\documentclass{subfile}

\begin{document}
	\section{ChMO}\label{sec:chmo}

		\begin{problem}[$2019$, problem $1$]
			Let $a,b,c,d,e\geq-1$ be real numbers such that $a+b+c+d+e=5$. Find the minimum and maximum value of
				\begin{align*}
					(a+b)(b+c)(c+d)(d+e)(e+a)
				\end{align*}
		\end{problem}

		\begin{problem}[$2018$, problem $6$]
			Let $n,k$ be natural numbers such that $n>k$ and $a_{1},\ldots,a_{n}$ be real numbers in the interval $(k-1,k)$. Let $x_{1},\ldots,x_{n}$ be positive real numbers such that for any subset $I$ of $\{1,2,\ldots,n\}$ with $k$ elements,
				\begin{align*}
					\sum_{i\in I}x_{i}
						& \leq\sum_{i\in I}a_{i}
				\end{align*}
			Find the maximum possible value of $x_{1}\cdots x_{n}$.
		\end{problem}

		\begin{problem}[$2017$, problem $6$]
			Given an integer $n\geq2$ and real numbers $a,b$ such that $0 < a < b$. Let $x_{1},\ldots,x_{n}$ be real numbers in the interval $[a,b]$. Find the maximum value of
				\begin{align*}
					\dfrac{\dfrac{x_{1}^{2}}{x_{2}}+\ldots+\dfrac{x_{n-1}^{2}}{x_{n}}+\dfrac{x_{n}^{2}}{x_{1}}}{x_{1}+\ldots+x_{n}}
				\end{align*}
		\end{problem}

		\begin{problem}[$2015$, problem $1$]
			Let $z_{1},\ldots,z_{n}$ be complex numbers such that $|z_{i}-1|\leq r$ for some real number $r\in(0,1)$. Show that
				\begin{align*}
					\left|\sum_{i=1}^{n}z_{i}\right|\cdot\left|\sum_{i=1}^{n}\dfrac{1}{z_{i}}\right|
						& \geq n^{2}(1-r^{2})
				\end{align*}
		\end{problem}

		\begin{problem}[$2011$, problem $1$]
			Let $a_{1},\ldots,a_{n}$ be real numbers. Prove that
				\begin{align*}
					\sum_{i=1}^{n}a_{i}^{2}-\sum_{i=1}^{n}a_{i}a_{i+1}
						& \leq \left\lfloor\dfrac{n}{2}\right\rfloor(M-m)
				\end{align*}
			where $a_{n+1}=a_{1},M=\max\{a_{1},\ldots,a_{n}\},m=\min\{a_{1},\ldots,a_{n}\}$.
		\end{problem}

		\begin{problem}[$2011$, problem $5$]
			Let $n\geq4$ be an integer and $a_{1},\ldots,a_{n},b_{1},\ldots,b_{n}$ be non-negative real numbers such that
				\begin{align*}
					a_{1}+\ldots+a_{n}
						& = b_{1}+\ldots+b_{n}>0
				\end{align*}
			Find the maximum of
				\begin{align*}
					\dfrac{\sum_{i=1}^{n}a_{i}(a_{i}+b_{i})}{\sum_{i=1}^{n}b_{i}(a_{i}+b_{i})}
				\end{align*}
		\end{problem}

		\begin{problem}[$2009$, problem $4$]
			Let $n>3$ be an integer and $a_{1},\ldots,a_{n}$ be real numbers satisfying $\min\{a_{i}-a_{j}\}\leq1$ for $1\leq i<j\leq n$. Find the minimum value of
				\begin{align*}
					\sum_{i=1}^{n}a_{i}^{3}
				\end{align*}
		\end{problem}

		\begin{problem}[$2008$, problem $3$]
			Let $n$ be a positive integer and $x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}$ be real numbers such that
				\begin{align*}
					x_{1}
						& \leq x_{2}\ldots\leq x_{n}\\
					y_{1}
						& \geq y_{2}\ldots\geq y_{n}\\
					\sum_{i=1}^{n}ix_{i}
						& = \sum_{i=1}^{n}iy_{i}
				\end{align*}
			Show that for any real number $\alpha$,
				\begin{align*}
					\sum_{i=1}^{n}x_{i}\lfloor i\alpha\rfloor
						& \geq\sum_{i=1}^{n}y_{i}\lfloor i\alpha\rfloor
				\end{align*}
		\end{problem}

		\begin{problem}[$2007$, problem $1$]
			Let $a,b,c$ be complex numbers and $|a+b|=m,|a-b|=n$. If $mn\neq0$, show that
				\begin{align*}
					\max\{|ac+b|,|a+bc|\}
						& \geq\dfrac{mn}{\sqrt{m^{2}+n^{2}}}
				\end{align*}
		\end{problem}

		\begin{problem}[$2006$, problem $1$]
			Let $a_{1},\ldots,a_{k}$ be real numbers such that $a_{1}+\ldots+a_{k}=0$. Prove that
				\begin{align*}
					\max\{a_{1}^{2},\ldots,a_{k}^{2}\}
						& \leq \dfrac{k}{3}\left((a_{1}-a_{2})^{2}+\ldots+(a_{k-1}-a_{k})^{2}\right)
				\end{align*}
		\end{problem}

		\begin{problem}[$2006$, problem $5$]
			Let $(a_{n})$ be a sequence such that $a_{1}=\frac{1}{2},a_{k+1}=-a_{k}+\frac{1}{2-a_{k}}$. Prove that
				\begin{align*}
					\left(\dfrac{n}{2(a_{1}+\ldots+a_{n})}-1\right)^{n}
						& \leq\left(\dfrac{a_{1}+\ldots+a_{n}}{n}\right)^{n}\left(\dfrac{1}{a_{1}}-1\right)\cdots\left(\dfrac{1}{a_{n}}-1\right)
				\end{align*}
		\end{problem}

