Secondary and Higher Secondary Marathon

For students of class 11-12 (age 16+)
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Phlembac Adib Hasan
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Re: Secondary and Higher Secondary Marathon

Unread post by Phlembac Adib Hasan » Mon Nov 12, 2012 8:41 pm

Problem 4: (Posted before)
If $a,b\in \mathbb {N}_0$, show that $\sqrt {a}+\sqrt {b}$ is rational iff both of $a$ and $b$ are integers perfect squares.
Source: Self-made.
Last edited by Phlembac Adib Hasan on Tue Nov 13, 2012 7:22 pm, edited 2 times in total.
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Fahim Shahriar
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Re: Secondary and Higher Secondary Marathon

Unread post by Fahim Shahriar » Tue Nov 13, 2012 10:31 am

Phlembac Adib Hasan wrote: Problem 4: (Posted before)
If $a,b\in \mathbb {N}_0$, show that $\sqrt {a}+\sqrt {b}$ is rational iff both of $a$ and $b$ are integers.
Source: Self-made.

Is it true ?
Name: Fahim Shahriar Shakkhor
Notre Dame College

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Phlembac Adib Hasan
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Re: Secondary and Higher Secondary Marathon

Unread post by Phlembac Adib Hasan » Tue Nov 13, 2012 12:07 pm

Fahim Shahriar wrote:
Phlembac Adib Hasan wrote: Problem 4: (Posted before)
If $a,b\in \mathbb {N}_0$, show that $\sqrt {a}+\sqrt {b}$ is rational iff both of $a$ and $b$ are integers.
Source: Self-made.

Is it true ?
Sorry, for the typo. Edited now.
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SANZEED
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Re: Secondary and Higher Secondary Marathon

Unread post by SANZEED » Tue Nov 13, 2012 11:14 pm

First if $a,b$ are both perfect squares,then $\sqrt a+\sqrt b$ must be an integer,therefore,rational. So the "only if" case is true.
Now let us assume that $\sqrt a+\sqrt b=\frac{m}{n}$. Now squaring both sides, we get
$a+2\sqrt a\cdot \sqrt b+b=\frac{m^2}{n^2}$.
This implies $2\sqrt a\cdot \sqrt b=\frac{m^2-n^2(a+b)}{n^2}\Rightarrow 2n^2\sqrt a\cdot \sqrt b=m^2-n^2(a+b)$.
So we must have $\sqrt {ab}\in \mathbb N$. Thus we must have $\sqrt {ab}=\frac{x}{y}$. Thus $ab=\frac{x^2}{y^2}$.
Now, $\sqrt a+\sqrt b=\sqrt a+\frac{x}{y\sqrt a}=\frac{x+ay}{y\sqrt a}=\frac{m}{n}$. Thus $\sqrt a$ must be rational.
So for some $u,v\in \mathbb Z$ we have that $a=\frac{u^2}{v^2}$. But we have that $a\in \mathbb N_{0}$. So $a$ must be a square integer i.e. $v|u$ . Similar steps for $b$.
$\color{blue}{\textit{To}} \color{red}{\textit{ problems }} \color{blue}{\textit{I am encountering with-}} \color{green}{\textit{AVADA KEDAVRA!}}$

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Fahim Shahriar
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Re: Secondary and Higher Secondary Marathon

Unread post by Fahim Shahriar » Thu Nov 15, 2012 6:47 pm

Problem 5:
The sum of three numbers is $6$, the sum of their squares is $8$, and the sum of their cubes is $5$. What is the sum of their fourth powers?

Source: অলিম্পিয়াড সমগ্র বই
Name: Fahim Shahriar Shakkhor
Notre Dame College

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Phlembac Adib Hasan
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Re: Secondary and Higher Secondary Marathon

Unread post by Phlembac Adib Hasan » Thu Nov 15, 2012 9:01 pm

Fahim Shahriar wrote:Problem 5:
The sum of three numbers is $6$, the sum of their squares is $8$, and the sum of their cubes is $5$. What is the sum of their fourth powers?

Source: অলিম্পিয়াড সমগ্র বই
assume the numbers are $a,b,c$
So $ab+bc+ca=1/2[(a+b+c)^2-a^2-b^2-c^2]=14$
Identity:
Newton's Identities
$a^4+b^4+c^4=(a+b+c)abc-(ab+bc+ca)(a^2+b^2+c^2)+(a+b+c)(a^3+b^3+c^3)$
So $a^4+b^4+c^4=0$
Let me know whether I am right.
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Fahim Shahriar
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Re: Secondary and Higher Secondary Marathon

Unread post by Fahim Shahriar » Fri Nov 16, 2012 9:35 am

Phlembac Adib Hasan wrote:Let me know whether I am right.

Yes. You're right.
Who is going to post the next one?
Name: Fahim Shahriar Shakkhor
Notre Dame College

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Phlembac Adib Hasan
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Re: Secondary and Higher Secondary Marathon

Unread post by Phlembac Adib Hasan » Fri Nov 16, 2012 6:51 pm

Problem 6: Find all integers $n$ such that $n^4+4$ is a prime.
Source: http://www.brilliantscholars.com/assess ... ry/114744/
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Tahmid Hasan
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Re: Secondary and Higher Secondary Marathon

Unread post by Tahmid Hasan » Fri Nov 16, 2012 8:30 pm

Phlembac Adib Hasan wrote:Problem 6: Find all integers $n$ such that $n^4+4$ is a prime.
Source: http://www.brilliantscholars.com/assess ... ry/114744/
Sophie Jermain kills it.
I'm posting a new problem.
Problem $7$:The interior bisector of $\angle A$ from $\triangle ABC$ intersects the side $BC$ and the circumcircle of $\triangle ABC$ at $D;M$, respectively. Let $\omega$ be a circle with center $M$ and radius $MB$. A line passing through $D$, intersects $\omega$ at $X;Y$. Prove that $AD$ bisects $\angle XAY$.
Source: Iran NMO-2004-5
বড় ভালবাসি তোমায়,মা

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Re: Secondary and Higher Secondary Marathon

Unread post by photon » Sat Nov 17, 2012 3:59 pm

Tahmid Hasan wrote: Sophie Jermain kills it.
i couldn't get that ..... i did problem 6 with factorising which results $1,-1$. i think this is general way.
solution to problem 7
$\angle BAM=\angle CAM \Rightarrow \angle BCM=\angle CBM$ , $BM=CM$.so $C$ point lies on the $\omega$ circle .
now , $XD.DY=BD.DC=AD.DM$
$\therefore \displaystyle \frac{XD}{AD}=\frac{DM}{DY}$ , $\Delta AXD \sim \Delta DMY$. $\angle XAD=\angle MYD$ . $\therefore A,X,M,Y $ are cyclic.
$MX=MY \Rightarrow \angle XYM=\angle MXY$
$\angle XAM=\angle XYM=\angle MXY=\angle MAY$ ,
$AD$ bisects $\angle XAY$.

someone post the next problem.
Try not to become a man of success but rather to become a man of value.-Albert Einstein

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