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Algebra

Blazing goIITian

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3 Jun 2009 15:21:04 IST

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2338

best questions in maths.

None

1)In a tournament of n participants, each pair plays one game (no ties). Prove that exactly one of the following situations occurs:

(i) The league can be partitioned into two nonempty groups such that each player in one of these groups has won against each player of the other.
(ii) All participants can be ranked 1 through $n$ so that $i-$ th player wins the game against the $(i + 1)$ st and the $n-$ th player wins against the first.

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apollo

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3 Jun 2009 15:24:37 IST

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2) An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o’clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. prove, that there is no time when the three hands are equally spaced around the clock, with each separating angle $\frac{2 \cdot \pi}{3}.$

apollo

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3 Jun 2009 15:25:01 IST

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3)Let $A$ be an $n \times n$ matrix whose elements are non-negative real numbers. Assume that $A$ is a non-singular matrix and all elements of $A^{-1}$ are non-negative real numbers. Prove that every row and every column of $A$ has exactly one non-zero element.

apollo

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3 Jun 2009 15:26:07 IST

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4)Let $f(x) = a \sin^2x + b \sin x + c,$ where $a, b,$ and $c$ are real numbers. Find all values of $a, b$ and $c$ such that the following three conditions are satisfied simultaneously:

(i) $f(x) = 381$ if $\sin x = \frac{1}{2}.$
(ii) The absolute maximum of $f(x)$ is $444.$
(iii) The absolute minimum of $f(x)$ is $364.$

apollo

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4 Jun 2009 20:43:24 IST

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5)In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?

apollo

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4 Jun 2009 20:44:13 IST

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6)Determine all positive roots of the equation $x^x = \frac{1}{\sqrt{2}}.$

apollo

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4 Jun 2009 20:45:04 IST

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7)If $x,y,z$ are real numbers satisfying relations
$x+y+z = 1 \quad \textrm{and} \quad \arctan x + \arctan y + \arctan z = \frac{\pi}{4},$
prove that $x^{2n+1} + y^{2n+1} + z^{2n+1} = 1$ holds for all positive integers $n$ .

apollo

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4 Jun 2009 20:45:57 IST

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8)Prove that for an arbitrary pair of vectors $f$ and $g$ in the space the inequality
$af^2 + bfg +cg^2 \geq 0$
holds if and only if the following conditions are fulfilled:
$a \geq 0, \quad c \geq 0, \quad 4ac \geq b^2.$

Sushant Mehta

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4 Jun 2009 20:56:41 IST

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Look, 381=a/4 +b/2 +c.Therefore,a+2b+4c=1524.Now, Sin is maximum at 90 degree i.e the maximum value is sin(90)=1.So,a+b+c=444.............eq 1And,Sin is minimum at -90 degree i.e. the minimum is Sin(-90)= -Sin(90)= -1.Hence, a+(-b)+c=364............eq2.or,a+c=b+364...............eq3.Now adding both equations,2(a+b)=804So, a+b=402....eq4Substituting equation 4 in equation 3,b=38.Solving further,c=323,a=77

Soumik

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5 Jun 2009 13:52:58 IST

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Rahul, I cud only read the 1st qn....the others are having technical errors....

1st one is a sitter...

See there can be only 2 conditions :-

1) All have lost to some other player, at least once

2) 1 of the guys have not suffered defeat.....

If 1st condition holds true then we can always frame such a sequence.....However if 2nd is true then...

Let (a_i,a_j) denote the winner between a_i & a_j.....

Let $a_{k$

So 1 of my groups $A\equiv \boxed{a_k,a_{k$

$N\equiv \boxed{a_x}\\ \ 1\le x\le n\\ \ x\ne k\\ \ x\ne k$

These 2 groups satisfy the 2nd condition....

