It is given the expression $y=\frac{x^2-2x+1}{x^2-2x+2}$, where $x$ is a variable. Prove that: (a) if $x_1$ and $x_2$ are two values of $x$, the $y_1$ and $y_2$ are the respective values of $y$ only if $x_1<x_2$, $y_1<y_2$; (b) when $x$ is varying $y$ attains all possible values for which $0\le y<1$.

Construct a triangle by its altitude , median and angle bisector originating from one vertex.

Let $A_1$ and $C_1$ be the feet of altitudes $AH$ and $CH$ of an acute-angled triangle $ABC$. Points $A_2$ and $C_2$ are the reflections of $A_1$ and $C_1$ about $AC$. Prove that the distance between the circumcenters of triangles $C_2HA_1$ and $C_1HA_2$ equals $AC$.

Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?

Let $x > 1$ be a non-integer number. Prove that \[\biggl( \frac{x+\{x\}}{[x]} - \frac{[x]}{x+\{x\}} \biggr) + \biggl( \frac{x+[x]}{ \{x \} } - \frac{ \{ x \}}{x+[x]} \biggr) > \frac 92 \]

Consider sequences $a$ of the form $a = (a_1, a_2, ... , a_{20})$ such that each term $a_i$ is either $0$ or $1$. For each such sequence $a$, we can produce a sequence $b = (b_1, b_2, ..., b_{20})$, where $$b_i\begin{cases} a_i + a_{i+1} & i = 1 \\ a_{i-1} + a_i + a_{i+1} & 1 < i < 20\\ a_{i-1} + a_i &i = 20 \end{cases}$$

Let $n, r$ be positive integers. Find the smallest positive integer $m$ satisfying the following condition. For each partition of the set $\{1, 2, \ldots ,m \}$ into $r$ subsets $A_1,A_2, \ldots ,A_r$, there exist two numbers $a$ and $b$ in some $A_i, 1 \leq i \leq r$, such that \[ 1 < \frac ab < 1 +\frac 1n.\]

Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$ has positive integer solutions.

Let $ABC$ be a triangle with circumcircle $w$. Let $D$ be the midpoint of arc $BC$ that contains $A$. Define $E$ and $F$ similarly. Let the incircle of $ABC$ touches $BC,CA,AB$ at $K,L,M$ respectively. Prove that $DK,EL,FM$ are concurrent.

In the plane, there are two circles $\Gamma_1, \Gamma_2$ intersecting each other at two points $A$ and $B$. Tangents of $\Gamma_1$ at $A$ and $B$ meet each other at $K$. Let us consider an arbitrary point $M$ (which is different of $A$ and $B$) on $\Gamma_1$. The line $MA$ meets $\Gamma_2$ again at $P$. The line $MK$ meets $\Gamma_1$ again at $C$. The line $CA$ meets $\Gamma_2 $ again at $Q$. Show that the midpoint of $PQ$ lies on the line $MC$ and the line $PQ$ passes through a fixed point when $M$ moves on $\Gamma_1$. [color=red][Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .][/color]

Let $S$ be a set consisting of all positive integers less than or equal to $100$. Let $P$ be a subset of $S$ such that there do not exist two elements $x,y\in P$ such that $x=2y$. Find the maximum possible number of elements of $P$. [i]Proposed by Nathan Ramesh

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be positive reals such that \[a_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n\] and \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n.\] Prove that \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.\]

Julia is placing identical $1$-by-$1$ tiles on the $2$-by-$2$ grid pictured, one piece at a time, so that every piece she places after the first is adjacent to, but not on top of, some piece she’s already placed. Determine the number of ways that Julia can complete the grid. [img]https://cdn.artofproblemsolving.com/attachments/4/6/4a585593b9301ddb0e4ac3ceced212c378c9f8.png[/img]

n \geq 6 is an integer. evaluate the minimum of f(n) s.t: any graph with n vertices and f(n) edge contains two cycle which are distinct( also they have no comon vertice)?

