Let $S_0=\{z\in\mathbb C:|z|=1,z\ne-1\}$ and $f(z)=\frac{\operatorname{Im}z}{1+\operatorname{Re}z}$. Prove that $f$ is a bijection between $S_0$ and $\mathbb R$. Find $f^{-1}$.

Inside a triangle $ABC$ three circles with radius $1$ are drawn. (Circles can be tangent to each other and to the sides of the triangle, but can not have any common internal points.) Find the biggest value of $r$ for which one can state that he can always draw a fourth circle inside the triangle of radius $r$, which does not intersect three already drawn circles.

Consider a circle $O_1$ with radius $R$ and a point $A$ outside the circle. It is known that $\angle BAC=60^\circ$, where $AB$ and $AC$ are tangent to $O_1$. We construct infinitely many circles $O_i$ $(i=1,2,\dots\>)$ such that for $i>1$, $O_i$ is tangent to $O_{i-1}$ and $O_{i+1}$, that they share the same tangent lines $AB$ and $AC$ with respect to $A$, and that none of the $O_i$ are larger than $O_1$. Find the total area of these circles. I know this problem was easy, but it still appeared in the TST, and so I posted it. It was kind of a disappointment for me.

Let $X$ be a set with $n$ elements and $0 \le k \le n$. Let $a_{n,k}$ be the maximum number of permutations of the set $X$ such that every two of them have at least $k$ common components (where a common component of $f$ and g is an $x \in X$ such that $f(x) = g(x)$). Let $b_{n,k}$ be the maximum number of permutations of the set $X$ such that every two of them have at most $k$ common components. (a) Show that $a_{n,k} \cdot b_{n,k-1} \le n!$. (b) Let $p$ be prime, and find the exact value of $a_{p,2}$.

Is there a number whose digits are only $1$'s and which is divided by $1999$?

For all $n\ge2$, let $a_n$ be the product of all coprime natural numbers less than $n$. Prove that (a) $n\mid a_n+1\Leftrightarrow n=2,4,p^\alpha,2p^\alpha$ (b) $n\mid a_n-1\Leftrightarrow n\ne2,4,p^\alpha,2p^\alpha$ Here $p$ is an odd prime number and $\alpha\in\mathbb N$.

A game is played on a $23\times 23$ board. The first player controls two white chips which start in the bottom left and top right corners. The second player controls two black ones which start in bottom right and top left corners. The players move alternately. In each move, a player moves one of the chips under control to a square which shares a side with the square the chip is currently in. The first player wins if he can bring the white chips to squares which share a side with each other. Can the second player prevent the first player from winning?

There are $n \ge 4$ cities on a round lake, between which $n -4$ people travel and one green drivers operate. Each ferry connects two non-adjacent cities, and itself do not cross two driving routes so that collisions can be avoided. In order to better adapt the transport routes to the needs of the passengers, the following change can be done: A new route can be assigned to any driver. The routes of the remaining drives must not cross and also must not be changed at the same time. Let Santa Marta and Cape Town be two non-adjacent cities. Show that you have finitely many route changes so that the Green Driver will operate between Santa Marta and Cape Town after these changes. Note: At no time may two trips between the same cities or one drive between two neighboring cities. [hide=original wording]An einem runden See liegen $n >= 4$ Stadte, zwischen denen $n - 4$ Personenfahren und eine grune Autofahre verkehren. Jede Fahre verbindet zwei nicht benachbarte Stadte, wobei sich keine zwei Fahrenrouten uberkreuzen, damit Kollisionen vermieden werden konnen. Um die Transportrouten besser den Bedurfnissen der Passagiere anzupassen, kann folgende Anderung vorgenommen werden: Einer beliebigen Fahre kann eine neue Route zugeordnet werden. Dabei durfen die Routen der restlichen Fahren nicht uberkreuzt und auch nicht gleichzeitig verandert werden. Seien Santa Marta und Kapstadt zwei nicht benachbarte Stadte. Zeige, dass man endlich viele Routenanderungen vornehmen kann, sodass die grune Autofahre nach diesen Anderungen zwischen Santa Marta und Kapstadt verkehrt. Bemerkung: Zu keinem Zeitpunkt durfen zwei Fahren zwischen denselben Stadten oder eine Fahre zwischen zwei benachbarten Stadten verkehren.[/hide]

Let $p > 3$ be a prime number, and let $F_p$ denote the (finite) set of residue classes modulo $p$. Let $S_d$ denote the set of $2$-variable polynomials $P(x, y)$ with coefficients in $F_p$, total degree $\le d$, and satisfying $P(x, y) = P(y,- x -y)$. Show that $$|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil}$$. [i]The total degree of a $2$-variable polynomial $P(x, y)$ is the largest value of $i + j$ among monomials $x^iy^j$ [/i] appearing in $P$.

Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$.

Find the range of the function $f(x)=\frac{\sqrt{x^2+1}}{x-1}$.

Rijul and Rohinee are playing a game on an $n\times n$ board alternating turns, with Rijul going first. In each turn, they fill an unfilled cell with a number from $1,2,\cdots, n^2$ such that no number is used twice. Rijul wins if there is any column such that the sum of all its elements is divisible by $n$. Rohinee wins otherwise. For what positive integers $n$ does he have a winning strategy? Proposed by Rohan Goyal

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

A rational number written in base eight is $\underline{a} \underline{b} . \underline{c} \underline{d}$, where all digits are nonzero. The same number in base twelve is $\underline{b} \underline{b} . \underline{b} \underline{a}$. Find the base-ten number $\underline{a} \underline{b} \underline{c}$.

Let $n$ be any natural number. Find all $n$-tuples of real numbers $x_1\le x_2\le ... \le x_n$, for which holds $$\left(\sum_{i=1}^n x_i\right)^2 \le n \sum_{i=1}^n x_i x_{n-i+1}.$$

[b]a)[/b] Let $ u $ be a polynom in $ \mathbb{Q}[X] . $ Prove that the function $ E_u:\mathbb{Q}[X]\longrightarrow\mathbb{Q}[X] $ defined as $ E_u(P)=P(u) $ is an endomorphism. [b]b)[/b] Let $ E $ be an injective endomorphism of $ \mathbb{Q} [X] . $ Show that there exists a nonconstant polynom $ v $ in $ \mathbb{Q}[X] $ such that $ E(P)=P(v) , $ for any $ P $ in $ \mathbb{Q}[X] . $ [b]c)[/b] Let $ A $ be an automorphism of $ \mathbb{Q}[X] . $ Demonstrate that there is a nonzero constant polynom $ w $ in $ \mathbb{Q}[X] $ which has the property that $ A(P)=P(w) , $ for any $ P $ in $ \mathbb{Q}[X] . $ [i]Marcel Țena[/i]

Let $ABCD$ be a convex quadrilateral with positive area such that every side has a positive integer length and $AC=BC=AD=25$. If $P_{max}$ and $P_{min}$ are the quadrilaterals with maximum and minimum possible perimeter, the ratio of the area of $P_{max}$ and $P_{min}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ for some positive integers $a,b,c$, where $a,c$ are relatively prime and $b$ is not divisible by the square of any integer. Find $a+b+c$. [i]Proposed by [b]FedeX333X [/b][/i]

Find all functions $f: \mathbb{R}^+\to\mathbb{R}$ such that $f(x+y)=f(x^2)+f(y^2)$ for any $x,y \in\mathbb{R}^+$

Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$

Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius $30$. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$.

Let $N>2^{5000}$ be a positive integer. Prove that if $1\leq a_1<\cdots<a_k<100$ are distinct positive integers then the number \[\prod_{i=1}^{k}\left(N^{a_i}+a_i\right)\] has at least $k$ distinct prime factors. Note. Results with $2^{5000}$ replaced by some other constant $N_0$ will be awarded points depending on the value of $N_0$. [i]Proposed by Evan Chen[/i]

Given an acute triangle $ABC$, let $AD$ be an altitude and $H$ the orthocenter. Let $E$ denote the reflection of $H$ with respect to $A$. Point $X$ is chosen on the circumcircle of triangle $BDE$ such that $AC\| DX$ and point $Y$ is chosen on the circumcircle of triangle $CDE$ such that $DY\| AB$. Prove that the circumcircle of triangle $AXY$ is tangent to that of $ABC$.

Prove that the integral part of the decimal representation of the number $(3+\sqrt{5})^n$ is odd, for every positive integer $n.$

Find all functions $f : R \to R$ such that $$f(f(x) + f(y)^2) = f(x)^2 +y^2f(y)^3.$$ Here $R$ denotes the usual real numbers.

Find the least positive integer whose digits add to a multiple of 27 yet the number itself is not a multiple of 27. For example, 87999921 is one such number.