Let $ x_1,x_2,..,x_n$ be non-negative numbers whose sum is $ 1$ . Show that there exist numbers $ a_1,a_2,\ldots ,a_n$ chosen from amongst $ 0,1,2,3,4$ such that $ a_1,a_2,\ldots ,a_n$ are different from $ 2,2,\ldots ,2$ and $ 2\leq a_1x_1\plus{}a_2x_2\plus{}\ldots\plus{}a_nx_n\leq 2\plus{}\frac{2}{3^n\minus{}1}$.

Two triangles $ ABC$and $ A'B'C'$ are positioned in the space such that the length of every side of $ \triangle ABC$ is not less than $ a$, and the length of every side of $ \triangle A'B'C'$ is not less than $ a'$. Prove that one can select a vertex of $ \triangle ABC$ and a vertex of $ \triangle A'B'C'$ so that the distance between the two selected vertices is not less than $ \sqrt {\frac {a^2 \plus{} a'^2}{3}}$.

Determine whether or not there exist two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that every $x$ with $0 < x < 1$ satisfies the following inequality: \[\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| < (1-x)^{2011}.\]

Let $x_1,\cdots, x_n$ be nonzero vectors of a vector space $V$ and $\varphi:V\to V$ be a linear transformation such that $\varphi x_1 = x_1$, $\varphi x_k = x_k - x_{k-1}$ for $k = 2, 3,\ldots,n$. Prove that the vectors $x_1,\ldots,x_n$ are linearly independent.

Do there exist polynomials $a(x)$, $b(x)$, $c(y)$, $d(y)$ such that \[1 + xy + x^2y^2= a(x)c(y) + b(x)d(y)\] holds identically?

A large cube is formed by stacking $27$ unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is [asy] size(120); defaultpen(linewidth(0.7)); pair slant = (2,1); for(int i = 0; i < 4; ++i) draw((0,i)--(3,i)^^(i,0)--(i,3)^^(3,i)--(3,i)+slant^^(i,3)--(i,3)+slant); for(int i = 1; i < 4; ++i) draw((0,3)+slant*i/3--(3,3)+slant*i/3^^(3,0)+slant*i/3--(3,3)+slant*i/3);[/asy] $\textbf{(A)}\ 16\qquad \textbf{(B)}\ 17 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 19 \qquad \textbf{(E)}\ 20$

Find all vectors of $n$ real numbers $(x_1,x_2,\ldots,x_n)$ such that \[ \left\{ \begin{array}{ccc} x_1 & = & \dfrac 1{x_2} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n } \\ x_2 & = & \dfrac 1{x_1} + \dfrac 1{x_3} + \cdots + \dfrac 1{x_n} \\ \ & \cdots & \ \\ x_n & = & \dfrac 1{x_1} + \dfrac 1{x_2} + \cdots + \dfrac 1{x_{n-1}} \end{array} \right. \] [i]Mircea Becheanu[/i]

