There are two sorts of people on an island: [i]knights[/i], who always talk truth, and [i]scoundrels[/i], who always lie. One day, the people establish a council consisting of $2003$ members. They sit around a round table, and during the council each member said: "Both my neighbors are scoundrels". In a later day, the council meets again, but one member could not come due to illness, so only $2002$ members were present. They sit around the round table, and everybody said: "Both my neighbors belong to the sort different from mine". Is the absent member a knight or a scoundrel?

Two circles $\omega , \Omega$ intersect in distinct points $A,B$ a line through $B$ intersects $\omega , \Omega$ in $C,D$ respectively such that $B$ lies between $C,D$ another line through $B$ intersects $\omega , \Omega$ in $E,F$ respectively such that $E$ lies between $B,F$ and $FE=CD$. Furthermore $CF$ intersects $\omega , \Omega$ in $P,Q$ respectively and $M,N$ are midpoints of the arcs $PB,QB$. Prove that $CNMF$ is a cyclic quadrilateral.

The solution set for inequality $|x|^3-2x^2-4|x|+3<0$ is________.

For which positive integers $n$ is $231^n-222^n-8 ^n -1$ divisible by $2007$?

Find the number of positive integers $n$ such that the highest power of $7$ dividing $n!$ is $8$.

Let $a_0, a_1, \ldots, a_{10}$ be integers such that, for each $i \in \{0,1,\ldots,2047\}$, there exists a subset $S \subseteq \{0,1,\ldots,10\}$ with \[ \sum_{j \in S} a_j \equiv i \pmod{2048}. \] Show that for each $i \in \{0,1,\ldots,10\}$, there is exactly one $j \in \{0,1,\ldots,10\}$ such that $a_j$ is divisible by $2^i$ but not by $2^{i+1}$. Note: $\sum_{j \in S} a_j$ is the summation notation, for instance, $\sum_{j \in \{2,5\}} a_j = a_2 + a_5$, while for the empty set $\varnothing$, one defines $\sum_{j \in \varnothing} a_j = 0$.

Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F=360^{\circ }$ and \[ \frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1. \] Prove that \[ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1. \]

Let $AH$, $BL$ and $CM$ be an altitude, a bisectrix and a median in triangle $ABC$. It is known that lines $AH$ and $BL$ are an altitude and a bisectrix of triangle $HLM$. Prove that line $CM$ is a median of this triangle.

Let points $M$ and $N$ lie on sides $AB$ and $BC$ of triangle $ABC$ in such a way that $MN||AC$. Points $M'$ and $N'$ are the reflections of $M$ and $N$ about $BC$ and $AB$ respectively. Let $M'A$ meet $BC$ at $X$, and let $N'C$ meet $AB$ at $Y$. Prove that $A,C,X,Y$ are concyclic.

For what a from the interval $[0,\pi]$ do there exist $a$ and $b$ that are not simultaneously equal to zero, for which the inequality $$a \cos x + b \cos 2x \le 0$$ is satisfied for all $x$ belonging to the segment $[a, \pi]$?

Solve the system of equations for real $x$ and $y$: \begin{eqnarray*} 5x \left( 1 + \frac{1}{x^2 + y^2}\right) &=& 12 \\ 5y \left( 1 - \frac{1}{x^2+y^2} \right) &=& 4 . \end{eqnarray*}

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Determine all positive real numbers $a,b,c,d$ such that $a+b+c+d=80$ and $$a+\frac{b}{1+a}+\frac{c}{1+a+b}+\frac{d}{1+a+b+c}=8.$$

Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that a) If $A\subseteq S$, then $A\in F$ if and only if $S\setminus A\notin F$; b) If $A\subseteq B\subseteq S$ and $B\in F$, then $A\in F$. Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$, $A\in F$ if and only if \[\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).\] [i]Evan O'Dorney.[/i]

There is three-dimensional space. For every integer $n$ we build planes $ x \pm y\pm z = n$. All space is divided on octahedrons and tetrahedrons. Point $(x_0,y_0,z_0)$ has rational coordinates but not lies on any plane. Prove, that there is such natural $k$ , that point $(kx_0,ky_0,kz_0)$ lies strictly inside the octahedron of partition.

