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

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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…

Some Number Theory Questions for RMO and INMO

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1) Let n \geq 2 and k be any positive integers. Prove that (n-1)^{2}\mid (n^{k}-1) if and only if (n-1) \mid k.

2) Prove that there are no positive integers a, b, n >1 such that (a^{n}-b^{n}) \mid (a^{n}+b^{n}).

3) If a and b>2 are any positive integers, prove that 2^{a}+1 is not divisible by 2^{b}-1.

4) The integers 1,3,6,10, \ldots, n(n+1)/2, …are called the triangular numbers because they are the numbers of dots needed to make successive triangular arrays of dots. For example, the number 10 can be perceived as the number of acrobats in a human triangle, 4 in a row at the bottom, 3 at the next level, then 2, then 1 at the top. The square numbers are 1, 4, 9, \ldots, n^{2}, \ldots The pentagonal numbers 1, 5, 12, 22, \ldots, (3n^{2}-n)/2, \ldots, can be seen in a geometric array in the following way: Start with n equally spaced dots P_{1}, P_{2}, \ldots, P_{n} on a straight line in a plane, with distance 1 between consecutive dots. Using P_{1}P_{2} as a base side, draw a regular pentagon in the plane. Similarly, draw n-2 additional regular pentagons on base sides P_{1}P_{3}, P_{1}P_{4}, \ldots, P_{1}P_{n}, all pentagons lying on the same side of the line P_{1}P_{n}. Mark dots at each vertex and at unit intervals along the sides of these pentagons. Prove that the total number of dots in the array is (3n^{2}-n)/2. In general, if regular k-gons are constructed on the sides P_{1}P_{2}, P_{1}P_{3}, …, P_{1}P_{n}, with dots marked again at unit intervals, prove that the total number of dots is 1+kn(n-1)/2 -(n-1)^{2}. This is the nth k-gonal number.

5) Prove that if m>n, then a^{2^{n}}+1 is a divisor of a^{2^{m}}-1. Show that if a, m, n are positive with m \neq n, then

( a^{2^{m}}+1, a^{2^{n}}+1) = 1, if a is even; and is 2, if a is odd.

6) Show that if (a,b)=1 then (a+b, a^{2}-ab+b^{2})=1 or 3.

7) Show that if (a,b)=1 and p is an odd prime, then ( a+b, \frac{a^{p}+b^{p}}{a+b})=p or 1.

8) Suppose that 2^{n}+1=xy, where x and y are integers greater than 1 and n>0. Show that 2^{a}\mid (x-1) if and only if 2^{a}\mid (y-1).

9) Prove that (n!+1, (n+1)!+1)=1.

10) Let a and b be positive integers such that (1+ab) \mid (a^{2}+b^{2}). Show that the integer (a^{2}+b^{2})/(1+ab) must be a perfect square.

Note that in the above questions, in general, (a,b) means the gcd of a and b.

More later,
Nalin Pithwa.

Pre-RMO training; a statement and its converse; logic and plane geometry

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I hope the following explanation is illuminating to my readers/students:

How to prove that two lines are parallel ? (Note that we talk of parallel lines only when they lie in the same plane; on the other hand: consider the following scenario — your study table and the floor on which it stands. Let us say you draw a straight line AB on your study table and another line PQ on the floor on which the study table is standing; then, even though lines AB and PQ never meet, we do not say that they are parallel because they lie in different planes. Such lines are called skew lines. They are dealt with in solid geometry or 3D geometry or vector spaces).

Coming back to the question — when can we say that two lines are parallel?

Answer:

Suppose that a transversal crosses two other lines.

1) If the corresponding angles are equal, then the lines are parallel.
2) If the alternate angles are equal, then the lines are parallel.
3) If the co-interior angles are supplementary, then the lines are parallel.

A STATEMENT AND ITS CONVERSE

Let us first consider the following statements:

A transversal is a line that crosses two other lines. If the lines crossed by a transversal are parallel, then the corresponding angles are equal; if the lines crossed by a transversal are parallel, then the alternate interior angles are equal; if the lines crossed by a transversal are parallel, then the co-interior angles are supplementary.

The statements given below are the converses of the statement given in the above paragraph; meaning that they are formed from the former statements by reversing the logic. For example:

STATEMENT: If the lines are parallel then the corresponding angles are equal.

CONVERSE: If the corresponding angles are equal, then the lines are parallel.

Pairs such as these, a statement and its converse, occur routinely through out mathematics, and are particularly prominent in geometry. In this case, both the statement and its converse are true. It is important to realize that a statement and its converse are, in general, quite different. NEVER ASSUME THAT BECAUSE A STATEMENT IS TRUE, SO ITS CONVERSE IS ALSO TRUE. For example, consider the following:

STATEMENT: If a number is a multiple of 4, then it is even.
CONVERSE: If a number is even, then it is a multiple of 4.

The first statement is clearly true. But, let us consider the number 18. It is even. But 18 is not a multiple of 4. So, the converse is not true always.

\it Here \hspace{0.1in} is \hspace{0.1in}an \hspace {0.1in}example \hspace{0.1in}from \hspace{0.1in}surfing

STATEMENT: If you catch a wave, then you will be happy.
CONVERSE: If you are happy, then you will catch a wave.

Many people would agree with the first statement, but everyone knows that its converse is plain silly — you need skill to catch waves.

Thus, the truth of a statement has little to do with its converse. Separate justifications (proofs) are required for the converse and its statements.

Regards,
Nalin Pithwa.

Reference: (I found the above beautiful, simple, lucid explanation in the following text): ICE-EM, year 7, book 1; The University of Melbourne, Australian Curriculum, Garth Gaudry et al.

In bubbles, she sees a mathematical universe: Abel Laureate, Prof Karen Uhlenbeck

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I was just skimming the biography “A Beautiful Mind” by Sylvia Nasar, about the life of mathematical genius, John Nash, Economics Nobel Laureate (and later Abel Laureate)…

Some math wisdom came to my mind: Good mathematicians look for analogies between theorems but the very best of them look for analogies within analogies; I was reading the following from the biography of John Nash: …It was the great HUngarian-born polymath John von Neumann who first recognized that social behaviour could be analyzed as games. Von Neumann’s 1928 article on parlour games was the first successful attempt to derive logical and mathematical rules about rivalries. Just as Blake saw the universe in a grain of sand, great scientists have often looked for clues to vast and complex problems in the small, familiar phenomena of daily life. Isaac Newton reached insights about the heavens by juggling wooden balls. Einstein contemplated a boat paddling upriver. Von Neumann pondered the game of poker.

\vdots

And, read how first woman Abel Laureate, genius mathematician Prof Karen Uhlenbeck sees a mathematical universe in bubbles…

Hats off to Prof Karen Uhlenbeck and NY Times author, Siobhan Roberts!!