In coordinate plane, a variable point $M$, starting from the origin $O(0, 0)$, moves on the line $l$ with slope $k$, where $k$ is an irrational number. [b](i)[/b] Prove that point $O(0, 0)$ is the only rational point (namely, the coordinates of which are both rationals) on the line $l.$ [b](ii)[/b] Prove that for any number $\varepsilon > 0$, there exist integers $m, n$ such that the distance between $l$ and the point $(m, n)$ is less than $\varepsilon.$

In how many ways can we construct a square with dimensions $3\times 3$ using $3$ white, $3$ green and $3$ red squares of dimensions $1\times 1$, such that in every horizontal and in every certical line, squares have different colours .

Let $D$ be an arbitrary point on the side $BC$ of the non-isosceles acute triangle $ABC$. The circle with center $D$ and radius $DA$ intersects the rays $AB^\to$ (after $B$) and $AC^\to$ (after $C$) at $M$ and $N$. Prove that the orthocenter of triangle $AMN$ lies on a fixed line, independent of the choice of $D$.

Prove that if the sum of $x^5,y^5$ and $z^5$, where $x,y$ and $z$ are integer numbers, is divisible by $25$ then the sum of some two of them is divisible by $25$.

For all positive integers $k$, define $f(k)=k^2+k+1$. Compute the largest positive integer $n$ such that \[2015f(1^2)f(2^2)\cdots f(n^2)\geq \Big(f(1)f(2)\cdots f(n)\Big)^2.\][i]Proposed by David Altizio[/i]

Compute the largest root of $x^4-x^3-5x^2+2x+6$.

Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$

Does there exist a second-degree polynomial $p(x, y)$ in two variables such that every non-negative integer $ n $ equals $p(k,m)$ for one and only one ordered pair $(k,m)$ of non-negative integers? [i]Proposed by Finland.[/i]

If circular arcs $ AC$ and $ BC$ have centers at $ B$ and $ A$, respectively, then there exists a circle tangent to both $ \stackrel{\frown}{AC}$ and $ \stackrel{\frown}{BC}$, and to $ \overline{AB}$. If the length of $ \stackrel{\frown}{BC}$ is $ 12$, then the circumference of the circle is [asy]unitsize(4cm); defaultpen(fontsize(8pt)+linewidth(.8pt)); dotfactor=3; pair O=(0,.375); pair A=(-.5,0); pair B=(.5,0); pair C=shift(-.5,0)*dir(60); draw(Arc(A,1,0,60)); draw(Arc(B,1,120,180)); draw(A--B); draw(Circle(O,.375)); dot(A); dot(B); dot(C); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N);[/asy]$ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 28$

Construct a parallelogram $ABCD$, if three points are marked on the plane: the midpoints of its altitudes $BH$ and $BP$ and the midpoint of the side $AD$.

Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]

[b]Problem:[/b]For a positive integer $ n$,let $ V(n; b)$ be the number of decompositions of $ n$ into a product of one or more positive integers greater than $ b$. For example,$ 36 \equal{} 6.6 \equal{}4.9 \equal{} 3.12 \equal{} 3 .3. 4$, so that $ V(36; 2) \equal{} 5$.Prove that for all positive integers $ n$; b it holds that $ V(n;b)<\frac{n}{b}$. :)

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

Let circles $\omega$ and $\Gamma$, centered at $O_1$ and $O_2$ and having radii $42$ and $54$ respectively, intersect at points $X,Y$, such that $\angle O_1XO_2 = 105^{\circ}$. Points $A$, $B$ lie on $\omega$ and $\Gamma$ respectively such that $\angle O_1XA = \angle AXB = \angle BXO_2$ and $Y$ lies on both minor arcs $XA$ and $XB$. Define $P$ to be the intersection of $AO_2$ and $BO_1$. Suppose $XP$ intersects $AB$ at $C$. What is the value of $\frac{AC}{BC}$? [i]Proposed by Puhua Cheng[/i]

In triangle $ABC,$ the points $D$ and $E$ are the intersections of the angular bisectors from $C$ and $B$ with the sides $AB$ and $AC,$ respectively. Points $F$ and $G$ on the extensions of $AB$ and $AC$ beyond $B$ and $C,$ respectively, satisfy $BF = CG = BC.$ Prove that $F G \parallel DE.$

Denote the number of different prime divisors of the number $n$ by $\omega (n)$, where $n$ is an integer greater than $1$. Prove that there exist infinitely many numbers $n$ for which $\omega (n)< \omega (n+1)<\omega (n+2)$ holds.