Last year, Master Cheung is famous for multi-rotation. This year, he comes to DAMO to make noodles for sweeping monk. One day, software engineer Xiao Li talks with Master Cheung about his job. Xiao Li mainly researches and designs the algorithm to adjust the paramter of different kinds of products. These paramters can normally be obtainly by minimising loss function $f$ on $\mathbb{R}^n$. In the recent project of Xiao Li, this loss function is obtained by other topics. For safety consideration and technique reasons, this topic makes Xiao Li difficult to find the interal details of the function. They only provide a port to calculate the value of $f(\text x)$ for any $\text x\in\mathbb{R}^n$. Therefore, Xiao Li must only use the value of the function to minimise $f$. Also, every times calculating the value of $f$ will use a lot of calculating resources. It is good to know that the dimension $n$ is not very high (around $10$). Also, colleague who provides the function tells Xiao Li to assume $f$ is smooth first. This problem reminds Master Cheung of his antique radio. If you want to hear a programme from the radio, you need to turn the knob of the radio carefully. At the same time, you need to pay attention to the quality of the radio received, until the quality is the best. In this process, no one knows the relationship between the angle of turning the knob and the quality of the radio received. Master Cheung and Xiao Li realizes that minimising $f$ is same as adjusting the machine with multiple knobs: Assume every weight of $\text x$ is controlled by a knob. $f(\text x)$ is a certain performance of the machine. We only need to adjust every knobs again and again and observes the value of $f$ in the same time. Maybe there is hope to find the best $\text x$. As a result, two people suggest an iteration algorithm (named Automated Forward/Backward Tuning, $\text{AFBT}$, to minimise $f$. In $k$-th iteration, the algorithm adjusts the individual weight of $\text{x}_k$ to $2n$ points $\{\text x_k\pm t_k\text e^i:i=1,...,n\}$, where $t_k$ is the step size; then, make $y_k$ be the smallest one among the value of the function of thosse points. Then check if $\text y_k$ sufficiently makes $f$ decrease; then, take $\text x_{k+1}=\text y_k$, then make the step size doubled. Otherwise, make $\text x_{k+1}=\text x_k$ and makes the step size decrease in half. In the algorithm, $\text e^i$ is the $i$-th coordinate vector in $\mathbb{R}^n$. The weight of $i$-th is $1$. Others are $0$; $\mathbf{1}(\cdot)$ is indicator function. If $f(\text x_k)-f(\text y_k)$ is at least the square of $t_k$, then take the value of $\mathbf{1}(f(\text k)-f(y_k)\ge t^2_k)$ as $1$. Otherwise, take it as $0$. $\text{AFBT}$ algorithm Input $\text{x}_0\in \mathbb{R}^n$, $t_0>0$. For $k=0, 1, 2, ...$, perform the following loop: 1: #Calculate loss function. 2: $s_k:=\mathbb{1}[f(\text{x}_k)-f(\text{y}_k)\ge t^2_k]$ #Is it sufficiently decreasing? Yes: $s_k=1$; No: $s_k=0$. 3: $\text{x}_{k+1}:=(1-s_k)\text{x}_k+s_k\text{y}_k$ #Update the point of iteration. 4: $t_{k+1}:=2^{2S_k-1}t_k$ #Update step size. $s_k=1$: Step size doubles; $s_k=0$: Step size decreases by half. Now, we made assumption to the loss function $f:\mathbb{R}^n\to \mathbb{R}$. Assumption 1. Let $f$ be a convex function. For any $\text{x}, \text{y}\in \mathbb{R}^n$ and $\alpha \in [0, 1]$, we have $f((1-\alpha)\text{x}+\text{y})\le (1-\alpha)f(\text{x})+\alpha f(\text{y})$. Assumption 2. $f$ is differentiable on $\mathbb{R}^n$ and $\nabla f$ is L-Lipschitz continuous on $\mathbb{R}^n$. Assumption 3. The level set of $f$ is bounded. For any $\lambda\in\mathbb{R}$, set $\{\text x\in \mathbb{R}^n:f(\text x)\le \lambda\}$ is all bounded. Based on assumption 1 and 2, we can prove that $\left\langle \nabla f(\text x),\text y-\text x \right\rangle \le f(\text y)-f(\text x)\le \left\langle \nabla f(\text x),\text y-\text x\right\rangle+\frac{L}{2}||\text x-\text y||^2$ You can refer to any convex analysis textbook for more properties of convex function. Prove that under the assumption 1-3, for $AFBT$, $\lim_{k \to \infty}f(\text{x}_k)=f^*$

Let $ABC$ be a triangle and $D$, $E$, $F$ be the midpoints of arcs $BC$, $CA$, $AB$ on the circumcircle. Line $\ell_a$ passes through the feet of the perpendiculars from $A$ to $DB$ and $DC$. Line $m_a$ passes through the feet of the perpendiculars from $D$ to $AB$ and $AC$. Let $A_1$ denote the intersection of lines $\ell_a$ and $m_a$. Define points $B_1$ and $C_1$ similarly. Prove that triangle $DEF$ and $A_1B_1C_1$ are similar to each other.

Prove that if a vector space is the union of some of it's proper subspaces, then number of these subspaces can not be less than the number of elements of the field of that vector space.

A marker is placed at the origin of an integer lattice. Calvin and Hobbes play the following game. Calvin starts the game and each of them takes turns alternatively. At each turn, one can choose two (not necessarily distinct) integers $a, b$, neither of which was chosen earlier by any player and move the marker by $a$ units in the horizontal direction and $b$ units in the vertical direction. Hobbes wins if the marker is back at the origin any time after the first turn. Prove or disprove that Calvin can prevent Hobbes from winning. Note: A move in the horizontal direction by a positive quantity will be towards the right, and by a negative quantity will be towards the left (and similar directions in the vertical case as well).

