}
11) \frac{a^{2}}{(a-b)(a-c)(x+a)} + \frac{b^{2}}{(b-c)(b-a)(x+b)} + \frac{c^{2}}{(c-a)(c-b)(x+c)}
12) a^{2}\frac{(a+b)(a+c)}{(a-b)(a-c)} + b^{2}\frac{(b+c)(b+a)}{(b-c)(b-a)} + c^{2}\frac{(c+a)(c+b)}{(c-a)(c-b)}
13) \frac{a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b)}{(b-c)^{3}+(c-a)^{3}+(a-b)^{3}}
14) \frac{a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)+2(c-a)(a-b)(b-c)}{(b-c)^{3}+(c-a)^{3}+(a-b)^{3}}
15) \frac{a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b)}{a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)}
16) \frac{a^{2}(b-c)^{3}+b^{2}(c-a)^{3}+c^{2}(a-b)^{3}}{(a-b)(b-c)(c-a)}
17) \frac{\frac{b-c}{a} + \frac{c-a}{b} + \frac{a-b}{c}}{\frac{1}{a}(\frac{1}{b^{2}}-\frac{1}{c^{2}})+\frac{1}{b}(\frac{1}{c^{2}}-\frac{1}{a^{2}})+\frac{1}{c}(\frac{1}{a^{2}}-\frac{1}{b^{2}})}^
18) \frac{a^{2}(\frac{1}{a^{2}}-\frac{1}{b^{2}})+b^{2}(\frac{1}{a^{2}}-\frac{1}{c^{2}})+c^{2}(\frac{1}{b^{2}}-\frac{1}{a^{2}})}{\frac{1}{bc}(\frac{1}{c}-\frac{1}{b})+\frac{1}{ca}(\frac{1}{a}-\frac{1}{c})+\frac{1}{ab}(\frac{1}{b}-\frac{1}{c})}
19) \frac{a}{(a-b)(a-c)(x-a)} + \frac{b}{(b-c)(b-a)(x-b)} + \frac{c}{(c-a)(c-b)(x-c)}

More later,
Nalin Pithwa

Tutorial on Basic Set Theory and Functions: for PRMO, RMO and IITJEE Mains maths

I) Prove that every function can be represented as a sum of an even function and an odd function.

II)Let A, B, C be subsets of a set S. Prove the following statements and illustrate them with Venn Diagrams:

2a) The famous DeMorgan’s laws in their basic forms: A^{'} \bigcup B^{'} = (A \bigcap B)^{'} and A^{'} \bigcap B^{'} = (A \bigcup B)^{'}. Assume that both sets A and B are subsets of Set S. In words, the first is: union of complements is the complement of intersection; the second is: intersection of two complements is the complement of the union of the two sets.

Sample Solution:

Let us say that we need to prove: A^{'}\bigcap B^{'}=(A \bigcup B)^{'}.

Proof: It must be shown that the two sets have the same elements; in other words, that each element of the set on LHS is an element of the set on RHS and vice-versa.

If x \in A^{'} \bigcap B^{'}, then x \in A^{'} and x \in B^{'}. This means that x \in S, and x \notin A and x \notin B. Since x \notin A and x \notin B, hence x \notin A \bigcup B. Hence, x \in (A \bigcup B)^{'}.

Conversely, if x \in (A \bigcup B)^{'}, then x \in S  and x \notin A \bigcup B. Therefore, x \notin A and x \notin B. Thus, x \in A^{'} and x \in Y^{'}, so that x \in A^{'} \bigcap B^{'}. QED.

2b) A \bigcap (B \bigcup C) = (A \bigcap B)\bigcup (A \bigcap C).

2c) A \bigcup (B \bigcap C) = (A \bigcup B) \bigcap (A \bigcup C)

III) Prove that if I and S are sets and if for each i \in I, we have X_{i} \subset S, then (\bigcap_{i \in I} X_{i})^{'} = \bigcup_{i \in I}(X_{i})^{'}.

