For $ n \in \mathbb{N}$, let $ f(n)\equal{}1^n\plus{}2^{n\minus{}1}\plus{}3^{n\minus{}2}\plus{}...\plus{}n^1$. Determine the minimum value of: $ \frac{f(n\plus{}1)}{f(n)}.$

Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be positive integers from set $\{1, 2,..., n\}$ such that every number from this set occurs exactly once. Is it possible that numbers $a_1$, $a_1 + a_2 ,..., a_1 + a_2 + ... + a_n$ all have different remainders upon division by $n$, if: $a)$ $n=7$ $b)$ $n=8$

In a convex quadrilateral $ABCD$ we draw the bisectors $\ell_a$, $\ell_b$, $\ell_c$, $\ell_d$ of external angles $A$, $B$, $C$, $D$ respectively. The intersection points of the lines $\ell_a$ and $\ell_b$, $\ell_b$ and $\ell_c$, $\ell_c$ and $\ell_d$, $\ell_d$ and $\ell_a$ are designated by $K$, $L$, $M$, $N$. It is known that $3$ perpendiculars drawn from $K$ on $AB$, from $L$ om $BC$, from $M$ on $CD$ intersect at one point. Prove that the quadrilateral $ABCD$ is cyclic.

From the triangle $ABC$, are gicen only the incenter $I$, the touchpoint $K$ of the inscribed circle with the side $AB$, as well as the center $I_a$ of the exscribed circle, that touches the side $BC$ . Construct a triangle equal in size to triangle $ABC$. (Gryhoriy Filippovskyi)

Let $a$ and $b$ be complex non-zero numbers and $z_1,z_2$ the roots of the polynomials $X^2+aX+b$. Show that $|z_1+z_2|=|z_1|+|z_2|$ if and only if there exists a real number $\lambda\ge 4$ such that $a^2=\lambda b$.

We are given an arbitrary $\bigtriangleup ABC$. a) Can we dissect $\bigtriangleup ABC$ in $4$ pieces, from which we can make two triangle similar to $\bigtriangleup ABC$ (each piece can be used only once)? Justify your answer! b) Is it possible that for every positive integer $n \ge 2$ , we are able to dissect $\bigtriangleup ABC$ in $2n$ pieces, from which we can make two triangles similar to $\bigtriangleup ABC$ (each piece can be used only once)? Justify your answer!

Let $ABC$ be a triangle such that $\angle BAC = 90^\circ$ and $AB < AC$. We divide the interior of the triangle into the following six regions: \begin{align*} S_1=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PB<PC \\ S_2=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PC<PB \\ S_3=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PB<PA<PC \\ S_4=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PB<PC<PA \\ S_5=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PC<PA<PB \\ S_6=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PC<PB<PA\end{align*} Suppose that the ratio of the area of the largest region to the area of the smallest non-empty region is $49 : 1$. Determine the ratio $AC : AB$.

Find all functions $f$ from positive integers to themselves, such that the followings hold. $1)$.for each positive integer $n$ we have $f(n)<f(n+1)<f(n)+2020$. $2)$.for each positive integer $n$ we have $S(f(n))=f(S(n))$ where $S(n)$ is the sum of digits of $n$ in base $10$ representation.

Two isosceles trapezoids are inscribed in a circle in such a way that each side of each trapezoid is parallel to a certain side of the other trapezoid . Prove that the diagonals of one trapezoid are equal to the diagonals of the other.

Find all functions $f : \mathbb{N} \to \mathbb{N}$ satisfying the following conditions: [list] [*]For every $n \in \mathbb{N}$, $f^{(n)}(n) = n$. (Here $f^{(1)} = f$ and $f^{(k)} = f^{(k-1)} \circ f$.) [*]For every $m, n \in \mathbb{N}$, $\lvert f(mn) - f(m) f(n) \rvert < 2017$. [/list]

Consider $m+1$ horizontal and $n+1$ vertical lines ($m,n\ge 4$) in the plane forming an $m\times n$ table. Cosider a closed path on the segments of this table such that it does not intersect itself and also it passes through all $(m-1)(n-1)$ interior vertices (each vertex is an intersection point of two lines) and it doesn't pass through any of outer vertices. Suppose $A$ is the number of vertices such that the path passes through them straight forward, $B$ number of the table squares that only their two opposite sides are used in the path, and $C$ number of the table squares that none of their sides is used in the path. Prove that \[A=B-C+m+n-1.\] [i]Proposed by Ali Khezeli[/i]

Let $ABC$ be an acute scalene triangle with altitudes $AE$ $(E \in BC)$ and $BD$ $(D \in AC)$. Point $M$ lies on $AC$, such that $AM = AE$ and $C,A$ and $M$ lie in this order. Point $L$ lies on $BC$, such that $BL=BD$ and $C,B$ and $L$ lie in this order. Let $P$ be the midpoint of $DE$. Prove that $EM,DL$ and the perpendicular from $P$ to $AB$ are concurrent.

Evaluate $ \int_0^\frac{\pi}{4} \frac{x\plus{}\sin x}{1\plus{}\cos x}\ dx$.

Given a cyclic quadrilateral $ABCD$. Suppose $E, F, G, H$ are respectively the midpoint of the sides $AB, BC, CD, DA$. The line passing through $G$ and perpendicular on $AB$ intersects the line passing through $H$ and perpendicular on $BC$ at point $K$. Prove that $\angle EKF = \angle ABC$.

$$F(x)=x^{6}+15x^{5}+85x^{4}+225x^{3}+274x^{2}+120x+1$$

(a) In the plane of the triangle $ABC$, find a point with the following property: its symmetrical points with respect to the midpoints of the sides of the triangle lie on the circumscribed circle. (b) Construct the triangle $ABC$ if it is known the positions of the orthocenter $H$, midpoint of the side $AB$ and the midpoint of the segment joining the feet of the heights through vertices $A$ and $B$.

Define the product $P_n=1! \cdot 2!\cdot 3!\cdots (n-1)!\cdot n!$ a) Find all positive integers $m$, such that $\frac {P_{2020}}{m!}$ is a perfect square. b) Prove that there are infinite many value(s) of $n$, such that $\frac {P_{n}}{m!}$ is a perfect square, for at least two positive integers $m$.