		\begin{problem}[$2005$, problem $1$]
			Let $\theta_{i}\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ for $1\leq i\leq 4$. Prove that, there exists a real number $x$ such that
				\begin{align*}
					\cos^{2}{\theta_{1}}\cos^{2}{\theta_{2}}-(\sin{\theta_{1}}\sin{\theta_{2}}-x)^{2}
						& \geq0\\
					\cos^{2}{\theta_{3}}\cos^{2}{\theta_{4}}-(\sin{\theta_{3}}\sin{\theta_{4}}-x)^{2}
						& \geq0
				\end{align*}
			if and only if
				\begin{align*}
					\sum_{i=1}^{4}\sin^{2}{\theta_{i}}
						& \leq 2\left(1+\prod_{i=1}^{4}\sin{\theta_{i}}+\prod_{i=1}^{4}\cos{\theta_{i}}\right)
				\end{align*}
		\end{problem}

		\begin{problem}[$2004$, problem $2$]
			Let $n\geq2$ be an integer and $a_{1},\ldots,a_{n}$ are positive integers such that $a_{1}<\ldots<a_{n}$ and
				\begin{align*}
					\dfrac{1}{a_{1}}+\ldots+\dfrac{1}{a_{n}}
						& \leq1
				\end{align*}
			Prove that for any real number $x$,
				\begin{align*}
					\left(\sum_{i=1}^{n}\dfrac{1}{a^{2}+x^{2}}\right)^{2}
						& \leq \dfrac{1}{2}\dfrac{1}{a_{1}(a_{1}-1)+x^{2}}
				\end{align*}
		\end{problem}

		\begin{problem}[$2003$, problem $3$]
			Let $n$ be a positive integer. Find the smallest positive real number $\lambda$ such that for any $x_{1},\ldots,x_{n}\in\left(0,\frac{\pi}{2}\right)$,
				\begin{align*}
					\prod_{i=1}^{n}\tan{x_{i}}
						& = 2^{\frac{n}{2}}
				\end{align*}
			implies
				\begin{align*}
					\sum_{i=1}^{n}\cos{x_{i}}
						& \leq \lambda
				\end{align*}
		\end{problem}

		\begin{problem}[$2003$, problem $3$]
			Let $a,b,c,d$ be positive real numbers such that $ab+cd=1$ and $x_{1},x_{2},x_{3},x_{4},y_{1},y_{2},y_{3},y_{4}$ are real numbers such that
				\begin{align*}
					x_{i}^{2}+y_{i}^{2}
						& = 1
				\end{align*}
			for $1\leq i\leq 4$. Prove that
				\begin{align*}
					\left(ax_{1}+bx_{2}+cx_{3}+cx_{4}\right)^{2}+(ay_{4}+by_{3}+cy_{2}+dy_{4})^{1}
						& \leq 2\left(\dfrac{a^{2}+b^{2}}{ab}+\dfrac{c^{2}+d^{2}}{cd}\right)
				\end{align*}
		\end{problem}

		\begin{problem}[$2002$, problem $1$]
			For every four points $P_{1},P_{2},P_{3},P_{4}$ on the plane, find the minimum value of
				\begin{align*}
					\dfrac{\sum_{1\leq i<j\leq 4}P_{i}P_{j}}{\min\limits_{1\leq i<j\leq 4}P_{i}P_{j}}
				\end{align*}
		\end{problem}

		\begin{problem}[$2002$, problem $3$]
			Let $c$ be a real number such that $c\in\left(\frac{1}{2},1\right)$. Find the least real number $M$ such that for every integer $n\geq2$ and real numbers $0\leq a_{1}\leq \ldots\leq a_{n}$, if
				\begin{align*}
					\dfrac{1}{n}\sum_{i=1}^{n}ia_{i}
						& = c\sum_{i=1}^{n}a_{i}
				\end{align*}
			then we always have that
				\begin{align*}
					\sum_{i=1}^{n}a_{i}
						& \leq M\sum_{i=1}^{m}a_{i}
				\end{align*}
			where $m=\lfloor cn\rfloor$.
		\end{problem}

		\begin{problem}[$1999$, problem $5$]
			Determine the maximum value of $\lambda$ such that
				\begin{align*}
					f(x)
						& \geq \lambda(x-a)^{3}
				\end{align*}
			for all non-negative real number $x$ where
				\begin{align*}
					f(x)
						& = x^{3}+ax^{2}+bx+c
				\end{align*}
			and has non-negative roots. Find the equality condition.
		\end{problem}

		\begin{problem}[$1998$, problem $2$]
			Given a positive integer $n>1$. Determine with proof if there exists $2n$ distinct positive integers $a_{1},\ldots,a_{n},b_{1},\ldots,b_{n}$ such that
				\begin{align*}
					a_{1}+\ldots+a_{n}
						& = b_{1}+\ldots+b_{n}\\
					n-1
						& > \sum_{i=1}^{n}\dfrac{a_{i}-b_{i}}{a_{i}+b_{i}}>n-1-\dfrac{1}{1998}
				\end{align*}
		\end{problem}

		\begin{problem}[$1998$, problem $6$]
			Let $n\geq2$ be a positive integer and $x_{1},\ldots,x_{n}$ be real numbers such that
				\begin{align*}
					\sum_{i=1}^{n}x_{i}^{2}+\prod_{i=1}^{n-1}x_{i}x_{i+1}
						& = 1
				\end{align*}
			For each $k$, find the maximum value of $|x_{k}|$.
		\end{problem}

		\begin{problem}[$1997$, problem $1$]
			Let $x_{1},\ldots,x_{1997}$ be real numbers satisfying
				\begin{align*}
					-\dfrac{1}{\sqrt{3}}
						& \leq x_{i}\leq\dfrac{1}{\sqrt{3}}\\
					x_{1}+\ldots+x_{1997}
						& = -318\sqrt{3}
				\end{align*}
			Determine the maximum value of
				\begin{align*}
					x_{1}^{12}+\ldots+x_{1997}^{12}
				\end{align*}
			with proof.
		\end{problem}

		\begin{problem}[$1997$, problem $6$]
			Let $(a_n)$ be a sequence of real numbers satisfying
				\begin{align*}
					a_{n+m}
						& \leq a_{n}+a_{m}
				\end{align*}
			for all non-negative integers $m,n$. Prove that, if $n\geq m$,
				\begin{align*}
					a_{n}
						& \leq ma_{1}+\left(\dfrac{n}{m}-1\right)a_{m}
				\end{align*}
		\end{problem}