Sample Solution: 

It must be shown that each element of the set on the LHS is an element of the set on RHS, and vice-versa.

If x \in (\bigcap_{i \in I} X_{i})^{'}, then x \in S and x \notin \bigcap_{i \in I} X_{i}. Therefore, x \notin X_{i}, for at least one j \in I. Thus, x \in (X_{i})^{'}, so that x \in \bigcup_{i \in I}(X_{i})^{'}.

Conversely, if x \in \bigcup_{i \in I}(X_{i})^{i}, then for some j \in I, we have x \in (X_{i})^{'}. Thus, x \in S and x \notin X_{i}. Since x \notin X_{i}, we have x \notin \bigcap_{i \in I}X_{i}. Therefore, x \in \bigcap_{i \in I}(X_{i})^{'}. QED.

IV) If A, B and C are sets, show that :

4i) (A-B)\bigcap C = (A \bigcap C)-B

4ii) (A \bigcup B) - (A \bigcap B)=(A-B) \bigcup (B-A)

4iii) A-(B-C)=(A-B)\bigcup (A \bigcap B \bigcap C)

4iv) (A-B) \times C = (A \times C) - (B \times C)

V) Let I be a nonempty set and for each i \in I let X_{i} be a set. Prove that

5a) for any set B, we have : B \bigcap \bigcup_{i \in I} X_{i} = \bigcup_{i \in I}(B \bigcap X_{i})

5b) if each X_{i} is a subset of a given set S, then (\bigcup_{i \in I}X_{i})^{'}=\bigcap_{i \in I}(X_{i})^{'}

VI) Prove that if f: X \rightarrow Y, g: Y \rightarrow Z, and Z \rightarrow W are functions, then : h \circ (g \circ f) = (h \circ g) \circ f

VII) Let f: X \rightarrow Y be a function, let A and B be subsets of X, and let C and D be subsets of Y. Prove that:

7i) f(A \bigcup B) = f(A) \bigcup f(B); in words, image of union of two sets is the union of two images;

7ii) f(A \bigcap B) \subset f(A) \bigcap f(B); in words, image of intersection of two sets is a subset of the intersection of the two images;

7iii) f^{-1}(C \bigcup D) = f^{-1}(C) \bigcup f^{-1}(D); in words, the inverse image of the union of two sets is the union of the images of the two sets.

7iv) f^{-1}(C \bigcap D)=f^{-1}(C) \bigcap f^{-1}(D); in words, the inverse image of intersection of two sets is intersection of the two inverse images.

7v) f^{-1}(f(A)) \supset A; in words, the inverse of the image of a set contains the set itself.

7vi) f(f^{-1}(C)) \subset C; in words, the image of an inverse image of a set is a subset of that set.

For questions 8 and 9, we can assume that the function f is f: X \rightarrow Y and a set A lies in domain X and a set C lies in co-domain Y.

8) Prove that a function f is 1-1 if and only if f^{-1}(f(A))=A for all A \subset X; in words, a function sends different inputs to different outputs iff a set in its domain is the same as the inverse of the image of that set itself.

9) Prove that a function f is onto if and only if f(f^{-1}(C))=C for all C \subset Y; in words, the image of a domain is equal to whole co-domain (which is same as range) iff a set in its domain is the same as the image of the inverse image of that set.

Cheers,

Nalin Pithwa

Check your talent: are you ready for math or mathematical sciences or engineering

At the outset, let me put a little sweetener also: All I want to do is draw attention to the importance of symbolic manipulation. If you can solve this tutorial easily or with only a little bit of help, I would strongly feel that you can make a good career in math or applied math or mathematical sciences or engineering.

On the other hand, this tutorial can be useful as a “miscellaneous or logical type of problems” for the ensuing RMO 2019.

I) Let S be a set having an operation * which assigns an element a*b of S for any a,b \in S. Let us assume that the following two rules hold:

i) If a, b are any objects in S, then a*b=a

ii) If a, b are any objects in S, then a*b=b*a

Show that S can have at most one object.

