As the figure is shown, place a $2\times 5$ grid table in horizontal or vertical direction, and then remove arbitrary one $1\times 1$ square on its four corners. The eight different shapes consisting of the remaining nine small squares are called [i]banners[/i]. [asy] defaultpen(linewidth(0.4)+fontsize(10));size(50); pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9); draw(B--C--H--J--N^^B--I^^D--N^^E--M^^F--L^^G--K); draw(Aa--Ca--Ha--Ja--Aa^^Ba--Ia^^Da--Na^^Ea--Ma^^Fa--La^^Ga--Ka); [/asy] [asy] defaultpen(linewidth(0.4)+fontsize(10));size(50); pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9); draw(B--Ca--Ea--M--N^^B--O^^C--E^^Aa--Ma^^Ba--Oa^^Da--N); draw(L--Fa--Ha--J--L^^Ga--K^^P--I^^F--H^^Ja--La^^Pa--Ia); [/asy] Here is a fixed $9\times 18$ grid table. Find the number of ways to cover the grid table completely with 18 [i]banners[/i].

For every positive integer $n$ we take the greatest divisor $d$ of $n$ such that $d\leq \sqrt{n}$ and we define $a_n=\frac{n}{d}-d$. Prove that in the sequence $a_1,a_2,a_3,...$, any non negative integer $k$ its in the sequence infinitely many times.

Two distinct lines pass through the center of three concentric circles of radii $3$, $2$, and $1$. The area of the shaded region in the diagram is $8/13$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is $180$ degrees.) [asy] defaultpen(linewidth(0.8)); pair O=origin; fill(O--Arc(O, 2, 20, 160)--cycle, mediumgray); fill(O--Arc(O, 1, 20, 160)--cycle, white); fill(O--Arc(O, 2, 200, 340)--cycle, mediumgray); fill(O--Arc(O, 1, 200, 340)--cycle, white); fill(O--Arc(O, 3, 160, 200)--cycle, mediumgray); fill(O--Arc(O, 2, 160, 200)--cycle, white); fill(O--Arc(O, 1, 160, 200)--cycle, mediumgray); fill(O--Arc(O, 3, -20, 20)--cycle, mediumgray); fill(O--Arc(O, 2, -20, 20)--cycle, white); fill(O--Arc(O, 1, -20, 20)--cycle, mediumgray); draw(Circle(origin, 1));draw(Circle(origin, 2));draw(Circle(origin, 3)); draw(5*dir(200)--5*dir(20)^^5*dir(160)--5*dir(-20));[/asy] $ \textbf{(A)} \frac{\pi}8\qquad \textbf{(B)}\frac{\pi}7\qquad \textbf{(C)}\frac{\pi}6\qquad \textbf{(D)}\frac{\pi}5\qquad \textbf{(E)}\frac{\pi}4 $

Consider $ABC$ with $AC > AB$ and incenter $I$. The midpoints of $\overline{BC}$ and $\overline{AC}$ are $M$ and $N$, respectively. If $\overline{AI}$ is perpendicular to $\overline{IN}$, then prove that $\overline{AI}$ is tangent to the circumscribed circle of $\vartriangle BMI$.

Show that there are infinitely many positive real numbers a which are not integers such that a(a-3{a}) is an integer.

Define the sequence $A_1,A_2,\ldots$ of matrices by the following recurrence: $$ A_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}, \quad A_{n+1} = \begin{pmatrix} A_n & I_{2^n} \\ I_{2^n} & A_n \\ \end{pmatrix} \quad (n=1,2,\ldots) $$ where $I_m$ is the $m\times m$ identity matrix. Prove that $A_n$ has $n+1$ distinct integer eigenvalues $\lambda_0< \lambda_1<\ldots <\lambda_n$ with multiplicities $\binom{n}{0},\binom{n}{1},\ldots,\binom{n}{n}$, respectively.

a) Prove that there exist integers $a, b, c$ not all zero and each of absolute value less than one million, such that $$ |a +b \sqrt{2} +c \sqrt{3} | <10^{-11} .$$ b) Let $ a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that $$ |a +b \sqrt{2} +c \sqrt{3} | >10^{-21} .$$

For each of the following, provide proof or a counterexample: a) Every $2\times2$ matrix with real entries can we written as the sum of the squares of two $2\times2$ matrices with real entries. b) Every $3\times3$ matrix with real entries can we written as the sum of the squares of two $3\times3$ matrices with real entries.