		\begin{problem}[$1996$, problem $2$]
			Let $n$ be a natural number. Suppose that $x_{0}=0$ and $x_{i}>0$ for all $i\in\{1,\ldots,n\}$. If $\sum_{i=1}^{n}x_{i}=1$, prove that
				\begin{align*}
					1
						& \leq \sum_{i=1}^{n}\dfrac{x_{i}}{\sqrt{1+x_{0}+\ldots+x_{i-1}}\sqrt{x_{i}+\ldots+x_{n}}}
							< \dfrac{\pi}{2}
				\end{align*}
		\end{problem}

		\begin{problem}[$1995$, problem $1$]
			Let $n\geq3$ be an integer and $a_{1},\ldots,a_{n},b_{1},\ldots,b_{n}$ be real numbers such that
				\begin{align*}
					a_{1}+\ldots+a_{n}
						& = b_{1}+\ldots+b_{n}\\
					0<a_{1}=a_{2},
						& a_{i}+a_{i+1}=a_{i+2}\\
					0<b_{1}\leq b_{2},
						& b_{i}+b_{i+1}\leq b_{i+2}
				\end{align*}
			Prove that $a_{n-1}+a_{n}\leq b_{n-1}+b_{n}$.
		\end{problem}

		\begin{problem}[$1995$, problem $5$]
			Let $a_{1},\ldots,a_{10}$ distinct natural numbers such that $a_{1}+\ldots+a_{10}=1995$. Find the minimum value of
				\begin{align*}
					a_{1}a_{2}+a_{2}a_{3}+\ldots+a_{9}a_{10}+a_{10}a_{1}
				\end{align*}
		\end{problem}

		\begin{problem}[$1993$, problem $2$]
			Given a positive integer $k$ and a positive real number $a$. Find the maximum value of
				\begin{align*}
					a^{k_1}+\ldots+a^{k_r}
				\end{align*}
			where $1\leq r\leq k$ and $k_{1}+\ldots+k_{r}=k$.
		\end{problem}

		\begin{problem}[$2019$ Team Selection Test, problem $3$]
			Let $n$ be a positive integer and $a_{1},\ldots,a_{n}$ be non-negative real numbers such that
				\begin{align*}
					a_{1}+\ldots+a_{n}
						& = 1
				\end{align*}
			Find the maximum possible value of
				\begin{align*}
					\sum_{1\leq i<j\leq n}\min\{(i-j)^{2},(n+i-j)^{2}a_{i}a_{j}\}
				\end{align*}
		\end{problem}

		\begin{problem}[$2019$ Team Selection Test, problem $7$]
			Let $x,y,z$ be complex numbers such that
				\begin{align*}
					|x|^{2}+|y|^{2}+|z|^{2}
						& = 1
				\end{align*}
			Prove that
				\begin{align*}
					\left|x^{3}+y^{3}+z^{3}-3xyz\right|
						& \leq1
				\end{align*}
		\end{problem}

		\begin{problem}[$2019$ Team Selection Test, problem $5$]
			Find all positive integer $n$ such that for any positive real numbers $a,b,c,x,y,z$ the following conditions hold:
				\begin{align*}
					\max\{a,b,c,x,y,z\}
						& = a\\
					a+b+c
						& = x+y+z\\
					abc
						& = xyz\\
					a^{n}+b^{n}+c^{n}
						& = x^{n}+y^{n}+z^{n}
				\end{align*}
		\end{problem}

		\begin{problem}[$2018$ Team Selection Test, problem $5$]
			Let $n,k$ be positive integers such that $n>4k$. Find the minimum value of $\lambda(n,k)=\lambda$ such that for any positive real numbers $a_{1},\ldots,a_{n}$, we have
				\begin{align*}
					\sum_{i=1}^{n}\dfrac{a_{i}}{\sqrt{a_{i}^{2}+a_{i+1}^{2}+\ldots+a_{i+k}^{2}}}
				\end{align*}
			where $a_{n+i}=a_i$ for $1\leq i\leq k$.
		\end{problem}

		\begin{problem}[$2018$ Team Selection Test, problem $3$]
			Let $H(n)$ be the \textit{harmonic sum}
				\begin{align*}
					1+\dfrac{1}{2}+\ldots+\dfrac{1}{n}
				\end{align*}
			Prove that there exists a positive constant $C$ such that
				\begin{align*}
					H(a_{1})+\ldots+H(a_{n})
						& \leq C\sqrt{\sum_{i=1}^{n}ia_{i}}
				\end{align*}
			for arbitrary positive integer $n$ and positive real numbers $a_{1},\ldots,a_{n}$.
		\end{problem}

		\begin{problem}[$2017$ Team Selection Test, problem $2$]
			Let $n$ be positive integers and $x>1$ be a real number. Prove that
				\begin{align*}
					\sum_{i=1}^{n}\dfrac{\{ix\}}{\lfloor ix\rfloor}
						& < \sum_{i=1}^{n}\dfrac{1}{2i-1}
				\end{align*}
			where $\{a\}$ and $\lfloor a\rfloor$ is the \textit{decimal} and \textit{integer} portion of $a$ respectively.
		\end{problem}

		\begin{problem}[$2017$ Team Selection Test, problem $5$]
			Let $m\geq2$ be a positive integer and $a_{1},\ldots,a_{m}$ be positive real numbers. Prove that
				\begin{align*}
					(m-1)^{m-1}\left(x_{1}^{m}+\ldots+x_{m}^{m}\right)
						& \geq (x_{1}+\ldots+x_{m})^{m}-m^{m}x_{1}\cdots x_{m}
				\end{align*}
			Find out when equality holds.
		\end{problem}

		\begin{problem}[$2016$ Team Selection Test, problem $2$]
			Find the smallest positive real number $\lambda$ such that for any complex numbers $z_{1},z_{2},z_{3}$ with $|z_{i}|<1$ and $z_{1}+z_{2}+z_{3}=0$,
				\begin{align*}
					|z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1}|^{2}+|z_{1}z_{2}z_{3}|^{2}
						& < \lambda
				\end{align*}
		\end{problem}