II) Let S be the set of all integers. For a, b in S define * by a*b=a-b. Verify the following:

a) a*b \neq b*a unless a=b.

b) (a*b)*c \neq a*{b*c} in general. Under what conditions on a, b, c is a*(b*c)=(a*b)*c?

c) The integer 0 has the property that a*0=a for every a in S.

d) For a in S, a*a=0

III) Let S consist of two objects \square and \triangle. We define the operation * on S by subjecting \square and \triangle to the following condittions:

i) \square * \triangle=\triangle = \triangle * \square

ii) \square * \square = \square

iii) \triangle * \triangle = \square

Verify by explicit calculation that if a, b, c are any elements of S (that is, a, b and c can be any of \square or \triangle) then:

i) a*b \in S

ii) (a*b)*c = a*(b*c)

iii) a*b=b*a

iv) There is a particular a in S such that a*b=b*a=b for all b in S

v) Given b \in S, then b*b=a, where a is the particular element in (iv) above.

This will be your own self-appraisal !!

Regards,

Nalin Pithwa

More PreRMO or PRMO practice questions 2019

  1. Find the remainder when 37^{100} is divided by 29.
  2. Compute the last two digits of 9^{1500}.
  3. Prove for any integers a and b that a^{2}+b^{2} never leaves a remainder of 3 when divided by 4.
  4. Prove that the equation a^{2}+b^{3}=3c^{2} has no solutions in non zero integers a, b and c.
  5. Prove that the square of any odd integer always leaves a remainder of 1 when divided by 8.

More later,

Nalin Pithwa

 

Pre RMO practice 2019

1) Determine whether the following functions are well-defined:

1a) f: Q \rightarrow Z defined by f(a/b)=a

1b) f: Q \rightarrow Q defined by f(a/b)=a^{2}/b^{2}

2) Determine whether the function f: R^{+} \rightarrow Z defined by mapping a real number r to the first digit to the right of the decimal point in a decimal expansion of r is well-defined.

3) Apply the Euclidean algorithm to obtain GCD of (57970,10353) and express it as a linear combination of 57970 and 10353.

4) For each of the following pairs of integers a and b, determine their greatest common divisor, their least common multiple, and write their greatest common divisor in the form ax+by for some integers x and y.

(a) a=20, b=13

(b) a=69, b=372

(c) a=792, b=275

(d) a=11391, b=5673

(e) a=1761, b=1567

(f) a=507885, b=60808

5) Prove that if the integer k divides the integers a and b then k divides as+bt for every pair of integers s and t.

6) Prove that if n is composite then there are integers a and b such that a divides ab but n does not divide either a or b.

7) Let a, b and N be fixed integers with a and b non-zero and let d= (a,b) be the greatest common divisor of a and b. Suppose x_{0} and y_{0} are particular solutions to ax+by=N. Prove for any integer r that integers x=x_{0}+\frac{b}{d}t and y=y_{0}-\frac{a}{d}t are also solutions to ax+by=N (this is in fact the general solution).

8) Determine the value \phi(n) for each integer n \leq 30 where \phi denotes the Euler-\phi function.

9) Prove the Well-Ordering Property of integers by induction and prove the minimal element is unique.

10) If p is a prime prove that there do not exist non-zero integers a and b such that a^{2}=pb^{2} (that is, \sqrt{p} is not a rational number).

11) Let p be a prime and n \in Z^{+}. Find a formula for the largest power of p which divides n! (it involves the greatest integer function).

12) Prove for any given positive integer N there exist only finitely many integers n with \phi(n)=N where \phi denotes Euler’s \phi-function.

13) Prove that if d divides n then $latex  \phi(d)$ divides \phi(n) where \phi denotes Euler’s \phi-function.

More later,

Hope this gives you some math meal to churn for the Pre RMO or PRMO or even the ensuing RMO of Homi Bhabha Science Foundation.

Regards,

Nalin Pithwa