Let $ABCD$ be a convex quadrilateral with positive area such that every side has a positive integer length and $AC=BC=AD=25$. If $P_{max}$ and $P_{min}$ are the quadrilaterals with maximum and minimum possible perimeter, the ratio of the area of $P_{max}$ and $P_{min}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ for some positive integers $a,b,c$, where $a,c$ are relatively prime and $b$ is not divisible by the square of any integer. Find $a+b+c$. [i]Proposed by [b]FedeX333X [/b][/i]

A natural number is written on the blackboard. In each step, we erase the units digit and add four times the erased digit to the remaining number, and write the result on the blackboard instead of the initial number. Starting with the number $13^{2006}$, is it possible to obtain the number $2006^{13}$ by repeating this step finitely many times?

Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in $3$, Anne and Carl in $5$. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?

Point $M$ lies on the diagonal $BD$ of parallelogram $ABCD$ such that $MD = 3BM$. Lines $AM$ and $BC$ intersect in point $N$. What is the ratio of the area of triangle $MND$ to the area of parallelogram $ABCD$?

In the plane are given $100$ points, such that no three of them are on the same line. The points are arranged in $10$ groups, any group containing at least $3$ points. Any two points in the same group are joined by a segment. a) Determine which of the possible arrangements in $10$ such groups is the one giving the minimal numbers of triangles. b) Prove that there exists an arrangement in such groups where each segment can be coloured with one of three given colours and no triangle has all edges of the same colour. [i]Vasile Pop[/i]

Let $a$, $b$ and $c$ be lengths of sides of triangle $ABC$. Prove that at least one of the equations $$x^2-2bx+2ac=0$$ $$x^2-2cx+2ab=0$$ $$x^2-2ax+2bc=0$$ does not have real solutions

Let $ABC$ be a triangle with $AB < AC$ inscribed in $(O)$. Tangent line at $A$ of $(O)$ cuts $BC$ at $D$. Take $H$ as the projection of $A$ on $OD$ and $E,F$ as projections of $H$ on $AB,AC$.Suppose that $EF$ cuts $(O)$ at $R,S$. Prove that $(HRS)$ is tangent to $OD$

Let $S$ be the unit circle in the complex plane (i.e. the set of all complex numbers with their moduli equal to $1$). We define function $f:S\rightarrow S$ as follow: $\forall z\in S$, $ f^{(1)}(z)=f(z), f^{(2)}(z)=f(f(z)), \dots,$ $f^{(k)}(z)=f(f^{(k-1)}(z)) (k>1,k\in \mathbb{N}), \dots$ We call $c$ an $n$-[i]period-point[/i] of $f$ if $c$ ($c\in S$) and $n$ ($n\in\mathbb{N}$) satisfy: $f^{(1)}(c) \not=c, f^{(2)}(c) \not=c, f^{(3)}(c) \not=c, \dots, f^{(n-1)}(c) \not=c, f^{(n)}(c)=c$. Suppose that $f(z)=z^m$ ($z\in S; m>1, m\in \mathbb{N}$), find the number of $1989$-[i]period-point[/i] of $f$.

For positive real numbers $ a$, $ b$, $ c$ and $ d$ such that $ a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2 \equal{} 1$ prove that $ a^2b^2cd \plus{} \plus{}ab^2c^2d \plus{} abc^2d^2 \plus{} a^2bcd^2 \plus{} a^2bc^2d \plus{} ab^2cd^2 \le 3/32,$ and determine the cases of equality.

A mouse walks on a plane. At time $i$, it could do nothing or turn right, then it moves $p_i$ meters forward, where $p_i$ is the $i$-th prime. Is it possible that the mouse moves back to the starting point?

Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

Let $A_1, A_2, ..., A_k$ be $5$-element subsets of set $\{1, 2, ..., 23\}$ such that, for all $1 \le i < j \le k$ set $A_i \cap A_j$ has at most three elements. Show that $k \le 2018$.

Let ${A, B, C}$ and ${O}$ be four points in plane, such that $\angle ABC>{{90}^{{}^\circ }}$ and ${OA=OB=OC}$.Define the point ${D\in AB}$ and the line ${l}$ such that ${D\in l, AC\perp DC}$ and ${l\perp AO}$. Line ${l}$ cuts ${AC}$at ${E}$ and circumcircle of ${ABC}$ at ${F}$. Prove that the circumcircles of triangles ${BEF}$and ${CFD}$are tangent at ${F}$.

Two people, $A$ and $B$, play the following game with a deck of 32 cards. With $A$ starting, and thereafter the players alternating, each player takes either 1 card or a prime number of cards. Eventually all of the cards are chosen, and the person who has none to pick up is the loser. Who will win the game if they both follow optimal strategy?

Which of the following numbers is largest? $ \text{(A)}\ 9.12344\qquad\text{(B)}\ 9.123\overline{4}\qquad\text{(C)}\ 9.12\overline{34}\qquad\text{(D)}\ 9.1\overline{234}\qquad\text{(E)}\ 9.\overline{1234} $

Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$ a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple. b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple. [i]There are exactly $n$ factors in the product on the left hand side.[/i]