		\begin{problem}[$2016$ Team Selection Test, problem $7$]
			Let $n>1$ be an integer and $\alpha$ be a real number such that $0<\alpha<2$ and $a_{1},\ldots,a_{n},c_{1},\ldots,c_{n}$ be positive real numbers. For a positive real number $y$, let
				\begin{align*}
					f(y)
						& = \left(\sum_{a_{i}<y}c_{i}a_{i}^{2}\right)^{\frac{1}{2}}+\left(\sum_{a_{i}>y}c_{i}a_{i}^{\alpha}\right)^{\frac{1}{\alpha}}
				\end{align*}
			If a positive real number $x$ satisfies $x\geq f(y)$ for some positive real number $y$, prove that $f(x)\leq 8^{\frac{1}{\alpha}}x$.
		\end{problem}

		\begin{problem}[$2015$ Team Selection Test, problem $4$]
			Let $n\geq2$ be an integer and $x_{1},x_{2},\ldots,$ be a non-decreasing monotonous sequence of positive real numbers such that $x_{1},\frac{x_{2}}{2},\frac{x_{3}}{3},\ldots$ is a non-increasing monotonous sequence. Prove that
				\begin{align*}
					\dfrac{\sum_{i=1}^{n}x_{i}}{n\sqrt[n]{\sum_{i=1}^{n}x_{i}}}
						& \leq \dfrac{n+1}{2\sqrt[n]{n!}}
				\end{align*}
		\end{problem}

		\begin{problem}[$2014$ Team Selection Test, problem $4$]
			Let $(x_{n})$ be a sequence of real numbers and $(y_{n})$ be a sequence such that $y_{1}=x_{1}$ and for $n\geq1$,
				\begin{align*}
					y_{n+1}
						& = x_{n+1}-\sqrt{\sum_{i=1}^{n}x_{i}^{2}}
				\end{align*}
			Find the smallest positive real number $\lambda$ such that for any $(x_{n})$ and positive integer $m$,
				\begin{align*}
					\dfrac{1}{m}\sum_{i=1}^{n}x_{i}^{2}
						& \leq \sum_{i=1}^{n}\lambda^{n-i}y_{i}^{2}
				\end{align*}
		\end{problem}

		\begin{problem}[$2014$ Team Selection Test, problem $5$]
			Let $n\geq2$ be a positive integer. Find the greatest constant $\lambda(n)$ such that for any non-zero complex numbers $z_{1},\ldots,z_{n}$,
				\begin{align*}
					\sum_{i=1}^{n}|z_{i}|^{2}
						& \geq \lambda(n)\min\limits_{1\leq i\leq n}|z_{i+1}-z_{i}|^{2}
				\end{align*}
			where $z_{n+1}=z_{1}$.
		\end{problem}

		\begin{problem}[$2013$ Team Selection Test, problem $4$]
			Let $n,k>1$ be integers and $a_{1},\ldots,a_{n}$ be non-negative real numbers such that $a_{1}\geq\ldots\geq a_{n}$ and
				\begin{align*}
					a_{1}+\ldots+a_{n}
						& = 1\\
					c_{1}+\ldots+c_{m}
						& \leq m^{k}
				\end{align*}
			for any positive integer $m\leq n$ and non-negative real numbers $c_{1},\ldots,c_{m}$. Find the maximum value of
				\begin{align*}
					c_{1}a_{1}^{k}+\ldots+c_{n}a_{n}^{k}
				\end{align*}
		\end{problem}

		\begin{problem}[$2013$ Team Selection Test, problem $17$]
			Let $n\geq2$ be an integer and $a_{1},\ldots,a_{n},b_{1},\ldots,b_{n}$ be non-negative real numbers. Prove that
				\begin{align*}
					\left(\dfrac{n}{n-1}\right)^{n-1}\left(\dfrac{1}{n}\sum_{i=1}^{n}a_{i}^{2}\right)+\left(\dfrac{1}{n}\sum_{i=1}^{n}b_{i}^{2}\right)
						& \geq \prod_{i=1}^{n}\sqrt[n]{a_{i}^{2}+b_{i}^{2}}
				\end{align*}
		\end{problem}

		\begin{problem}[$2012$ Team Selection Test, problem $1$]
			Let $n$ be a positive integer and $x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}$ be complex numbers such that $|x_{i}|=|y_{i}|$ for $1\leq i\leq n$. Let
				\begin{align*}
					x
						& = \dfrac{1}{n}\sum_{i=1}^{n}x_{i}\\
					y
						& = \dfrac{1}{n}\sum_{i=1}^{n}y_{i}\\
					z_{i}
						& = xy_{i}-yx_{i}-x_{i}y_{i}
				\end{align*}
			Prove that
				\begin{align*}
					\sum_{i=1}^{n}|z_{i}|
						& \leq n
				\end{align*}
		\end{problem}

		\begin{problem}[$2012$ Team Selection Test, problem $4$]
			Let $m,n>1$ be integers and $r,s$ are real numbers such that $r<s$. Let $(a_{ij})$ be a $m\times n$ non-zero matrix such that $a_{ij}\geq0$. Find the maximum value of
				\begin{align*}
					\dfrac{\left(\sum_{j=1}^{n}\left(\sum_{i=1}^{m}a_{ij}^{s}\right)^{\frac{r}{s}}\right)^{\frac{1}{r}}}{\left(\sum_{i=1}^{m}\left(\sum_{j=1}^{n}a_{ij}^{r}\right)^{\frac{s}{r}}\right)^{\frac{1}{s}}}
				\end{align*}
		\end{problem}

		\begin{problem}[$2011$ Team Selection Test, problem $6$]
			Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_{1},\ldots,x_{2n}$ satisfying
				\begin{align*}
					\dfrac{1}{2n}\sum_{i=1}^{2n}(x_{i}+2)^{n}
						& \geq \prod_{i=1}^{2n}x_{i}
				\end{align*}
			the following inequality is also true:
				\begin{align*}
					\dfrac{1}{2n}\sum_{i=1}^{2n}(x_{i}+1)^{n}
						& \geq \lambda\prod_{i=1}^{2n}x_{i}
				\end{align*}
		\end{problem}

		\begin{problem}[$2011$ Team Selection Test, problem $7$]
			Let $n\geq 3$ be an integer. Find the largest real number $M$ such that for any positive real numbers $x_{1},\ldots,x_{n}$ there is an arrangement $y_{1},\ldots,y_{n}$ such that
				\begin{align*}
					\sum_{i=1}^{n}\dfrac{y_{i}^{2}}{y_{i+1}^{2}-y_{i+1}y_{i+2}+y_{i+2}^{2}}
						& \geq M
				\end{align*}
			where $y_{n+2}=y_{2},y_{n+1}=y_{1}$.
		\end{problem}

