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阿尔巴尼亚不等式集锦

2015-01-27 12:10:43| 分类：奥赛 | 标签： |举报 |字号大中小订阅

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1. Given the equation $x^4-x^3-1=0$

(a) Find the number of its real roots.
(b) We denote by $S$ the sum of the real roots and by $P$ their product. Prove that $P< - \frac{11}{10}$ and $S> \frac {6}{11}$ .

Albania BMO TST 2009

2. Let's consider the inequality $a^3+b^3+c^3<k(a+b+c)(ab+bc+ca)$ where $a,b,c$ are the sides of a triangle and $k$ a real number.

a) Prove the inequality for $k=1$ .
b) Find the smallest value of $k$ such that the inequality holds for all triangles.

Albania BMO TST 2010

3. Let $a\geq 2$ be a real number; with the roots $x_{1}$ and $x_{2}$ of the equation $x^2-ax+1=0$ we build the sequence with $S_{n}=x_{1}^n + x_{2}^n$ .

a)Prove that the sequence $\frac{S_{n}}{S_{n+1}}$ , where $n$ takes value from $1$ up to infinity, is strictly non increasing.
b)Find all value of $a$ for the which this inequality hold for all natural values of $n$ $\frac{S_{1}}{S_{2}}+\cdots +\frac{S_{n}}{S_{n+1}}>n-1$

Albania BMO TST 2010

4. Find the greatest value of the expression $\frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9}$ where $x$ , $y$ , $z$ are nonnegative real numbers such that $x+y+z=1$ .

Albania Team Selection Test 2012

5. Let $a,b,c,d$ be positive real numbers such that $abcd=1$ .Find with proof that $x=3$ is the minimal value for which the following inequality holds : $a^x+b^x+c^x+d^x\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}$

Albania IMO TST 2013

6. Let $n$ be a natural number such that $n \geq 2$ . Show that

$\frac {1}{n + 1} \left( 1 + \frac {1}{3} + \cdot \cdot \cdot + \frac {1}{2n - 1} \right) > \frac {1}{n} \left( \frac {1}{2...$

Albania BMO TST 2014

7. In a convex quadrilateral $ABCD$ , $\angle ABC$ and $\angle BCD$ are $\geq 120^o$ . Prove that $|AC|$ + $|BD| \geq |AB|+|BC|+|CD|$ . (With $|XY|$ we understand the length of the segment $XY$ ).

Albanian Mathematical Olympiad 12 GRADE 2011--Question 3

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