Check your mathematical induction concepts

Discuss the following “proof” of the (false) theorem:

If n is any positive integer and S is a set containing exactly n real numbers, then all the numbers in S are equal:

PROOF BY INDUCTION:

Step 1:

If n=1, the result is evident.

Step 2: By the induction hypothesis the result is true when n=k; we must prove that it is correct when n=k+1. Let S be any set containing exactly k+1 real numbers and denote these real numbers by a_{1}, a_{2}, a_{3}, \ldots, a_{k}, a_{k+1}. If we omit a_{k+1} from this list, we obtain exactly k numbers a_{1}, a_{2}, \ldots, a_{k}; by induction hypothesis these numbers are all equal:

a_{1}=a_{2}= \ldots = a_{k}.

If we omit a_{1} from the list of numbers in S, we again obtain exactly k numbers a_{2}, \ldots, a_{k}, a_{k+1}; by the induction hypothesis these numbers are all equal:

a_{2}=a_{3}=\ldots = a_{k}=a_{k+1}.

It follows easily that all k+1 numbers in S are equal.

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Comments, observations are welcome 🙂

Regards,

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

Some random problems in algebra (part b) for RMO and INMO training

1) Solve in real numbers the system of equations:

y^{2}+u^{2}+v^{2}+w^{2}=4x-1

x^{2}+u^{2}+v^{2}+w^{2}=4y-1

x^{2}+y^{2}+v^{2}+w^{2}=4u-1

x^{2}+y^{2}+u^{2}+w^{2}=4v-1

x^{2}+y^{2}+u^{2}+v^{2}=4w-1

Hints: do you see some quadratics ? Can we reduce the number of variables? …Try such thinking on your own…

2) Let a_{1}, a_{2}, a_{3}, a_{4}, a_{5} be real numbers such that a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=0 and \max_{1 \leq i <j \leq 5} |a_{i}-a_{j}| \leq 1. Prove that a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2} \leq 10.

3) Let a, b, c be positive real numbers. Prove that

\frac{1}{2a} + \frac{1}{2b} + \frac{1}{2d} \geq \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a}

More later

Nalin Pithwa.

Some random assorted (part A) problems in algebra for RMO and INMO training

You might want to take a serious shot at each of these. In the first stage of attack, apportion 15 minutes of time for each problem. Do whatever you can, but write down your steps in minute detail. In the last 5 minutes, check why the method or approach does not work. You can even ask — or observe, for example, that if surds are there in an equation, the equation becomes inherently tough. So, as a child we are tempted to think — how to get rid of the surds ?…and so on, thinking in math requires patience and introversion…

So, here are the exercises for your math gym today:

1) Prove that if x, y, z are non-zero real numbers with x+y+z=0, then

\frac{x^{2}+y^{2}}{x+y} + \frac{y^{2}+z^{2}}{y+z} + \frac{z^{2}+x^{2}}{x+z} = \frac{x^{3}}{yz} + \frac{y^{3}}{zx} + \frac{z^{3}}{xy}

2) Let a b, c, d be complex numbers with a+b+c+d=0. Prove that

a^{3}+b^{3}+c^{3}+d^{3}=3(abc+bcd+adb+acd)

3) Let a, b, c, d be integers. Prove that a+b+c+d divides

2(a^{4}+b^{4}+c^{4}+d^{4})-(a^{2}+b^{2}+c^{2}+d^{2})^{2}+8abcd

4) Solve in complex numbers the equation:

(x+1)(x+2)(x+3)^{2}(x+4)(x+5)=360

5) Solve in real numbers the equation:

\sqrt{x} + \sqrt{y} + 2\sqrt{z-2} + \sqrt{u} + \sqrt{v} = x+y+z+u+v

6) Find the real solutions to the equation:

(x+y)^{2}=(x+1)(y-1)

7) Solve the equation:

\sqrt{x + \sqrt{4x + \sqrt{16x + \sqrt{\ldots + \sqrt{4^{n}x+3}}}}} - \sqrt{x}=1

8) Prove that if x, y, z are real numbers such that x^{3}+y^{3}+z^{3} \neq 0, then the ratio \frac{2xyz - (x+y+z)}{x^{3}+y^{3}+z^{3}} equals 2/3 if and only if x+y+z=0.

9) Solve in real numbers the equation:

\sqrt{x_{1}-1} = 2\sqrt{x_{2}-4}+ \ldots + n\sqrt{x_{n}-n^{2}}=\frac{1}{2}(x_{1}+x_{2}+ \ldots + x_{n})

10) Find the real solutions to the system of equations:

\frac{1}{x} + \frac{1}{y} = 9

(\frac{1}{\sqrt[3]{x}} + \frac{1}{\sqrt[3]{y}})(1+\frac{1}{\sqrt[3]{x}})(1+\frac{1}{\sqrt[3]{y}})=18

More later,
Nalin Pithwa

PS: if you want hints, do let me know…but you need to let me know your approach/idea first…else it is spoon-feeding…