		\begin{problem}[$2010$ Team Selection Test, problem $1$]
			Let $n$ be a positive integer. The real numbers $a_{0},\ldots,a_{n},b_{0},\ldots,b_{n}$ satisfy $a_{i}+a_{i+1}\geq 0$ for $1\leq i\leq 2n-1$ and $a_{2i+1}\leq 0$ for $1\leq i\leq n-1$. For any integers $p,q$ such that $0\leq p\leq q\leq n$, we have
				\begin{align*}
					\sum_{i=2p}^{2q}b_{i}
						& > 0
				\end{align*}
			Prove that
				\begin{align*}
					\sum_{i=0}^{2n}(-1)^{i}a_{i}b_{i}
						& \geq 0
				\end{align*}
			Determine when equality holds.
		\end{problem}

		\begin{problem}[$2010$ Team Selection Test, problem $5$]
			Find all positive real numbers $\lambda$ such that for all integer $n\geq 2$ and all positive real numbers $a_{1},\ldots,a_{n}$ such that $a_{1}+\ldots+a_{n}=n$,
				\begin{align*}
					\sum_{i=1}^{n}\dfrac{1}{a_{i}}-\lambda\prod_{i=1}^{n}\dfrac{1}{a_{i}}
						& \leq n-\lambda
				\end{align*}
		\end{problem}

		\begin{problem}[$2010$ Team Selection Test, problem $7$]
			Let $n\geq 2$ be an integer and $a$ be a positive real number. Find the smallest positive real number $M(n,a)=M$ such that
				\begin{align*}
					\sum_{i=1}^{n}\dfrac{1}{a+S-x_{i}}
						& \geq M
				\end{align*}
			where $S=\sum_{i=1}^{n}x_{i}$ for any positive real numbers $x_{1},\ldots,x_{n}$.
		\end{problem}

		\begin{problem}[$2010$ Team Selection Test $2$]
			Let $n\geq2$ be an integer and $x_{1},\ldots,x_{n}$ be real numbers in the interval $[0,1]$. Prove that there exists real numbers $a_{0},\ldots,a_{n}$ such that
				\begin{align*}
					a_{0}+a_{n}
						& = 0\\
					|a_{i}|
						& \leq 1\\
					|a_{i}-a_{i-1}|
						& = x_{i}
				\end{align*}
		\end{problem}

		\begin{problem}[$2009$ Team Selection Test $2$]
			Let $n\geq2$ be an integer. Find the maximum constant $\lambda(n)$ so that if a sequence of real numbers $a_{0},a_{1},\ldots$ satisfies $0=a_{0}\leq a_{1}\leq\ldots\leq a_{n}$ and for $1\leq i\leq n-1$, $2a_{i}\geq a_{i-1}+a_{i+1}$ then
				\begin{align*}
					\left(\sum_{i=1}^{n}ia_{i}\right)^{2}
						& \geq \lambda(n)\sum_{i=1}^{n}a_{i}^{2}
				\end{align*}
		\end{problem}

		\begin{problem}[$2009$ Team Selection Test, problem $5$]
			Let $m>1$ be an integer and $n$ be an odd integer such that $3\leq n<2m$. Consider a matrix $(a_{ij})$ such that for any $1\leq j\leq n$, $a_{1j},a_{2j},\ldots,a_{mj}$ is a permutation of $1,2,\ldots,m$ and  for any $1\leq i\leq m$ and $1\leq j\leq n-1$, $|a_{ij}-a_{i(j+1)}|\leq1$. Find the minimum value of
				\begin{align*}
					\max\limits_{1<i<m}\sum_{j=1}^{n}a_{ij}
				\end{align*}
		\end{problem}

		\begin{problem}[$2009$ Quiz $1$, problem $3$]
			Let $m,n$ be positive integers and $x_{1},\ldots,x_{m},y_{1},\ldots,y_{n}$ be positive real numbers. Prove that
				\begin{align*}
					2XY\sum_{i=1}^{m}\sum_{j=1}^{n}|x_{i}-y_{j}|
						& \geq X^{2}\sum_{i=1}^{n}\sum_{j=1}^{n}|y_{i}-y_{j}|+Y^{2}\sum_{i=1}^{m}\sum_{j=1}^{m}|x_{i}-x_{j}|
				\end{align*}
		\end{problem}

		\begin{problem}[$2009$ Quiz $5$, problem $3$]
			Let $a_{1},a_{2},a_{3},a_{4}$ be non-negative real numbers such that $a_{1}+a_{2}+a_{3}+a_{4}=1$. Prove that
				\begin{align*}
					\max\left\{\sum_{i=1}^{4}\sqrt{a_{i}^{2}+a_{i}a_{i-1}+a_{i-1}^{2}+a_{i-1}a_{i-2}},\sum_{i=1}^{4}\sqrt{a_{i}^{2}+a_{i}a_{i+1}+a_{i+1}a_{i+2}}\right\}
						& \geq 2
				\end{align*}
			where $a_{i+4}=a_i$.
		\end{problem}

		\begin{problem}[$2008$ Team Selection Test, problem $5$]
			Let $m,n>1$ be integers and $(a_{ij})$ be a non-zero matrix of non-negative real numbers. Find the minimum and maximum value of
				\begin{align*}
					\dfrac{m\sum_{i=1}^{m}\left(\sum_{j=1}^{n}a_{ij}\right)^{2}+n\sum_{j=1}^{n}\left(\sum_{i=1}^{m}a_{ij}\right)^{2}}{\left(\sum_{i=1}^{m}\sum_{j=1}^{n}a_{ij}\right)^{2}+mn\sum_{i=1}^{m}\sum_{j=1}^{n}a_{ij}^{2}}
				\end{align*}
		\end{problem}

