# 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

# Patterns in the primes: Clay Math, James Maynard, 2015

(shared from Clay Math website and shared for my readers. Many thanks to James Maynard and Clay Math :-))

Patterns in primes

# 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

# Tutorial on Euclidean algorithm for GCD: pre RMO or PRMO 2019

For each of the following pairs of integers a and n, show that a is relatively prime to n, or that a and n are coprime to each other:

1. a=13, n=20
2. a=69, n=89
3. a=1891, n=3797
4. a=6003722857, n=77695236973 (the Euclidean algorithm only requires 3 steps for these integers!).

Cheers,

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