Given a quadrilateral $ABCD$ such that $AB^2 + CD^2 = AD^2 + BC^2$, prove that $AC \perp BD$.

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 $\overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{OR}$.

Let $v_1, v_2, \ldots, v_m$ be vectors in $\mathbb{R}^n$, such that each has a strictly positive first coordinate. Consider the following process. Start with the zero vector $w = (0, 0, \ldots, 0) \in \mathbb{R}^n$. Every round, choose an $i$ such that $1 \le i \le m$ and $w \cdot v_i \le 0$, and then replace $w$ with $w + v_i$. Show that there exists a constant $C$ such that regardless of your choice of $i$ at each step, the process is guaranteed to terminate in (at most) $C$ rounds. The constant $C$ may depend on the vectors $v_1, \ldots, v_m$.

Find the sum of the indexes of the singular points other than zero of the vector field \[z\overline{z}^2+z^4+2\overline{z}^4\]

If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' \equal{} \frac {a_k \plus{} a_{k \plus{} 1}}2$ the [i]average sequence[/i] of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - [i]average sequence[/i] of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called [i]Good[/i]. Prove that if $ \{x_k\}$ is a [i]good[/i] sequence, then $ \{x_k^2\}$ is also [i]good[/i].

[i]Introductory part [/i] We call an $n$-tuple $x = (x_1, x_2, ... , x_n)$, with $x_k \in R$ (or respectively with all $x_k \in Z$) a real vector (or respectively an integer vector). The set of all real vectors (respectively all integer vectors) is usually denoted by $R^n$ (respectively $Z^n$). A vector $x$ is null if and only if $x_k = 0$, for all $k \in \{1, 2,... , n\}$. Also let $U_n$ be the set of all real vectors $x = (x_1, x_2, ... , x_n)$, such that $x^2_1 + x^2_2 + ...+ x^2_n = 1$. For two vectors $x = (x_1, ... , x_n), y = (y_1, ..., y_n)$ we define the scalar product as the real number $x\cdot y = x_1y_1 + x_2y_2 +...+ x_ny_n$. We define the norm of the vector $x$ as $||x|| =\sqrt{x^2_1 + x^2_2 + ...+ x^2_n}$ [i]The problem[/i] Let $A(k, r) = \{x \in U_n |$ for all $z \in Z^n$ we have either $|x \cdot z| \ge \frac{k}{||z||^r}$ or $z$ is null $\}$. Prove that if $r > n - 1$ the we can find a positive number $k$ such that $A(k, r)$ is not empty, and if $r < n - 1$ we cannot find such a positive number $k$.

Let $n\in\mathbb{N}^*$ and $v_1,v_2,\ldots ,v_n$ be vectors in the plane with lengths less than or equal to $1$. Prove that there exists $\xi_1,\xi_2,\ldots ,\xi_n\in\{-1,1\}$ such that \[ | \xi_1v_1+\xi_2v_2+\ldots +\xi_nv_n|\le\sqrt{2}\]

Let $n\geq 2$ be an integer and denote by $H_{n}$ the set of column vectors $^{T}(x_{1},\ x_{2},\ \ldots, x_{n})\in\mathbb{R}^{n}$, such that $\sum |x_{i}|=1$. Prove that there exist only a finite number of matrices $A\in\mathcal{M}_{n}(\mathbb{R})$ such that the linear map $f: \mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ given by $f(x)=Ax$ has the property $f(H_{n})=H_{n}$. [hide="Comment"]In the contest, the problem was given with a) and b): a) Prove the above for $n=2$; b) Prove the above for $n\geq 3$ as well.[/hide]

Decompose a $5$-dimensional real linear space into the irreducible invariant subspaces of the group generated by cyclic permutations of the basis vectors.

Suppose tetrahedron $PABC$ has volume $420$ and satisfies $AB = 13$, $BC = 14$, and $CA = 15$. The minimum possible surface area of $PABC$ can be written as $m+n\sqrt{k}$, where $m,n,k$ are positive integers and $k$ is not divisible by the square of any prime. Compute $m+n+k$. [i]Ray Li[/i]

Let $k$ and $N$ be positive real numbers which satisfy $k\leq N$. For $1\leq i \leq k$, there are subsets $A_i$ of $\{1,2,3,\ldots,N\}$ that satisfy the following property. For arbitrary subset of $\{ i_1, i_2, \ldots , i_s \} \subset \{ 1, 2, 3, \ldots, k \} $, $A_{i_1} \triangle A_{i_2} \triangle ... \triangle A_{i_s}$ is not an empty set. Show that a subset $\{ j_1, j_2, .. ,j_t \} \subset \{ 1, 2, ... ,k \} $ exist that satisfies $n(A_{j_1} \triangle A_{j_2} \triangle \cdots \triangle A_{j_t}) \geq k$. ($A \triangle B=A \cup B-A \cap B$)

Suppose a $3\times 3$ matrix $A$ satisfies $\mathbf{v}^t A \mathbf{v} > 0$ for any vector $\mathbf{v} \in\mathbb{R}^3 -\{0\}$. (Note that $A$ may not be a symmetric matrix.) (1) Prove that $\det(A)>0$. (2) Consider diagonal matrix $D=\text{diag}(-1,1,1)$. Prove that there's exactly one negative real among eigenvalues of $AD$.

Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of triangle $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle{BAP}= \angle{CAQ}$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcenter of triangle $BCR$. Prove that $AO \perp PQ.$

Let $m,n\in\mathbb{Z_{+}}$ be such numbers that set $\{1,2,\ldots,n\}$ contains exactly $m$ different prime numbers. Prove that if we choose any $m+1$ different numbers from $\{1,2,\ldots,n\}$ then we can find number from $m+1$ choosen numbers, which divide product of other $m$ numbers.