		\begin{problem}[$2008$ Team Selection Test, problem $6$]
			Find the maximum constant $M$ such that for any integer $n\geq 3$, there exists two sequences of positive real numbers $a_{1},\ldots,a_{n}$ and $b_{1},\ldots,b_{n}$ satisfying
				\begin{align*}
					\sum_{i=1}^{n}b_{i}
						& = 1\\
					2b_{i}
						& \geq b_{i-1}+b_{i+1}\\
					a_{k}^{2}
						& \geq 1+\sum_{i=1}^{k}a_{i}b_{i}
				\end{align*}
			and $a_{n}=M$.
		\end{problem}

		\begin{problem}[$2008$ Quiz $2$ problem $3$]
			Let $z_{1},z_{2},\ldots,z_{3}$ be complex numbers such that $|z_{i}|\leq 1$ and $\omega_{1},\omega_{2}$ are the roots of the equation
				\begin{align*}
					(z-z_{1})(z-z_{2})+(z-z_{2})(z-z_{3})+(z-z_{3})(z-z_{1})
						& = 0
				\end{align*}
			Prove that
				\begin{align*}
					\min\{|z_{j}-\omega_{1}|,|z_{j}-\omega_{2}\}
						& \leq 1
				\end{align*}
		\end{problem}

		\begin{problem}[$2008$ Quiz $4$ problem $2$]
			Let $x,y,z$ be positive real numbers. Prove that
				\begin{align*}
					\dfrac{xy}{z}+\dfrac{yz}{x}+\dfrac{zx}{y}
						& > 2\sqrt[3]{x^{3}+y^{3}+z^{3}}
				\end{align*}
		\end{problem}

		\begin{problem}[$2008$ Quiz $5$, problem $2$]
			Let $n\geq 2$ be an integer and $a_{1},\ldots,a_{n}$ be real numbers not all zero. Determine the necessary and sufficient condition so that there exists a sequence of integers $x_{1},\ldots,x_{n}$ which satisfies
				\begin{align*}
					0
						& < x_{1} <\ldots<x_{n}\\
					a_{1}x_{1}+\ldots+a_{n}x_{n}
						& \geq 0
				\end{align*}
		\end{problem}

		\begin{problem}[$2008$ Quiz $5$, problem $3$]
			Let $n$ be a positive integer and $x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}$ be real numbers such that $0<x_{1}\leq\frac{x_{2}}{2}\leq\ldots\leq\frac{x_{n}}{n}$ and $0<y_{n}\leq y_{n-1}\leq\ldots\leq y_{1}$. Prove that
				\begin{align*}
					\left(\sum_{i=1}^{n}x_{i}y_{i}\right)^{2}
						& \leq \left(\sum_{i=1}^{n}y_{i}\right)\left(\sum_{i=1}^{n}\left(x_{i}^{2}-\dfrac{1}{4}x_{i}x_{i+1}\right)y_{i}\right)
				\end{align*}
			where $x_{0}=0$.
		\end{problem}

		\begin{problem}[$2007$ Team Selection Test, problem $5$]
			Let $n>1$ be a positive integer and $x_{1},\ldots,x_{n}$ be real numbers satisfying $A=\left|\sum_{i=1}^{n}x_{i}\right|\neq0$ and $B=\max\limits_{1\leq i\leq n}|x_{i}-x_{j}|\neq0$. Prove that for any $n$ vectors $\overrightarrow{\alpha_{i}}$ in the plane, there exists a permutation $k_{1},\ldots,k_{n}$ of the numbers $1,\ldots,n$ such that
				\begin{align*}
					\left|\sum_{i=1}^{n}a_{i}x_{k_{i}}\overrightarrow{\alpha_{i}}\right|
						& \geq \dfrac{AB}{2A+B}\max_{1\leq i\leq n}|\alpha_{i}|
				\end{align*}
		\end{problem}

		\begin{problem}[$2007$ Quiz $2$, problem $1$]
			Let $u,v,w$ be positive real numbers such that
				\begin{align*}
					u+v+w+\sqrt[3]{uvw}
						& = 4
				\end{align*}
			Prove that
				\begin{align*}
					\sqrt{\dfrac{uv}{w}}+\sqrt{\dfrac{vw}{u}}+\sqrt{\dfrac{wu}{v}}
						& \geq u+v+w
				\end{align*}
		\end{problem}

		\begin{problem}[$2007$ Quiz $4$, problem $1$]
			Let $a_{1},\ldots,a_{n}$ be positive real numbers satisfying $a_{1}+\ldots+a_{n}=1$. Prove that
				\begin{align*}
					(a_{1}a_{2}+\ldots+a_{n}a_{1})\left(\dfrac{a_{1}}{a_{2}^{2}+a_{2}}+\ldots+\dfrac{a_{n}^{2}}{a_{1}^{2}+a_{1}}\right)
						& \geq \dfrac{n}{n+1}
				\end{align*}
		\end{problem}

		\begin{problem}[$2007$ Quiz $5$, problem $3$]
			Find the smallest constant $k$ such that
				\begin{align*}
					\dfrac{x}{\sqrt{x+y}}+\dfrac{y}{\sqrt{y+z}}+\dfrac{z}{\sqrt{z+x}}
						& \leq k\sqrt{x+y+z}
				\end{align*}
		\end{problem}

		\begin{problem}[$2006$ Team Selection Test, problem $3$]
			Let $n$ be a positive integer and $a_{1},\ldots,a_{n}$ be real numbers. Prove that there exists real numbers $b_{1},\ldots,b_{n}$ such that $a_{i}-b_{i}$ is a positive integer for $1\leq i\leq n$ and
				\begin{align*}
					\sum_{1\leq i<j\leq n}(b_{i}-b_{j})^{2}
						& \leq \dfrac{n^{2}-1}{12}
				\end{align*}
		\end{problem}

		\begin{problem}[$2006$ Team Selection Test, problem $8$]
			Let $n$ be a positive integer and $x_{1},\ldots,x_{n}$ be positive real numbers such that $x_{1}+\ldots+x_{n}=1$. Prove that
				\begin{align*}
					\left(\sum_{i=1}^{n}\sqrt{x_{i}}\right)\left(\sum_{i=1}^{n}\dfrac{1}{\sqrt{1+x_{i}}}\right)
						& \leq\dfrac{n^{2}}{n+1}
				\end{align*}
		\end{problem}

