Find constants $A > B$ such that $\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}$ is independent of $x$, where $f(x) = \frac{1 + Ax}{1 + Bx}$ for all real $x \ne - \frac{1}{B}$. Put $a_0 = 1$, $a_{n+1} = \frac{1}{1 + 2a_n}$. Find an expression for an by considering $f(a_0), f(a_1), ...$.

Let $A$ be the sum of the first $2k+1$ positive odd integers, and let $B$ be the sum of the first $2k+1$ positive even integers. Show that $A+B$ is a multiple of $4k+3$.

Let $n$ be a positive integer. Does $n^2$ has more positive divisors of the form $4k+1$ or of the form $4k-1$?

Let $ABCD$ be an isosceles trapezoid with $2DA=2AB=2BC=CD$. A point $P$ lies in the interior of $ABCD$ such that $BP=1$, $CP=2$, $DP=4$. Find the area of $ABCD$.

Let's consider words over the alphabet $\{a,b\}$ to be sequences of $a$ and $b$ with finite length. We say $u \leq v$ if $u$ is a subword of $v$ if we can get $u$ erasing some letter of $v$ (for example $aba \leq abbab$). We say that $u$ differentiates the words $x$ and $y$ if $u \leq x$ but $u \not\leq y$ or vice versa. Let $m$ and $l$ be positive integers. We say that two words are $m-$equivalents if there does not exist some $u$ with length smaller than $m$ that differentiates $x$ and $y$. a) Show that, if $2m \leq l$, there exists two distinct words with length $l$ \ $m-$equivalents. b) Show that, if $2m > l$, any two distinct words with length $l$ aren't $m-$equivalent.

Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$. \\ \\ Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.

Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation \[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

An geometric progression starting at $a_0 = 3$ has an even number of terms. Suppose the difference between the odd indexed terms and even indexed terms is $39321$ and that the sum of the first and last term is $49155$. Find the common ratio of this geometric progression.

Let $A, B,C$ be colinear points in this order, $\omega$ an arbitrary circle passing through $B$ and $C$, and $l$ an arbitrary line different from $BC$, passing through A and intersecting $\omega$ at $M$ and $N$. The bisectors of the angles $\angle CMB$ and $\angle CNB$ intersect $BC$ at $P$ and $Q$. Prove that $AP\cdot AQ = AB \cdot AC$.

Consider a point $ M$ found in the same plane with the triangle $ ABC$, but not found on any of the lines $ AB,BC$ and $ CA$. Denote by $ S_1,S_2$ and $ S_3$ the areas of the triangles $ AMB,BMC$ and $ CMA$, respectively. Find the locus of $ M$ satisfying the relation: $ (MA^2\plus{}MB^2\plus{}MC^2)^2\equal{}16(S_1^2\plus{}S_2^2\plus{}S_3^2)$

Three of the below entries, with labels $a$, $b$, $c$, are blatantly incorrect (in the United States). What is $a^2+b^2+c^2$? 041. The Gentleman's Alliance Cross 042. Glutamine (an amino acid) 051. Grant Nelson and Norris Windross 052. A compact region at the center of a galaxy 061. The value of \verb+'wat'-1+. (See \url{https://www.destroyallsoftware.com/talks/wat}.) 062. Threonine (an amino acid) 071. Nintendo Gamecube 072. Methane and other gases are compressed 081. A prank or trick 082. Three carbons 091. Australia's second largest local government area 092. Angoon Seaplane Base 101. A compressed archive file format 102. Momordica cochinchinensis 111. Gentaro Takahashi 112. Nat Geo 121. Ante Christum Natum 122. The supreme Siberian god of death 131. Gnu C Compiler 132. My TeX Shortcut for $\angle$.

We call a permutation of the set of real numbers $\{a_1,\cdots,a_n\}$, $n\in\mathbb{N}$ [i]average increasing[/i] if the arithmetic mean of its first $k$ elements for $k=1,\cdots ,n$ form a strictly increasing sequence. 1) Depending on $n$, determine the smallest number that can be the last term of some average increasing permutation of the numbers $\{1,\cdots,n\}$; 2) Depending on $n$, determine the lowest position (in some general order) that the number $n$ can be achieved in some average increasing permutation of the numbers $\{1,\cdots,n\}.$ [i] Proposed by David Hruska, Czech Republic[/i]

A $50$-card deck consists of $4$ cards labeled " i" for $i = 1, 2,..., 12$ and $2$ cards labeled "$13$". If Bob randomly chooses $2$ cards from the deck without replacement, what is the probability that his $2$ cards have the same label?

For any number $n\in\mathbb{N},n\ge 2$, denote by $P(n)$ the number of pairs $(a,b)$ whose elements are of positive integers such that \[\frac{n}{a}\in (0,1),\quad \frac{a}{b}\in (1,2)\quad \text{and}\quad \frac{b}{n}\in (2,3). \] $a)$ Calculate $P(3)$. $b)$ Find $n$ such that $P(n)=2002$.

Prove that in a finite group G the number of subgroups with index n is at most $| G |^{2 \log_2 n}$.

Determine the value of the parameter $m$ such that the equation $(m-2)x^2+(m^2-4m+3)x-(6m^2-2)=0$ has real solutions, and the sum of the third powers of these solutions is equal to zero.

Given is a triangle $ABC$ with side lengths $a, b,c$. Three spheres touch each other in pairs and also touch the plane of the triangle at points $A,B$ and $C$, respectively. Determine the radii of these spheres.

Let $N$ denote the numbers of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \le 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.

If $a, b, c$ are side lengths of a triangle, prove that \[(a + b)(b + c)(c + a) \geq 8(a + b - c)(b + c - a)(c + a - b).\]

For which $k$ the number $N = 101 ... 0101$ with $k$ ones is a prime?

In a triangle $ABC$, points $D,E$ lie on sides $AB,AC$ respectively. The lines $BE$ and $CD$ intersect at $F$. Prove that if $\color{white}\ .\quad\ \color{black}\ \quad BC^2=BD\cdot BA+CE\cdot CA,$ then the points $A,D,F,E$ lie on a circle.

In the triangle $ABC$, the side $BC$ is equal to $a$. Point $F$ is midpoint of $AB$, $I$ is the point of intersection of the bisectors of triangle $ABC$. It turned out that $\angle AIF = \angle ACB$. Find the perimeter of the triangle $ABC$. (Grigory Filippovsky)

Let $ABC$ be a triangle, $O$ its circumcenter and $R=1$ its circumradius. Let $G_1,G_2,G_3$ be the centroids of the triangles $OBC, OAC$ and $OAB.$ Prove that the triangle $ABC$ is equilateral if and only if $$AG_1+BG_2+CG_3=4$$