Soumik

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5 Jun 2009 14:00:56 IST

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I don't think roots (Except $x=\frac{1}{2}$ ) can be exactly determined for $x^x= rac{1}{sqrt{2}}$

However interval has to be (0,1).....Obvious from graph itself (Though drawing the graph was not easy either)

Seems that's the only real root of the eqn.

Rahul Duggal

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6 Jun 2009 13:53:36 IST

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6. not too sure

another way of checking the domain is x^x is trivially greater than 1 for x>1 whereas

so an observation here tells us that x is an integral power of

checking for we find they are solutions.

now we can note

but x^xis strictly increasing for x= 1/2ⁿ where n z⁺ and n>2

®µD®A

Blazing goIITian

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6 Jun 2009 14:12:43 IST

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5) is the answer 36 and 6 ?

Rahul Duggal

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6 Jun 2009 14:33:19 IST

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the LHS of the inequality can be intepreted as a quadratic eqn in f (a0)

as the inequality is 0, we want the curve above x axis or atmost 1 intersection with x axis so a>0

to justify the slackness in the inequality we have . assuming f,g are not null vectors for if they are the inequality is trivially true

4ac > b². now as b²>0 and a>0 it is justified to say c>0

i've proved a>0. the equality can be confirmed if f,g are null vectors

apollo

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7 Jun 2009 00:13:33 IST

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rahul and soumik , that was good work.

RUDRA, can u kindly show your working please ? ( u r right RUDRA). but i want the working

apollo

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7 Jun 2009 00:24:06 IST

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The trigonometric relation is equivalent to

$\frac{x+y+z-xyz}{1-xy-xz-yz}=1$
or $xyz-xy-xz-yz-x-y-z+1=0$ . But since $x+y+z=1$ ,
$0=xyz-xy-xz-yx+x+y+z-1=(x-1)(y-1)(z-1)$ .
Hence one of $x, y, z$ is $1$ .

WLOG suppose $x=1$ . Now $y=-z$

and

hence $x^{2n+1}+y^{2n+1}+z^{2n+1}=1+y^{2n+1}-y^{2n+1}=1$

apollo

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7 Jun 2009 00:27:24 IST

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Taking $\ln$ both sides, we have that

$\additional \displaystyle x\ln x=\frac{1}{2}\cdot\ln \frac{1}{2}} \\\Rightarrow \frac{x}{\frac{1}{2}}=\frac{\ln\frac{1}{2}}{\...$

but since $a\geq b \Rightarrow \ln a\geq \ln b$ , we must have one of $\displaystyle \max \left\{\frac{x}{\frac{1}{2}},\frac{\ln\frac{1}{2}}{\ln x}\right\}\geq 1 \geq \min \left\{\frac{x}{\frac{1}...$ , so we must have $x=\frac{1}{2}$

am i rite? or is there any fallacy in my method ?

apollo

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7 Jun 2009 00:31:18 IST

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OK CONTINUING WITH THE QUESTIONS ,

9)The professor tells Roger the product of two positive integers and Kevin their sum. At first, nobody of them knows the number of the other.

One of them says: You can't possibly guess my number.
Then the other says: You are wrong, the number is 136.

Which number did the professor tell them respectively? Give reasons for your claim.

apollo

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7 Jun 2009 00:33:03 IST

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10)

Two people, $A$ and $B$ , play the following game: $A$ start choosing a positive integrer number and then, each player in it's turn, say a number due to the following rule:

If the last number said was odd, the player add $7$ to this number;
If the last number said was even, the player divide it by $2$ .

The winner is the player that repeats the first number said. Find all numbers that $A$ can choose in order to win. Justify your answer.

apollo

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7 Jun 2009 00:34:13 IST

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11)

Consider the set $S$ of $100$ numbers: $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$ .
Any two numbers, $a$ and $b$ , are eliminated in $S$ , and the number $a+b+ab$ is added. Now, there are $99$ numbers on $S$ .
After doing this operation $99$ times, there's only $1$ number on $S$ . What values can this number take?

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