		\begin{problem}[$2006$ Team Selection Test, problem $11$]
			Given positive real numbers $x,y,z$ such that $x+y+z=1$. Prove that
				\begin{align*}
					\dfrac{xy}{\sqrt{xy+yz}}+\dfrac{yz}{\sqrt{yz+zx}}+\dfrac{zx}{\sqrt{zx+xy}}
						& \leq\dfrac{\sqrt{2}}{2}
				\end{align*}
		\end{problem}

		\begin{problem}[$2005$ Team Selection Test, problem $4$]
			Let $a_{1},\ldots,a_{6};b_{1},\ldots,b_{6};c_{1},\ldots,c_{6}$ be permutations of $1,\ldots,6$. Find the minimum value of $\sum_{i=1}^{6}a_{i}b_{i}c_{i}$.
		\end{problem}

		\begin{problem}[$2005$ Quiz $1$, problem $2$]
			Let $a,b,c$ be non-negative real numbers such that $ab+bc+ca=\frac{1}{3}$. Prove that
				\begin{align*}
					\dfrac{1}{a^{2}-bc+1}+\dfrac{1}{b^{2}-ca+1}+\dfrac{1}{c^{2}-ab+1}
						& \leq 3
				\end{align*}
		\end{problem}

		\begin{problem}[$2005$ Quiz $2$, problem $3$]
			Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove that
				\begin{align*}
					\dfrac{1}{(1+a)^{2}}+\dfrac{1}{(1+b)^{2}}+\dfrac{1}{(1+c)^{2}}+\dfrac{1}{(1+d)^{2}}
						& \geq 1
				\end{align*}
		\end{problem}

		\begin{problem}[$2004$ Quiz $4$, problem $2$]
			Find the greatest positive real number $k$ such that for any positive real numbers $a,b,c,d$,
				\begin{align*}
					(a+b+c)\left(3^{4}(a+b+c+d)^{5}+2^{4}(a+b+c+2d)^{5}\right)
						& \geq kabcd^{3}
				\end{align*}
		\end{problem}

		\begin{problem}[$2003$ Quiz $1$, problem $1$]
			$x,y,z$ are positive real numbers such that $x+y+z=xyz$. Find the minimum value of
				\begin{align*}
					x^{7}(yz-1)+y^{7}(zx-1)+z^{7}(xy-1)
				\end{align*}
		\end{problem}

		\begin{problem}[$2003$ Quiz $2$, problem $3$]
			Let $n$ be a positive integer and the roots of
				\begin{align*}
					f(z)
						& = z^{n}+a_{1}z^{n-1}+\ldots+a_{n}
				\end{align*}
			are $z_{1},\ldots,z_{n}$ where $a_{1},\ldots,a_{n}$ are complex numbers. If $\sum_{i=1}^{n}|a_{i}|^{2}\leq 1$, then prove that $\sum_{i=1}^{n}|z_{i}|^{2}\leq n$.
		\end{problem}

		\begin{problem}[$2003$ Quiz $6$, problem $1$]
			Let $n$ be a positive integer and $a_{1},\ldots,a_{n},x$ be real numbers.
				\begin{align*}
					g(x)
						& = \sum_{i=1}^{n}a_{i}\cos{ix}
				\end{align*}
			If $g(x)\geq -1$ for all real number $x$, then prove that $\sum_{i=1}^{n}a_{i}\leq n$.
		\end{problem}

		\begin{problem}[$2003$ Quiz $8$, problem $3$]
			Let $n\geq 2$ be an integer and $a_{1},\ldots,a_{n}$ be positive real numbers not all of which are equal such that
				\begin{align*}
					\sum_{i=1}^{n}\dfrac{1}{a_{i}^{2n}}
						& = 1
				\end{align*}
			Prove that
				\begin{align*}
					\left(\sum_{i=1}^{n}a_{i}^{2n}\right)-n^{2}\sum_{1\leq i < j\leq n}\left(\dfrac{a_{i}}{a_{j}}-\dfrac{a_{j}}{a_{i}}\right)^{2}
						& > n^{2}
				\end{align*}
		\end{problem}

		\begin{problem}[$2002$ Team Selection Test, problem $6$]
			Let
				\begin{align*}
					f(x_{1},x_{2},x_{3})
						& = -2(x_{1}^{3}+x_{2}^{3}+x_{3}^{3})+3x_{1}^{2}(x_{2}+x_{3})+3x_{2}^{2}(x_{3}+x_{1})+3x_{3}^{2}(x_{1}+x_{2})-12x_{1}x_{2}x_{3}
				\end{align*}
			For real numbers $r,s,t$, define
				\begin{align*}
					g(r,s,t)
						& = \max\limits_{t\leq x_{3}\leq t+2}|f(r, r+2, x_{3})+s|
				\end{align*}
			Find the minimum value of $g(r,s,t)$.
		\end{problem}

		\begin{problem}[$2001$ Team Selection Test, problem $4$]
			Let $n>3$ be an integer. The real numbers $x_{1},\ldots,x_{n+2}$ satisfy the condition $0<x_{1}<\ldots<x_{n+2}$. Find the minimum possible value of
				\begin{align*}
					\dfrac{\left(\sum_{i=1}^{n}\dfrac{x_{i+1}}{x_{i}}\right)\left(\sum_{i=1}^{n}\dfrac{x_{i+2}}{x_{i+1}}\right)}{\left(\sum_{i=1}^{n}\dfrac{x_{i+1}x_{i+2}}{x_{k+1}^{2}+x_{k}x_{k+2}}\right)\left(\sum_{i=1}^{n}\dfrac{x_{i+1}^{2}+x_{i}x_{i+2}}{x_{i}x_{i+1}}\right)}
				\end{align*}
		\end{problem}

		\begin{problem}[$2001$ Team Selection Test, problem $6$]
			Find the maximum value of
				\begin{align*}
					\max\limits_{1\leq x\leq 3}|x^{3}-ax^{2}-bx-c|
				\end{align*}
			where $a,b,c$ runs through all real numbers.
		\end{problem}

		\begin{problem}[$1999$ Team Selection Test, problem $1$]
			Let $n$ be a positive integer and $x_{1},\ldots,x_{n}$ be non-negative real numbers such that $x_{1}+\ldots+x_{n}=1$. Find the largest possible value of $\sum_{i=1}^{n}(x_{i}^{4}-x_{j}^{5})$.
		\end{problem}

