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PostPosted: Sun, 29 Aug 2010 11:14:09 UTC 
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Hello everyone.
I would like to start 'marathon' of the exercises from different competition in the secondary school level. (ages 16-19)

let's start with inequality :)
Exercise 1.
Let a_i\ (i=1,2,...,n) be the real positive numbers which \prod_{i=1}^{n}a_i=1. Prove that
\prod_{i=1}^{n}(1+a_i) \ge 2^n.

Enjoy!


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PostPosted: Sun, 29 Aug 2010 14:24:56 UTC 
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
ghostbuster9 wrote:
Hello everyone.
I would like to start 'marathon' of the exercises from different competition in the secondary school level. (ages 16-19)

let's start with inequality :)
Exercise 1.
Let a_i\ (i=1,2,...,n) be the real positive numbers which \prod_{i=1}^{n}a_i=1. Prove that
\prod_{i=1}^{n}(1+a_i) \ge 2^n.

Enjoy!


Generalisation: \prod(r+a_i)\geq (1+r)^n for all r\geq 0.

Spoiler:
Apply AM-GM on each degree piece.


Exercise 2. (1992 IMO Q5, rephrased)
Let S\in\mathbb{Z}^3=\{(x,y,z)\colon x,y,z\in\mathbb{Z}^3\} be a finite set, and let S_x,S_y,S_z be the projection of S along x onto yz coordinates, along y onto zx coordinates, and along z onto xy-coordinates respectively. Prove that \lvert S\rvert^2\leq \lvert S_x\rvert\cdot \lvert S_y\rvert\cdot\lvert S_z\rvert.

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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