Three equilateral triangles $ABC, CDE, EHK$ (the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors $\overrightarrow{AD}$ and $\overrightarrow{DK}$ are equal. Prove that the triangle $BHD$ is also equilateral

Let $\alpha_{1}, \alpha_{2}, \cdots , \alpha_{n}$ be numbers in the interval $[0, 2\pi]$ such that the number $\displaystyle\sum_{i=1}^n (1 + \cos \alpha_{i})$ is an odd integer. Prove that \[\displaystyle\sum_{i=1}^n \sin \alpha_i \ge 1\]

Let $\mathbb{M}$ be the vector space of $m \times p$ real matrices. For a vector subspace $S\subseteq \mathbb{M}$, denote by $\delta(S)$ the dimension of the vector space generated by all columns of all matrices in $S$. Say that a vector subspace $T\subseteq \mathbb{M}$ is a $\emph{covering matrix space}$ if \[ \bigcup_{A\in T, A\ne \mathbf{0}} \ker A =\mathbb{R}^p \] Such a $T$ is minimal if it doesn't contain a proper vector subspace $S\subset T$ such that $S$ is also a covering matrix space. [list] (a) (8 points) Let $T$ be a minimal covering matrix space and let $n=\dim (T)$ Prove that \[ \delta(T)\le \dbinom{n}{2} \] (b) (2 points) Prove that for every integer $n$ we can find $m$ and $p$, and a minimal covering matrix space $T$ as above such that $\dim T=n$ and $\delta(T)=\dbinom{n}{2}$[/list]

Let $P$ be a polynomial with each root is real and each coefficient is either $1$ or $-1$. The degree of $P$ can be at most ? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

Given 5 nonzero vectors in three-dimensional Euclidean space, prove that the sum of their pairwise angles is at most $6\pi$.

Triangle $ABC$ is inscribed in circle $\Omega$. The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$), respectively. Let $M$ be the midpoint of side $BC$. The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. Let $N$ be the midpoint of segment $PQ$, and let $H$ be the foot of the perpendicular from $L$ to line $ND$. Prove that line $ML$ is tangent to the circumcircle of triangle $HMN$.

Let $P_0, P_1,\ldots, P_{2021}$ points on the unit circle of centre $O$ such that for each $n\in \{1,2,\ldots, 2021\}$ the length of the arc from $P_{n-1}$ to $P_n$ (in anti-clockwise direction) is in the interval $\left[\frac{\pi}2,\pi\right]$. Determine the maximum possible length of the vector: \[\overrightarrow{OP_0}+\overrightarrow{OP_1}+\ldots+\overrightarrow{OP_{2021}}.\] [i]Mihai Iancu[/i]

Given parallelogram $OABC$ in the coodinate with $O$ the origin and $A,B,C$ be lattice points. Prove that for all lattice point $P$ in the internal or boundary of $\triangle ABC$, there exists lattice points $Q,R$(can be the same) in the internal or boundary of $\triangle OAC$ with $\overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{OR}$.

On a board are drawn the points $A,B,C,D$. Yetti constructs the points $A^\prime,B^\prime,C^\prime,D^\prime$ in the following way: $A^\prime$ is the symmetric of $A$ with respect to $B$, $B^\prime$ is the symmetric of $B$ wrt $C$, $C^\prime$ is the symmetric of $C$ wrt $D$ and $D^\prime$ is the symmetric of $D$ wrt $A$. Suppose that Armpist erases the points $A,B,C,D$. Can Yetti rebuild them? $\star \, \, \star \, \, \star$ [b]Note.[/b] [i]Any similarity to real persons is purely accidental.[/i]

Given that $a,b,c,d,e$ are real numbers such that $a+b+c+d+e=8$, $a^2+b^2+c^2+d^2+e^2=16$. Determine the maximum value of $e$.

There are $n > 1$ points on the plane. Two take turns connecting more an unconnected pair of points by a vector of one of two possible directions. If after the next move of a player the sum of all drawn vectors is zero, then the second one wins; if it's another move is impossible, and there was no zero sum, then the first one wins. Who wins when played correctly?

A lamp is situated at point $A$ and shines inside the cube. A (massive) square is hung on the midpoints of the 4 vertical faces. What's the area of its shadow? [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=285[/img]

$f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is a bijective map, that Image of every $n-1$-dimensional affine space is a $n-1$-dimensional affine space. 1) Prove that Image of every line is a line. 2) Prove that $f$ is an affine map. (i.e. $f=goh$ that $g$ is a translation and $h$ is a linear map.)

Define in the plane the sequence of vectors $v_1, v_2, \ldots$ with initial values $v_1 = (1, 0)$, $v_2 = (-1/\sqrt{2}, 1/\sqrt{2})$ and satisfying the relationship $$v_n=\frac{v_{n-1}+v_{n-2}}{\lVert v_{n-1}+v_{n-2}\rVert},$$ for $n \geq 3$. Show that the sequence is convergent and determine its limit. [b]Note:[/b] The expression $\lVert v \rVert$ denotes the length of the vector $v$.