Given acute angle triangle $ ABC$. Let $ CD$be the altitude , $ H$ be the orthocenter and $ O$ be the circumcenter of $ \triangle ABC$ The line through point $ D$ and perpendicular with $ OD$ , is intersect $ BC$ at $ E$. Prove that $ \angle DHE = \angle ABC$.

Let $ a,b,c,d$ be the positive integers such that $ a > b > c > d$ and $ (a + b - c + d) | (ac + bd)$ . Prove that if $ m$ is arbitrary positive integer , $ n$ is arbitrary odd positive integer, then $ a^n b^m + c^m d^n$ is composite number

Given a circumscribed trapezium $ ABCD$ with circumcircle $ \omega$ and 2 parallel sides $ AD,BC$ ($ BC<AD$). Tangent line of circle $ \omega$ at the point $ C$ meets with the line $ AD$ at point $ P$. $ PE$ is another tangent line of circle $ \omega$ and $ E\in\omega$. The line $ BP$ meets circle $ \omega$ at point $ K$. The line passing through the point $ C$ paralel to $ AB$ intersects with $ AE$ and $ AK$ at points $ N$ and $ M$ respectively. Prove that $ M$ is midpoint of segment $ CN$.

Find all function $ f: R^+ \rightarrow R^+$ such that for any $ x,y,z \in R^+$ such that $ x+y \ge z$ , $ f(x+y-z) +f(2\sqrt{xz})+f(2\sqrt{yz}) = f(x+y+z)$

Given positive integers$ m,n$ such that $ m < n$. Integers $ 1,2,...,n^2$ are arranged in $ n \times n$ board. In each row, $ m$ largest number colored red. In each column $ m$ largest number colored blue. Find the minimum number of cells such that colored both red and blue.

Find the maximum number $ C$ such that for any nonnegative $ x,y,z$ the inequality $ x^3 + y^3 + z^3 + C(xy^2 + yz^2 + zx^2) \ge (C + 1)(x^2 y + y^2 z + z^2 x)$ holds.

Given an integer $ a$. Let $ p$ is prime number such that $ p|a$ and $ p \equiv \pm 3 (mod8)$. Define a sequence $ \{a_n\}_{n = 0}^\infty$ such that $ a_n = 2^n + a$. Prove that the sequence $ \{a_n\}_{n = 0}^\infty$ has finitely number of square of integer.

The quadrilateral $ ABCD$ inscribed in a circle wich has diameter $ BD$. Let $ A',B'$ are symmetric to $ A,B$ with respect to the line $ BD$ and $ AC$ respectively. If $ A'C \cap BD = P$ and $ AC\cap B'D = Q$ then prove that $ PQ \perp AC$

Given positive integers $ m,n > 1$. Prove that the equation $ (x + 1)^n + (x + 2)^n + ... + (x + m)^n = (y + 1)^{2n} + (y + 2)^{2n} + ... + (y + m)^{2n}$ has finitely number of solutions $ x,y \in N$

How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3?

Let $ a_1,a_2,...,a_n$ is permutaion of $ 1,2,...,n$. For this permutaion call the pair $ (a_i,a_j)$ wrong pair if $ i<j$ and $ a_i >a_j$.Let number of inversion is number of wrong pair of permutation $ a_1,a_2,a_3,..,a_n$. Let $ n \ge 2$ is positive integer. Find the number of permutation of $ 1,2,..,n$ such that its number of inversion is divisible by $ n$.

Click for solution Group the $ n!$ permutations into $ (n - 1)!$ classes of $ n$ elements each where two permutations $ p, q$ are in the same class if and only if $ 1 \leq i < j \leq n - 1$ implies that $ i$ occurs before $ j$ in $ p$ if and only if $ i$ occurs before $ j$ in $ q$. In other words, group permutations by what they look like if you erase $ n$. For a permutation $ p$ in a given class, all inversions fall into two categories: there are $ k$ inversions ($ k$ dependent only on the choice of class) among the entries $ 1, 2, \ldots, n - 1$ of the permutation and there are some other number of inversions between $ n$ and the entries of $ p$ that follow it. As a result, there is one permutation in the class with $ k$ inversions (when $ n$ appears last), one with $ k + 1$ inversions (when $ n$ appears next-to-last), etc., and one permutation with $ k + (n - 1)$ inversions (when $ n$ appears first). Exactly one of these numbers is divisible by $ n$, so exactly $ (n - 1)!$ permutations have inversion number divisible by $ n$. (In fact, we proved the stronger result that for any $ i$, there are exactly $ (n - 1)!$ permutations in $ S_n$ whose inversion number is congruent to $ i$ modulo $ n$.)

Let $ \Omega$ is circle with radius $ R$ and center $ O$. Let $ \omega$ is a circle inside of the $ \Omega$ with center $ I$ radius $ r$. $ X$ is variable point of $ \omega$ and tangent line of $ \omega$ pass through $ X$ intersect the circle $ \Omega$ at points $ A,B$. A line pass through $ X$ perpendicular with $ AI$ intersect $ \omega$ at $ Y$ distinct with $ X$.Let point $ C$ is symmetric to the point $ I$ with respect to the line $ XY$.Find the locus of circumcenter of triangle $ ABC$ when $ X$ varies on $ \omega$