1,Cho 3 số a,b,c thỏa mãn 
$1\leq a,b,c \leq 3 $
$a+b+c=6 $
CMR: $a^2+b^2+c^2 \leq 14 $

2,Tìm $x,y$ nguyên dương thỏa mãn 
$x+y+z > 11$ 
$8x+9y+10z=100 $

3,CMR: $21[a+(1/b)] + 3[b+(1/a)]\geq 80 $
$(a \geq 3,b \geq 3) $

4, giải hệ 
$x+y=\sqrt{4z-1}  $
$y+z=\sqrt{4x-1} $
$x+z=\sqrt{4y-1}$

Câu hỏi này được treo giải thưởng trị giá +1000 vỏ sò bởi nambttvqht@gmail.com, đã hết hạn vào lúc 17-10-14 05:51 PM

1. Không mất tính tổng quát, giả sử: $a\le b\le c \Rightarrow 1\le a\le2;2\le c\le3$.
Ta có:
$(a-1)(a-2)\le0 \Leftrightarrow a^2\le3a-2$
$(c-2)(c-3)\le0 \Leftrightarrow c^2\le5c-6$
*) Nếu $1\le b\le 2 \Rightarrow b^2\le3b-2$
$\Rightarrow a^2+b^2+c^2\le 3(a+b+c)+2c-10\le3.6+2.3-10=14$
*) Nếu $2\le b\le 3 \Rightarrow c^2\le5c-6$
$\Rightarrow a^2+b^2+c^2\le 5(a+b+c)-2a-14\le5.6-2.1-14=14$
Dấu bằng xảy ra chẳng hạn khi: $a=1;b=2;c=3$.

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