Three (not necessarily distinct) points in the plane which have integer coordinates between $ 1$ and $2020$, inclusive, are chosen uniformly at random. The probability that the area of the triangle with these three vertices is an integer is $a/b$ in lowest terms. If the three points are collinear, the area of the degenerate triangle is $0$. Find $a + b$.

The numbers -2, 4, 6, 9 and 12 are rearranged according to these rules: 1. The largest isn't first, but it is in one of the first three places. 2. The smallest isn't last, but it is in one of the last three places. 3. The median isn't first or last. What is the average of the first and last numbers? $\textbf{(A)}\: 3.5 \qquad\textbf{(B)} \:5 \qquad\textbf{(C)} \:6.5 \qquad\textbf{(D)} \:7.5 \qquad\textbf{(E)} \:8$

The sum of the squares of the roots of the equation $x^2+2hx=3$ is $10$. The absolute value of $h$ is equal to $\textbf{(A) }-1\qquad\textbf{(B) }\textstyle\frac{1}{2}\qquad\textbf{(C) }\textstyle\frac{3}{2}\qquad\textbf{(D) }2\qquad \textbf{(E) }\text{None of these}$

Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$ Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $% P^{\prime }.$

Consider the polynomial f(x) = cx(x - 2) where $c$ is a positive real number. For any $n \in Z^+$, the notation $g_n(x)$ is a composite function $n$ times of $f$ and assume that the equation $g_n(x) = 0$ has all of the $2^n$ solutions are real numbers. 1. For $c = 5$, find in terms of $n$, the sum of all the solutions of $g_n(x)$, of which each multiple (if any) is counted only once. 2. Prove that $c\ge 1$.

Let $m$ and $n$ be positive integers. A sequence of points $(A_0,A_1,\ldots,A_n)$ on the Cartesian plane is called [i]interesting[/i] if $A_i$ are all lattice points, the slopes of $OA_0,OA_1,\cdots,OA_n$ are strictly increasing ($O$ is the origin) and the area of triangle $OA_iA_{i+1}$ is equal to $\frac{1}{2}$ for $i=0,1,\ldots,n-1$. Let $(B_0,B_1,\cdots,B_n)$ be a sequence of points. We may insert a point $B$ between $B_i$ and $B_{i+1}$ if $\overrightarrow{OB}=\overrightarrow{OB_i}+\overrightarrow{OB_{i+1}}$, and the resulting sequence $(B_0,B_1,\ldots,B_i,B,B_{i+1},\ldots,B_n)$ is called an [i]extension[/i] of the original sequence. Given two [i]interesting[/i] sequences $(C_0,C_1,\ldots,C_n)$ and $(D_0,D_1,\ldots,D_m)$, prove that if $C_0=D_0$ and $C_n=D_m$, then we may perform finitely many [i]extensions[/i] on each sequence until the resulting two sequences become identical.

We place randomly the numbers $1,2, \dots ,2006$ around a circle. A move consists of changing two neighbouring numbers. After a limited numbers of moves all the numbers are diametrically opposite to their starting position. Show that we changed at least once two numbers which had the sum $2007$.

Let $ ABC$ an acute triangle with circumcircle center $ O.$ The line $ (BO)$ intersects the circumcircle again in $ D,$ and the extension of the altitude from $ A$ intersects the circle in $ E.$ Prove that the quadrilateral $ BECD$ and the triangle $ ABC$ have the same area.

Consider in a plane $ P$ the points $ O,A_1,A_2,A_3,A_4$ such that \[ \sigma(OA_iA_j) \geq 1 \quad \forall i, j \equal{} 1, 2, 3, 4, i \neq j.\] where $ \sigma(OA_iA_j)$ is the area of triangle $ OA_iA_j.$ Prove that there exists at least one pair $ i_0, j_0 \in \{1, 2, 3, 4\}$ such that \[ \sigma(OA_iA_j) \geq \sqrt{2}.\]