TurkeyTsT 2008

[dektep]

เป็นข้อสอบ Team selection test ปี 2008 ของตุรกีครับ
1. In an $ABC$ triangle such that $m(\angle B) > m(\angle C)$, the internal and external bisectors of vertice $A$ intersects $BC$ respectively at points $D$ and $E$. $P$ is a variable point on $EA$ such that $A$ is on $[EP]$. $DP$ intersects $AC$ at $M$ and $ME$ intersects $AD$ at $Q$. Prove that all $PQ$ lines have a common point as $P$ varies.

2. A graph has $30$ vertices, $105$ edges and $4822$ unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph.

3.The equation $x^3 - ax^2 + bx - c = 0$ has three (not necessarily different) positive real roots. Find the minimal possible value of $\frac {1 + a + b + c}{3 + 2a + b} - \frac {c}{b}$.

4.The sequence $(x_n)$ is defined as; $x_1 = a$, $x_2 = b$ and for all positive integer $n$, $x_{n + 2} = 2008x_{n + 1} - x_n$. Prove that there are some positive integers $a,b$ such that $1 + 2006x_{n + 1}x_n$ is a perfect square for all positive integer $n$.

5.$D$ is a point on the edge $BC$ of triangle $ABC$ such that $AD = \frac {BD^2}{AB + AD} = \frac {CD^2}{AC + AD}$. $E$ is a point such that $D$ is on $[AE]$ and $CD = \frac {DE^2}{CD + CE}$. Prove that $AE = AB + AC$.

6. There are $n$ voters and $m$ candidates. Every voter makes a certain arrangement list of all candidates (there is one person in every place $1,2,...m$) and votes for the first $k$ people in his/her list. The candidates with most votes are selected and say them winners. A poll profile is all of this $n$ lists.
If $a$ is a candidate, $R$ and $R'$ are two poll profiles. $R'$ is $a - good$ for $R$ if and only if for every voter; the people which in a worse position than $a$ in $R$ is also in a worse position than $a$ in $R'$. We say positive integer $k$ is monotone if and only if for every $R$ poll profile and every winner $a$ for $R$ poll profile is also a winner for all $a - good$ $R'$ poll profiles. Prove that $k$ is monotone if and only if $k > \frac {m(n - 1)}{n}$.

[dektep]

3.

hint

$a=x+y+z , b=xy+yz+zx ,c=xyz ; x,y,z > 0$ และ $x,y,z$ เป็นรากของสมการ

[Erken]

3. ตอบ $\frac{1}{3}$ ครับ

[nooonuii]

ข้อความเดิมเขียนโดยคุณ dektep

4.The sequence $(x_n)$ is defined as; $x_1 = a$, $x_2 = b$ and for all positive integer $n$, $x_{n + 2} = 2008x_{n + 1} - x_n$. Prove that there are some positive integers $a,b$ such that $1 + 2006x_{n + 1}x_n$ is a perfect square for all positive integer $n$.

Hint:

มองหา invariant ให้เจอ แล้วจะรู้ว่า $a,b$ ควรเป็นอะไร

My Solution :

ให้ $a=1,b=2008$ จะำพิสูจน์ว่า $$1+2006x_nx_{n+1}=(x_{n+1}-x_n)^2$$
Proof :
$(x_{n+1}-x_n)^2-2006x_nx_{n+1}-1=x_{n+1}^2-2008x_nx_{n+1}+x_n^2-1$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=(x_{n+1})(x_{n+1}-2008x_n)+x_n^2-1$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=(2008x_n-x_{n-1})(-x_{n-1})+x_n^2-1$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=x_n^2-2008x_{n-1}x_n+x_{n-1}^2-1$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\vdots$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=x_2^2-2008x_1x_2+x_1^2-1$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=0$

[murderer@IPST]

1.

Hint

จวบกันที่Bใช้เมเนลอสในสามเหลี่ยมPEDกับเส้นAMC

[tatari/nightmare]

5.เราต่อ DA ออกไปทาง A และกำหนดจุด B',C' บนส่วนต่อที่ทำให้ $AB=AB',AC=AC'$ จากที่ $AD(AB+AD)=BD^2$ และ $AB+AD=AB'+AD=B'D$
เราจึงได้ $(AD)(B'D)=BD^2$ ซึ่งก็คือ power นั่นเอง
ใช้ idea เดิม เราจะได้ว่า $\angle B+\angle C=60$ แล้วก็จบ........#