A $\text{5.0 kg}$ block with a speed of $\text{8.0 m/s}$ travels $\text{2.0 m}$ along a horizontal surface where it makes a head-on, perfectly elastic collision with a $\text{15.0 kg}$ block which is at rest. The coefficient of kinetic friction between both blocks and the surface is $0.35$. How far does the $\text{15.0 kg}$ block travel before coming to rest? (A) $\text{0.76 m}$ (B) $\text{1.79 m}$ (C) $\text{2.29 m}$ (D) $\text{3.04 m}$ (E) $\text{9.14 m}$

GM Bisain's IQ is so high that he can move around in $10$ dimensional space. He starts at the origin and moves in a straight line away from the origin, stopping after $3$ units. How many lattice points can he land on? A lattice point is one with all integer coordinates. [i]2019 CCA Math Bonanza Lightning Round #4.2[/i]

Let $f(x) = x^4 + 2x^3 - x - 1$. (a) Prove that $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients. (b) Find the exact values of the 4 roots of $f(x)$.

Let $n$ be a positive integer and let $0\leq k\leq n$ be an integer. Show that there exist $n$ points in the plane with no three on a line such that the points can be divided into two groups satisfying the following properties. $\text{(i)}$ The first group has $k$ points and the distance between any two distinct points in this group is irrational. $\text{(ii)}$ The second group has $n-k$ points and the distance between any two distinct points in this group is an integer. $\text{(iii)}$ The distance between a point in the first group and a point in the second group is irrational.

[list=a] [*]Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $g\colon\mathbb{R}\rightarrow\mathbb{R}$, $g(x)=f(x)+f(2x)$, and $h\colon\mathbb{R}\rightarrow\mathbb{R}$, $h(x)=f(x)+f(4x)$, are continuous functions. Prove that $f$ is also continuous. [*]Give an example of a discontinuous function $f\colon\mathbb{R} \rightarrow\mathbb{R}$, with the following property: there exists an interval $I\subset\mathbb{R}$, such that, for any $a$ in $I$, the function $g_{a} \colon\mathbb{R}\rightarrow\mathbb{R}$, $g_{a}(x)=f(x)+f(ax)$, is continuous.[/list]

The vertices of a regular $2005$-gon are colored red, white and blue. Whenever two vertices of different colors stand next to each other, we are allowed to recolor them into the third color. (a) Prove that there exists a finite sequence of allowed recolorings after which all the vertices are of the same color. (b) Is that color uniquely determined by the initial coloring?

[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity. [b](b)[/b]What is the smallest area possible of pentagons with integral coordinates. Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.

Consider the set $A = \{1, 2, ..., n\}$, where $n \in N, n \ge 6$. Show that $A$ is the union of three pairwise disjoint sets, with the same cardinality and the same sum of their elements, if and only if $n$ is a multiple of $3$.

Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$. [asy] pair A,B,C,D,E,F,G; B=origin; A=5*dir(60); C=(5,0); E=0.6*A+0.4*B; F=0.6*A+0.4*C; G=rotate(240,F)*A; D=extension(E,F,B,dir(90)); draw(D--G--A,grey); draw(B--0.5*A+rotate(60,B)*A*0.5,grey); draw(A--B--C--cycle,linewidth(1.5)); dot(A^^B^^C^^D^^E^^F^^G); label("$A$",A,dir(90)); label("$B$",B,dir(225)); label("$C$",C,dir(-45)); label("$D$",D,dir(180)); label("$E$",E,dir(-45)); label("$F$",F,dir(225)); label("$G$",G,dir(0)); label("$\ell$",midpoint(E--F),dir(90)); [/asy]

Let $AB\varGamma\varDelta$ be a rhombus. (a) Prove that you can construct a circle $(c)$ which is inscribed in the rhombus and is tangent to its sides. (b) The points $\varTheta,H,K,I$ are on the sides $\varDelta\varGamma,B\varGamma,AB,A\varDelta$ of the rhombus respectively, such that the line segments $KH$ and $I\varTheta$ are tangent on the circle $(c)$. Prove that the quadrilateral defined from the points $\varTheta,H,K,I$ is a trapezium.

Let $\vartriangle ABC$ be a triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\omega$. $AA'$, $BB'$ and $CC'$ are altitudes of $\vartriangle ABC$ with $A'$ in $BC$, $B'$ in $AC$ and $C'$ in $AB$. $P$ is a point on the segment $AA'$. The perpenicular line to $B'C'$ from $P$ intersects $BC$ at $K$. $AA'$ intersects $\omega$ at $M \ne A$. The lines $MK$ and $AO$ intersect at $Q$. Prove that $\angle CBQ = \angle PBA$.