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All of the solution will be post in comment, so anyone welcome to write the solution, the official (by me) solution will be post as soon as the problem solved.


If the problem is homework, then the official solution will post at least in the same day as it need to be submit. 00:00 is possible.



Monday, February 22, 2010

Town Selection, Past


  1. Banyaknya solusi real $x$ dari persamaan
    $$3^{\left[ 1/2+\log _{3}\left( \cos x-\sin x\right) \right] }+2^{\log_{2}\left( \cos x+\sin x\right) }=\sqrt{2}$$
    adalah ...

  2. Ada berapa banyak segitiga siku-siku yang kelilingnya $2009$ dan sisi-sisinya bilangan bulat serta jari-jari lingkaran dalamnya juga bilangan bulat?

  3. Cari semua solusi bulat $a,b$ dari persamaan
    $$\frac{1}{a}+\frac{1}{b}=\frac{1}{2010}.$$

  4. Tentukan nilai $x,y,z$ real yang memenuhi
    $$\begin{eqnarray*}
    \dfrac{xy}{x+y} &=&\dfrac{1}{2} \\
    \dfrac{yz}{y+z} &=&\dfrac{1}{3} \\
    \dfrac{zx}{z+x} &=&\dfrac{1}{7}.
    \end{eqnarray*}$$

  5. Diketahui segitiga $ABC$ dengan sisi $a,b$, dan $c$. Jika $\angle BAC=\frac{\pi}{3}$ dan $a=1$, buktikan bahwa $b+c\leq 2.$

  6. Diberikan bilangan real $x,y$ yang memenuhi
    $$x^{2}+5y^{2}+1=2y\left( x+1\right) .$$
    Tentukan nilai $|x-y| .$

  7. Tentukan jumlah
    $$\frac{1}{2009^{-2009}+1}+\frac{1}{2009^{-2008}+1}+\cdots +\frac{1}{2009^{2008}+1}+\frac{1}{2009^{2009}+1}.$$

  8. Bapak dan Ibu $T$ mengadakan pertemuan di rumah mereka dan mengundang empat pasang suami-istri lainnya. Dalam pertemuan itu sejumlah, tetapi tidak semua, orang berjabat tangan. Tidak ada dua orang yang berjabatan lebih dari satu kali, dan setiap orang tidak berjabatan tangan dengan suami/ istrinya sendiri. Tuan dan nyonya rumah berjabatan dengan beberapa orang. Pada akhir acara Bapak $T$ menanyai kesembilan orang yang hadir (tidak termasuk dirinya sendiri) banyaknya jabat tangan yang dilakukan mereka. Ternyata kesembilan orang ini memberi jawaban yang berbeda. Tentukan banyaknya jabat tangan yang dilakukan oleh Ibu $T.$

  9. Diketahui $x,y,z,t$ adalah bilangan real yang tidak sama dengan nol
    dan memenuhi
    $$\begin{eqnarray*}
    x+y+z &=&t \\
    \frac{1}{x}+\frac{1}{y}+\frac{1}{z} &=&\frac{1}{t} \\
    x^{3}+y^{3}+z^{3} &=&1000^{3}
    \end{eqnarray*}$$
    Tentukan nilai $x+y+z+t$.

  10. Jika diketahui $\dfrac{9x+2y}{9z+2t}=\dfrac{2x+9y}{2z+9t}$ dan $\dfrac{%
    12x+y}{12z+t}=2010,$ hitunglah $\dfrac{24x+13y}{24x+13t}.$

Monday, February 15, 2010

Algebra - Durbin, John, R. Modern Algebra

12.12. Prove that if $a$ dan $b$ are integers, not both zero, then there are infinitely many pairs of integers $m,n$ such that $(a,b)=am+bn$.

12.13. Prove that if $c$ is a positive integer, then $(ac,bc)=(a,b)\cdot c$.

12.15. Prove that if $p$ is a prime and $a$ is an integer, and $a$ is not divisible by p, then $(a,p)=1$.

12.22. Prove that if $a,b,c$ are integers, not all zero, then they have a greatest common divisors, which can be written as a linear combination of $a,b,$ and $c$.

13.16. Prove that if $n$ is an integer, then $\sqrt{n}$ is rational iff $n$ is a perfect square.

13.19. Prove that if $a$ and $b$ are positive integers, then
$$(a,b)[a,b]=ab$$
when [a,b] is least common multiple.

Saturday, February 13, 2010

2010 Pieces of Paper

Given $11$ pieces of paper. In every step, choose several paper, then cut it into $11$ pieces of paper from each piece (which is choosen). Can we obtain $2010$ pieces of paper after several step?

Friday, February 12, 2010

Colouring

A circle is divided into $n$ equal sectors. We color the sectors in $n-1$ colors using each of the colors at least once. How many such colorings are there?

Wednesday, February 10, 2010

Polynomial and Complex Number

Let $\alpha$ be a root of polynomial
$$P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1 x+a_0$$
where $a_i \in [0,1]$, for $i=1,2,\dots,n-1$. Prove that
$$Re(\alpha)<\frac{1+\sqrt{5}}{2}.$$

Proposed by Bogdan Enescu,Romania

Tuesday, February 9, 2010

Some Notes on Root of Unity

Let $a_,\dots,a_n$ be positive real numbers and let $\omega$ be a primitive nth root of unity. If the sides of an equiangular polygon have lengths $a_,\dots,a_n$ (in counterclockwise order) then
$$1+\omega+\omega^2+\dots+\omega^{n-1}=0$$;
$$a_1 + a_2 \omega +a_3 \omega ^2+ \dots + a_n \omega^{n-1}=0$$.

Monday, February 8, 2010

Lemma 1 - test post

For any positif real numbers $a_1,a_2,\dots,a_{n}$, and any real numbers $x_1,x_2,\dots,x_{n}$, the inequality

$$ \sum_{1\leq i,j\leq n} x_{i}x_{j}\cdot min(a_{i},a_{j})\geq 0 $$
holds.


$\Sigma$