Find the maximum value of real number $k$ such that \[\frac{a}{1+9bc+k(b-c)^2}+\frac{b}{1+9ca+k(c-a)^2}+\frac{c}{1+9ab+k(a-b)^2}\geq \frac{1}{2}\] holds for all non-negative real numbers $a,\ b,\ c$ satisfying $a+b+c=1$.

In the figure below, $E$ is the midpoint of the arc $ABEC$ and the segment $ED$ is perpendicular to the chord $BC$ at $D$. If the length of the chord $AB$ is $l_1$, and that of the segment $BD$ is $l_2$, determine the length of $DC$ in terms of $l_1, l_2$. [asy] unitsize(1 cm); pair A=2dir(240),B=2dir(190),C=2dir(30),E=2dir(135),D=foot(E,B,C); draw(circle((0,0),2)); draw(A--B--C); draw(E--D); draw(rightanglemark(C,D,E,8)); label("$A$",A,.5A); label("$B$",B,.5B); label("$C$",C,.5C); label("$E$",E,.5E); label("$D$",D,dir(-60)); [/asy]

Let $\mathbb{Q}$ denote the set of rational numbers. Determine all functions $f:\mathbb{Q}\longrightarrow\mathbb{Q}$ such that, for all $x, y \in \mathbb{Q}$, $$f(x)f(y+1)=f(xf(y))+f(x)$$ [i]Nicolás López Funes and José Luis Narbona Valiente, Spain[/i]

Prove that $ \binom{m+n}{\min (m,n)}\le \sqrt{\binom{2m}{m}\cdot \binom{2n}{n}} , $ for nonnegative $ m,n. $ [i]Gheorghe Stoica[/i]

Let $x,y,z$ be positive real numbers. Prove that \[\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\geqslant\frac{z(x+y)}{y(y+z)}+\frac{x(y+z)}{z(z+x)}+\frac{y(z+x)}{x(x+y)}.\]

In a triangle $\triangle ABC$ with $\angle ABC < \angle BCA$, we define $K$ as the excenter with respect to $A$. The lines $AK$ and $BC$ intersect in a point $D$. Let $E$ be the circumcenter of $\triangle BKC$. Prove that \[\frac{1}{|KA|} = \frac{1}{|KD|} + \frac{1}{|KE|}.\]

The coefficient of $x^7$ in the polynomial expansion of \[ (1+2x-x^2)^4 \] is $ \textbf{(A)}\ -8 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ -12 \qquad\textbf{(E)}\ \text{none of these} $

On Misha's new phone, a passlock consists of six circles arranged in a $2\times 3$ rectangle. The lock is opened by a continuous path connecting the six circles; the path cannot pass through a circle on the way between two others (e.g. the top left and right circles cannot be adjacent). For example, the left path shown below is allowed but the right path is not. (Paths are considered to be oriented, so that a path starting at $A$ and ending at $B$ is different from a path starting at $B$ and ending at $A$. However, in the diagrams below, the paths are valid/invalid regardless of orientation.) How many passlocks are there consisting of all six circles? [asy] size(270); defaultpen(linewidth(0.8)); real r = 0.3, rad = 0.1, shift = 3.7; pen th = linewidth(5)+gray(0.2); for(int i=0; i<= 2;i=i+1) { for(int j=0; j<= 1;j=j+1) { fill(circle((i,j),r),gray(0.8)); fill(circle((i+shift,j),r),gray(0.8)); } draw((0,1)--(2-rad,1)^^(2,1-rad)--(2,rad)^^(2-rad,0)--(0,0),th); draw(arc((2-rad,1-rad),rad,0,90)^^arc((2-rad,rad),rad,270,360),th); draw((shift+1,0)--(shift+1,1-2*rad)^^(shift+1-rad,1-rad)--(shift+rad,1-rad)^^(shift+rad,1+rad)--(shift+2,1+rad),th); draw(arc((shift+1-rad,1-2*rad),rad,0,90)^^arc((shift+rad,1),rad,90,270),th); } [/asy]

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}$$

A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$. What is the least number of dominoes that are entirely inside some square $2 \times 2$?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a continuous function such that for any $a,b\in \mathbb{R}$, with $a<b$ such that $f(a)=f(b)$, there exist some $c\in (a,b)$ such that $f(a)=f(b)=f(c)$. Prove that $f$ is monotonic over $\mathbb{R}$.

A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.

Let $a, b, c$ be distinct real numbers such that $a+b+c=0$ and $$ a^{2}-b=b^{2}-c=c^{2}-a. $$ Evaluate all the possible values of $a b+a c+b c$. [i]Proposed by Nguyen Anh Vu, Vietnam[/i]

Define the sequence $(x_{n})$: $x_{1}=\frac{1}{3}$ and $x_{n+1}=x_{n}^{2}+x_{n}$. Find $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$, wehere $[$ $]$ denotes the integer part.

In triangle $ABC$, let $O$ be the center of the circumcircle, and let $H$ be the orthocenter. Let $P$ be the center of the circumcircle of triangle $BOC$, and $Q$ be the center of the circumcircle of triangle $BHC$. Prove that $OP \cdot OQ = OA^2$.

Let $x, y$ be two integers. Prove that if $2013$ divides $x^{1433} + y^{1433}$ then $2013$ divides $x^7 + y^7$.

Let $n\ge 3$. Suppose $a_1, a_2, ... , a_n$ are $n$ distinct in pairs real numbers. In terms of $n$, find the smallest possible number of different assumed values by the following $n$ numbers: $$a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1$$

Line $l:2x+y=10$, line $l'$ passes $(-10,0)$, and $l'\perp l$, then the coordinate of the intersection of $l$ and $l'$ is________.

Find the minimum real $x$ that satisfies $$\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots$$

Find coordinates of a set of eight non-collinear planar points so that each has an integral distance from others.

Construct the $ \triangle ABC $, with $ AC <BC $, if the circumcircle is known, and the points $ D, E, F $ in it, where they intersect, respectively, the altitude, the median and the angle bisector that they start from the vertex $ C $.

Find all pairs of integers $(x,y)$ satisfying $x^2 +3y^2 = 1998x$.

Find $a$ and $b$ for which the largest and smallest is values of the function $y=\frac{x^2+ax+b}{x^2-x+1}$ are equal to the $2$ and $-3$ respectively.

Consider the sum $$S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|.$$ The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$, satisfying $2c < d$. Find the value of $c + d$.

The columns and the row of a $3n \times 3n$ square board are numbered $1,2,\ldots ,3n$. Every square $(x,y)$ with $1 \leq x,y \leq 3n$ is colored asparagus, byzantium or citrine according as the modulo $3$ remainder of $x+y$ is $0,1$ or $2$ respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are $3n^2$ tokens of each color. Suppose that one can permute the tokens so that each token is moved to a distance of at most $d$ from its original position, each asparagus token replaces a byzantium token, each byzantium token replaces a citrine token, and each citrine token replaces an asparagus token. Prove that it is possible to permute the tokens so that each token is moved to a distance of at most $d+2$ from its original position, and each square contains a token with the same color as the square.