Consider the sequence $(a_k)_{k\ge 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k\ge 1$, if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then \[a_{k+1} = \frac{m + 18}{n+19}.\] Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\frac{t}{t+1}$ for some positive integer $t$.

In the diagram below, how many rectangles can be drawn using the grid lines which contain none of the letters $N$, $I$, $M$, $O$? [asy] size(4cm); for(int i=0;i<6;++i)draw((i,0)--(i,5)^^(0,i)--(5,i)); label("$N$", (1.5, 2.5)); label("$I$", (2.5, 3.5)); label("$M$", (3.5, 2.5)); label("$O$", (2.5, 1.5)); [/asy] [i]Proposed by Michael Tang[/i]

Let be a natural number $ n. $ Show that: [b]a.[/b] $ \left( 1+1/n \right)^k<1+k/n+k^2/n^2, $ for all naturals $ k $ with $ 1\le k\le n. $ [b]b.[/b] $ 3^n\cdot n!>(1+n)^n. $

The measures (in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of positive integers. Let $m^{\circ}$ be the measure of the largest interior angle of the hexagon. The largest possible value of $m^{\circ}$ is $ \textbf{(A)}\ 165^{\circ}\qquad\textbf{(B)}\ 167^{\circ}\qquad\textbf{(C)}\ 170^{\circ}\qquad\textbf{(D)}\ 175^{\circ}\qquad\textbf{(E)}\ 179^{\circ} $

Let $P,Q,R$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$. Suppose that $BQ$ and $CR$ meet at $A', AP$ and $CR$ meet at $B'$, and $AP$ and $BQ$ meet at $C'$, such that $AB' = B'C', BC' =C'A'$, and $CA'= A'B'$. Compute the ratio of the area of $\triangle PQR$ to the area of $\triangle ABC$.

How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1 \le x \le 4$ and $1 \le y \le 4$? $ \textbf{(A)}\ 496 \qquad\textbf{(B)}\ 500 \qquad\textbf{(C)}\ 512 \qquad\textbf{(D)}\ 516 \qquad\textbf{(E)}\ 560 $

Determine all functions $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ satisfying $f(x+y+f(y))=4030x-f(x)+f(2016y), \forall x,y \in \mathbb{R}^+$.

For a number $n$ in base $10$, let $f(n)$ be the sum of all numbers possible by removing some digits of $n$ (including none and all). For example, if $n = 1234$, $f(n) = 1234 + 123 + 124 + 134 + 234 + 12 + 13 + 14 + 23 + 24 + 34 + 1 + 2 + 3 + 4 = 1979$; this is formed by taking the sums of all numbers obtained when removing no digit from $n$ (1234), removing one digit from $n$ (123, 124, 134, 234), removing two digits from $n$ (12, 13, 14, 23, 24, 34), removing three digits from $n$ (1, 2, 3, 4), and removing all digits from $n$ (0). If $p$ is a 2011-digit integer, prove that $f(p)-p$ is divisible by $9$. Remark: If a number appears twice or more, it is counted as many times as it appears. For example, with the number $101$, $1$ appears three times (by removing the first digit, giving $01$ which is equal to $1$, removing the first two digits, or removing the last two digits), so it is counted three times.

Find all function $f: \mathbb{R}^+ \rightarrow \mathbb{R}$,such that the inequality $$f(x) + f\left(\frac{y}{x}\right) \leqslant \frac{x^3}{y^2} + \frac{y}{x^3}$$ holds for all positive reals $x,y$ and for every positive real $x$, there exist positive reals $y$, such that the equality holds.

Solve in the real numbers the equation $ \lfloor ax \rfloor -\lfloor (1+a)x \rfloor = (1+a)(1-x) . $ [i]Dumitru Acu[/i]