Secondary and Higher Secondary Marathon

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

Unread post by Tahmid Hasan » Tue Dec 18, 2012 6:25 pm

zadid xcalibured wrote:I have a solution for problem 18.So let's move on to the next one.Someone post problems.
Post your solution then.
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zadid xcalibured
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Re: Secondary and Higher Secondary Marathon

Unread post by zadid xcalibured » Tue Dec 18, 2012 8:04 pm

Solution $18$:let $BE$ and $AC$ intersect at $Q$.As $\angle{PDB}=\angle{DEB}=\angle{BQP}$ ,$PBQD$ is cyclic.And $PB=PD$ implies $\angle{BQP}=\angle{DQP}$ which implies $\angle{DQP}=\angle{CQE}$ which eventually implies $\triangle{ADQ}$ and $\triangle{CQE}$ are congruent as $AC$ and $DE$ are parallel.

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*Mahi*
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Re: Secondary and Higher Secondary Marathon

Unread post by *Mahi* » Tue Dec 18, 2012 8:21 pm

$BB' || AC$
rest of it should be clear form the picture.
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SANZEED
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Re: Secondary and Higher Secondary Marathon

Unread post by SANZEED » Tue Dec 18, 2012 9:08 pm

Problem- $\boxed {18}$
Let $B'\in \omega$ so that $BB'\parallel AC$. Then $\angle EBC=\angle EAC=\angle DCA=\angle DBA$ since $ACDE$ is a cyclic trapezoid. Again $ACBB'$ is also a cyclic trapezoid. So $\angle BCA=\angle B'AC$. Now between $\triangle BCF$ and $\triangle B'AF$, $\angle FBC=\angle FB'C, \angle FCB=\angle FAB', BC=B'A$. Thus $\triangle BCF$ and $\triangle B'AF$ are equivalent. Thus $AF=CF$.
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zadid xcalibured
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Re: Secondary and Higher Secondary Marathon

Unread post by zadid xcalibured » Wed Dec 19, 2012 3:20 pm

Problem $19$:Let $f (x) = a_0x^n + a_1x^{n−1} + ⋯ + a_n$ be a polynomial with real coefficients such that $a_0 ≠ 0$ and for all real $x$,$f (x) f (2x^2) = f (2x^3+x)$.Prove that $f(x)$ has no real root.
Last edited by zadid xcalibured on Wed Dec 19, 2012 11:11 pm, edited 1 time in total.

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SANZEED
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Re: Secondary and Higher Secondary Marathon

Unread post by SANZEED » Wed Dec 19, 2012 10:44 pm

zadid xcalibured wrote:Problem $19$:Let $f (x) = a_0x_n + a_1x_{n−1} + ⋯ + a_n$ be a polynomial with real coefficients such that $a_0 ≠ 0$ and for all real $x$,$f (x) f (2x^2) = f (2x^3+x)$.Prove that $f(x)$ has no real root.
Shouldn't it be $x^i$ ?
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zadid xcalibured
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Re: Secondary and Higher Secondary Marathon

Unread post by zadid xcalibured » Wed Dec 19, 2012 11:05 pm

Oops.Edited.

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Phlembac Adib Hasan
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Re: Secondary and Higher Secondary Marathon

Unread post by Phlembac Adib Hasan » Thu Dec 20, 2012 12:26 pm

zadid xcalibured wrote:Problem $19$:Let $f (x) = a_0x^n + a_1x^{n−1} + ⋯ + a_n$ be a polynomial with real coefficients such that $a_0 ≠ 0$ and for all real $x$,$f (x) f (2x^2) = f (2x^3+x)$.Prove that $f(x)$ has no real root.
Let $\exists q:q\neq 0$ and $f(q)=0$. Then the equation will imply $f(q^3+q)=0$ and so on. Since $x^3+x=x$ is possible only for $x=0$, so this will imply $f(x)$ has infinitely many roots in $\mathbb {R}$ which is impossible.

If $q=0$, then $a_n=0$. So $f(x)=x^kP(x)$. where $P(0)\neq 0$. Now in the given equation we find $x^{3k}P(x)P(2x^2)=x^k(2x^2+1)^kP(2x^3+x)$. Since we assumed $P(x)$ has no zero root, so RHS has free $x$ to the power $k$ where LHS has $x^{3k}$. So a contradiction and so $f(0)\neq 0$. Therefore $f(x)$ has no real root.
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Fahim Shahriar
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Re: Secondary and Higher Secondary Marathon

Unread post by Fahim Shahriar » Thu Dec 20, 2012 1:36 pm

I'm going to post problem no. 20.

$\boxed {20}$
A series is formed in the following manner:
$A(1) = 1$;
$A(n) = f(m)$ numbers of $f(m)$ followed by $f(m)$ numbers of $0$;
$m$ is the number of digits in $A(n-1)$
Find $A(30)$. Here $f(m)$ is the remainder when $m$ is divided by $9$.

Source:BD National Math Olympiad 2010, Category: Secondary/Higher Secondary
Name: Fahim Shahriar Shakkhor
Notre Dame College

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zadid xcalibured
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Re: Secondary and Higher Secondary Marathon

Unread post by zadid xcalibured » Fri Dec 21, 2012 12:47 am

This sequence must be periodic with period at most $9$.Actually this is periodic with period $6$.So $A(30)=A(6)=77777770000000$

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