Let $ n > 4$ be a non-negative integer. Given is the in a circle inscribed convex $ n$-gon $ A_0A_1A_2\dots A_{n \minus{} 1}A_n$ $ (A_n \equal{} A_0)$ where the side $ A_{i \minus{} 1}A_i \equal{} i$ (for $ 1 \le i \le n$). Moreover, let $ \phi_i$ be the angle between the line $ A_iA_{i \plus{} 1}$ and the tangent to the circle in the point $ A_i$ (where the angle $ \phi_i$ is less than or equal $ 90^o$, i.e. $ \phi_i$ is always the smaller angle of the two angles between the two lines). Determine the sum $ \Phi \equal{} \sum_{i \equal{} 0}^{n \minus{} 1} \phi_i$ of these $ n$ angles.

is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?

Three groups $A, B$ and $C$ of mathematicians from different countries have invited to a ceremony. We have formed meetings such that three mathematicians participate in every meeting and there is exactly one mathematician from each group in every meeting. Also every two mathematicians have participated in exactly one meeting with each other. [b](a)[/b] Prove that if this is possible, then number of mathematicians of the groups is equal. [b](b)[/b] Prove that if there exist $3$ mathematicians in each group, then that work is possible. [b](c)[/b] Prove that if number mathematicians of the groups be equal, then that work is possible.

There are 12 gold stars arranged in a circle on a blue background. Giorgi wants to label each star with one of the letters $G$, $E$, or $O$, such that no two consecutive stars have the same letter. Determine the number of distinct ways Giorgi can label the stars. [img]https://i.imgur.com/qIxdJ8j.jpeg[/img] [i]Proposed by Giorgi Arabidze, Georgia [/i]

Four orange lights are located at the points $(2,0)$, $(4,0)$, $(6,0)$ and $(8,0)$ in the $xy$-plane. Four yellow lights are located at the points $(1,0)$, $(3,0)$, $(5,0)$, $(7,0)$. Sparky chooses one or more of the lights to turn on. In how many ways can he do this such that the collection of illuminated lights is symmetric around some line parallel to the $y$-axis? [i]Proposed by Evan Chen[/i]

Let $P$ be the product of any three consecutive positive odd integers. The largest integer dividing all such $P$ is: $\textbf{(A)}\ 15 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 1 $

Let $ABC$ arbitrary triangle ($AB \neq BC \neq AC \neq AB$) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove, that 1) $\angle OIH > 90^0$(2 points) 2)$\angle OIH >135^0$(7 points) balls for 1) and 2) not additive.

Rothschild the benefactor has a certain number of coins. A man comes, and Rothschild wants to share his coins with him. If he has an even number of coins, he gives half of them to the man and goes away. If he has an odd number of coins, he donates one coin to charity so he can have an even number of coins, but meanwhile another man comes. So now he has to share his coins with two other people. If it is possible to do so evenly, he does so and goes away. Otherwise, he again donates a few coins to charity (no more than 3). Meanwhile, yet another man comes. This goes on until Rothschild is able to divide his coins evenly or until he runs out of money. Does there exist a natural number $N$ such that if Rothschild has at least $N$ coins in the beginning, he will end with at least one coin?

The sequence of integers $a_1,a_2,a_3 ,.. $such that $a_1 = 1$, $a_2 = 2$ and for every natural $n \ge 1$ $$a_{n+2}=\begin{cases} 2001a_{n+1} - 1999a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,even\,\,number} /\\ a_{n+1}-a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,odd\,\,number} \end{cases}$$ Is there such a natural $m$ that $a_m= 2000$?

Express $\sum_{k=1}^\infty\frac{6^k}{(3^{k+1}-2^{k+1})(3^k-2^k)}$ as a rational number.

A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy? $ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15 $

Let $a_{n}=|n(n+1)-19|$ for $n=0, 1, 2, ...$ and $n \neq 4$. Prove that if for every $k<n$ we have $\gcd(a_{n}, a_{k})=1$, then $a_{n}$ is a prime number.

Let $\triangle ABC$ be an equilateral triangle and $P$ a point on $\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$.

Let $ABC$ be a triangle. $F$ and $L$ are two points on the side $[AC]$ such that $AF=LC< AC/2$. Find the mesure of the angle $\angle FBL$ knowing that $AB^{2}+BC^{2}=AL^{2}+LC^{2}$.

Circles $\omega_1$ and $\omega_2$ are externally tangent to each other. Circle $\Omega$ is placed such that $\omega_1$ is internally tangent to $\Omega$ at $X$ while $\omega_2$ is internally tangent to $\Omega$ at $Y$. Line $\ell$ is tangent to $\omega_1$ at $P$ and $\omega_2$ at $Q$ and furthermore intersects $\Omega$ at points $A$ and $B$ with $AP<AQ$. Suppose that $AP=2$, $PQ=4$, and $QB=3$. Compute the length of line segment $\overline{XY}$.

Let $ABC$ be a scalene triangle whose incircle is tangent to $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Lines $BE$ and $CF$ meet at $G$. Prove that there exists a point $X$ on the circumcircle of triangle $EFG$ such that the circumcircles of triangles $BCX$ and $EFG$ are tangent, and \[\angle BGC = \angle BXC + \angle EDF.\] [i]Kornpholkrit Weraarchakul[/i]

Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.

Kait rolls a fair $6$-sided die until she rolls a $6$. If she rolls a $6$ on the $N$th roll, she then rolls the die $N$ more times. What is the probability that she rolls a $6$ during these next N times?

Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.

Let $ABCD$ be a square. Point $P, Q, R, S$ are chosen on the sides $AB$, $BC$, $CD$, $DA$, respectively, such that $AP + CR \ge AB \ge BQ + DS$. Prove that $$area \,\, (PQRS) \le \frac12 \,\, area \,\, (ABCD)$$ and determine all cases when equality holds.

A rectangle has sides of length $\sin x$ and $\cos x$ for some $x$. What is the largest possible area of such a rectangle?

Define the regions $M, N$ in the Cartesian Plane as follows: \begin{align*} M &= \{(x, y) \in \mathbb R^2 \mid 0 \leq y \leq \text{min}(2x, 3-x)\} \\ N &= \{(x, y) \in \mathbb R^2 \mid t \leq x \leq t+2 \} \end{align*} for some real number $t$. Denote the common area of $M$ and $N$ for some $t$ be $f(t)$. Compute the algebraic form of the function $f(t)$ for $0 \leq t \leq 1$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 5)[/i]

For which maximal $N$ there exists an $N$-digit number with the following property: among any sequence of its consecutive decimal digits some digit is present once only? Alexey Glebov

In the arithmetic triangle below each number (apart from those in the first row) is the sum of the two numbers immediately above. $0 \, 1\, 2\, 3 \,4\, ... \,1991 \,1992\, 1993$ $\,\,1\, 3\, 5 \,7\, ......\,\,\,\,3983 \,3985$ $\,\,\,4 \,8 \,12\, .......... \,\,\,7968$ ······································· Prove that the bottom number is a multiple of $1993$.

In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.