		\begin{problem}[$1998$ Team Selection Test, problem $3$]
			For a fixed real number $\theta\in\left[0,\frac{\pi}{2}\right]$, find the smallest positive real number $a$ for which
				\begin{align*}
					\dfrac{\sqrt{a}}{\cos{\theta}}+\dfrac{\sqrt{a}}{\sin{\theta}}
						& > 1
				\end{align*}
			and there exists $x\in\left[1-\frac{\sqrt{a}}{\sin{\theta}},\cos{\theta}\right]$ such that
				\begin{align*}
					\left((1-x)\sin{\theta}-\sqrt{a-x^{2}\cos^{2}{\theta}}\right)^{2}+\left(x\cos{\theta}-\sqrt{a-(1-x)^{2}\sin{\theta}}\right)^{2}
						& \leq a
				\end{align*}
		\end{problem}

		\begin{problem}[$1996$ Team Selection Test, problem $5$]
			Let $n\geq 4$ be an integer and $\alpha_{1},\ldots,\alpha_{n},\beta_{1},\ldots,\beta_{n}$ be real numbers such that $\sum_{i=1}^{n}\alpha_{i}^{2}<1$ and $\sum_{i=1}^{n}\beta_{i}^{2}<1$.
				\begin{align*}
					A^{2}
						& = 1-\sum_{i=1}^{n}\alpha_{i}^{2}\\
					B^{2}
						& = 1-\sum_{i=1}^{n}\beta_{i}^{2}\\
					W
						& = \dfrac{1}{2}\left(1-\sum_{i=1}^{n}\alpha_{i}\beta_{i}\right)^{2}
				\end{align*}
			Find all real number $\lambda$ such that
				\begin{align*}
					x^{n}+\lambda(x^{n-1}+\ldots+x^{3}+Wx^{2}+ABx+1)
						& = 0
				\end{align*}
			only has real roots.
		\end{problem}

		\begin{problem}[$1995$ Team Selection Test, problem $4$]
			Let $n$ be a positive integer and $r_{1},\ldots,r_{n},s_{1},\ldots,s_{n},t_{1},\ldots,t_{n},u_{1},\ldots,u_{n},v_{1},\ldots,v_{n}$ be $5n$ real numbers.
				\begin{align*}
					R
						& = \dfrac{1}{n}\sum_{i=1}^{n}r_{i}\\
					S
						& = \dfrac{1}{n}\sum_{i=1}^{n}s_{i}\\
					T
						& = \dfrac{1}{n}\sum_{i=1}^{n}t_{i}\\
					U
						& = \dfrac{1}{n}\sum_{i=1}^{n}u_{i}\\
					V
						& = \dfrac{1}{n}\sum_{i=1}^{n}v_{i}
				\end{align*}
			Prove that
				\begin{align*}
					\prod_{i=1}^{n}\dfrac{r_{i}s_{i}t_{i}u_{i}v_{i}+1}{r_{i}s_{i}t_{i}u_{i}v_{i}-1}
						& \geq \left(\dfrac{RSTUV+1}{RSTUV-1}\right)^{n}
				\end{align*}
		\end{problem}

		\begin{problem}[$1993$ Team Selection Test, problem $2$]
			Let $n\geq 2$ be an integer and $a,b,c,d$ be positive integers such that $\frac{a}{b}+\frac{c}{d}<1$ and $a+c\leq n$. Find the maximum value of $\frac{a}{b}+\frac{c}{d}$ for a fixed $n$.
		\end{problem}
\end{document}


\section{Question 2}
System is:
$$
G(s) = \frac{-s + 3}{s(s+2)(s^2+2s+4)}
$$
\subsection{part a}
Integral plus Time Delay:
$$
G = e^{-1.15s}\dfrac{0.375}{s(0.378s +  1)}
$$
\begin{figure}[H]
    \caption{system and ITD step responde}
    \centering
    \includegraphics[width=12cm]{../Figure/Q2/IDT1.png}
\end{figure}
Now we use foipdt function.
\begin{figure}[H]
    \caption{step responde with foipdt PD controller}
    \centering
    \includegraphics[width=12cm]{../Figure/Q2/foipdt_PD.png}
\end{figure}
\begin{figure}[H]
    \caption{step responde with foipdt PID controller}
    \centering
    \includegraphics[width=12cm]{../Figure/Q2/foipdt_PID.png}
\end{figure}
\subsection{part b}
Integral plus Time Delay:
$$
G = e^{-1.33s}\dfrac{0.375}{s}
$$
\begin{figure}[H]
    \caption{system and ITD step responde}
    \centering
    \includegraphics[width=12cm]{../Figure/Q2/IDT2.png}
\end{figure}
Now we use ipdtctrl function.
\begin{figure}[H]
    \caption{step responde with ipdtctrl for ISE PD controller}
    \centering
    \includegraphics[width=12cm]{../Figure/Q2/pd1.png}
\end{figure}
\begin{figure}[H]
    \caption{step responde with ipdtctrl for ITSE PD controller}
    \centering
    \includegraphics[width=12cm]{../Figure/Q2/pd2.png}
\end{figure}
\begin{figure}[H]
    \caption{step responde with ipdtctrl for ISTSE PD controller}
    \centering
    \includegraphics[width=12cm]{../Figure/Q2/pd3.png}
\end{figure}

\begin{figure}[H]
    \caption{step responde with ipdtctrl for ISE PID controller}
    \centering
    \includegraphics[width=12cm]{../Figure/Q2/pid1.png}
\end{figure}
\begin{figure}[H]
    \caption{step responde with ipdtctrl for ITSE PID controller}
    \centering
    \includegraphics[width=12cm]{../Figure/Q2/pid2.png}
\end{figure}
\begin{figure}[H]
    \caption{step responde with ipdtctrl for ISTSE PID controller}
    \centering
    \includegraphics[width=12cm]{../Figure/Q2/pid3.png}
\end{figure}

\subsection{conclusion}
Firt ITD function doesn't fit well but PD controller work very good but PID controller work very bad too slow for setteling time and has a high overshoot. system has integrator so we don't need to have integrator in controller and just make system slow and pendulous.
Second ITD fits very good bad PD and PID controller don't work well.
PD is far better than PID and PID is too slow and pendulous.
In final PD controller used first method is the best fot this system.