In a deck of $n > 1$ cards, some digits from $1$ to$8$are written on each card. A digit may occur more than once, but at most once on a certain card. On each card at least one digit is written, and no two cards are denoted by the same set of digits. Suppose that for every $k=1,2,\dots,7$ digits, the number of cards that contain at least one of them is even. Find $n$.

A finite sequence of numbers $(a_1,\cdots,a_n)$ is said to be alternating if $$a_1>a_2~,~a_2<a_3~,~a_3>a_4~,~a_4<a_5~,~\cdots$$ $$\text{or ~}~~a_1<a_2~,~a_2>a_3~,~a_3<a_4~,~a_4>a_5~,~\cdots$$ How many alternating sequences of length $5$ , with distinct numbers $a_1,\cdots,a_5$ can be formed such that $a_i\in\{1,2,\cdots,20\}$ for $i=1,\cdots,5$ ?

Semicircles $POQ$ and $ROS$ pass through the center of circle $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? [asy] import graph; size(7.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-6.27,xmax=10.01,ymin=-5.65,ymax=10.98; draw(circle((0,0),2)); draw((-3,0)--(3,0),EndArrow(6)); draw((0,-3)--(0,3),EndArrow(6)); draw(shift((0.01,1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,179.76,359.76)); draw(shift((-0.01,-1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,-0.38,179.62)); draw((-1.4,1.43)--(1.41,1.41)); draw((-1.42,-1.41)--(1.4,-1.42)); label("$ P(-1,1) $",(-2.57,2.17),SE*lsf); label("$ Q(1,1) $",(1.55,2.21),SE*lsf); label("$ R(-1,-1) $",(-2.72,-1.45),SE*lsf); label("$S(1,-1)$",(1.59,-1.49),SE*lsf); dot((0,0),ds); label("$O$",(-0.24,-0.35),NE*lsf); dot((1.41,1.41),ds); dot((-1.4,1.43),ds); dot((1.4,-1.42),ds); dot((-1.42,-1.41),ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] $ \textbf{(A)}\ \frac{\sqrt 2}4 \qquad\textbf{(B)}\ \frac 12 \qquad\textbf{(C)}\ \frac{2}{\pi} \qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{\sqrt 2}{2} $

Prove that, for all integer $a>1$, the prime divisors of $5a^4-5a^2+1$ have the form $20k\pm1,k\in\mathbb{Z}$. [i]Proposed by Géza Kós.[/i]

The $8 \times 8$ board is covered with the same shape as in the picture to the right (each of the shapes can be rotated $90^o$) so that any two do not overlap or extend beyond the edge of the chessboard. Determine the largest possible number of fields of this chessboard can be covered as described above. [img]https://cdn.artofproblemsolving.com/attachments/e/5/d7f44f37857eb115edad5ea26400cdca04e107.png[/img]

Three equal circles have a common point $M$ and intersect in pairs at points $A, B, C$. Prove that that $M$ is the orthocenter of triangle $ABC$.

A pair of integers is special if it is of the form $(n, n-1)$ or $(n-1, n)$ for some positive integer $n$. Let $n$ and $m$ be positive integers such that pair $(n, m)$ is not special. Show $(n, m)$ can be expressed as a sum of two or more different special pairs if and only if $n$ and $m$ satisfy the inequality $ n+m\geq (n-m)^2 $. Note: The sum of two pairs is defined as $ (a, b)+(c, d) = (a+c, b+d) $.

The parabola $ y \equal{} ax^2 \plus{} bx \plus{} c$ has vertex $ (p,p)$ and $ y$-intercept $ (0, \minus{} p)$, where $ p\neq 0$. What is $ b$? $ \textbf{(A) } \minus{} p \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } p$

A point $D$ lie on the lateral side $BC$ of an isosceles triangle $ABC$. The ray $AD$ meets the line passing through $B$ and parallel to the base $AC$ at point $E$. Prove that the tangent to the circumcircle of triangle $ABD$ at $B$ bisects $EC$.

There are distinct quadratics $e(x)$, $p(x)$, $h(x)$, $r(x)$, $a(x)$, and $m(x)$ with leading coefficient $1$, such that their roots are $2$ distinct values from the set $\{3, 4, 5, 6\}$. James takes three of these quadratics, sums two, and subtracts the last. Given that this new quadratic has a root at $0$, find its other root.

[b]8.[/b] Let $f(x)$ be a convex function defined on the interval $[0, \frac {1}{2}]$ with $f(0)=0$ and $f(\frac{1}{2})=1$; Let further $f(x)$ be differentiable in $(0, \frac {1}{2})$, and differentiable at $0$ and $\frac{1}{2}$ from the right and from the left, respectively. Finally, let $f'(0)>1$. Extend $f(x)$ to $[0.1]$ in the following manner: let $f(x)= f(1-x)$ if $x \in (\frac {1} {2}, 1]$. Show that the set of the points $x$ for shich the terms of the sequence $x_{n+1}=f(x_n)$ ($x_0=x; n = 0, 1, 2, \dots $) are not all different is everywhere dense in $[0,1]$; [b](R. 10)[/b]

Prove the following inequality. \[ \sqrt[3]{\int_0^{\frac {\pi}{4}} \frac {x}{\cos ^ 2 x\cos ^ 2 (\tan x)\cos ^ 2(\tan (\tan x))\cos ^ 2(\tan (\tan (\tan x)))}dx}<\frac{4}{\pi}\] Last Edited. Sorry, I have changed the problem. kunny

Two players by turn paint the circles on the given picture each with his colour. At the end, the rest of the area of each of small triangles is painted by the colour of the majority of vertices of this triangle. The winner is one who gets larger area of his colour (the area of circles is taken into account). Does any of them have winning strategy? If yes, then who wins? \[ \begin{picture}(60,60) \put(5,3){\put(3,0){\line(6,0){8}} \put(17,0){\line(6,0){8}} \put(31,0){\line(6,0){8}} \put(45,0){\line(6,0){8}} \put(10,14){\line(6,0){8}} \put(24,14){\line(6,0){8}} \put(38,14){\line(6,0){8}} \put(17,28){\line(6,0){8}} \put(31,28){\line(6,0){8}} \put(24,42){\line(6,0){8}} \put(1,2){\line(1,2){5}} \put(15,2){\line(1,2){5}} \put(29,2){\line(1,2){5}} \put(43,2){\line(1,2){5}} \put(8,16){\line(1,2){5}} \put(22,16){\line(1,2){5}} \put(36,16){\line(1,2){5}} \put(15,30){\line(1,2){5}} \put(29,30){\line(1,2){5}} \put(22,44){\line(1,2){5}} \put(13,2){\line( \minus{} 1,2){5}} \put(27,2){\line( \minus{} 1,2){5}} \put(41,2){\line( \minus{} 1,2){5}} \put(55,2){\line( \minus{} 1,2){5}} \put(20,16){\line( \minus{} 1,2){5}} \put(34,16){\line( \minus{} 1,2){5}} \put(48,16){\line( \minus{} 1,2){5}} \put(27,30){\line( \minus{} 1,2){5}} \put(41,30){\line( \minus{} 1,2){5}} \put(34,44){\line( \minus{} 1,2){5}} \put(0,0){\circle{6}} \put(14,0){\circle{6}} \put(28,0){\circle{6}} \put(42,0){\circle{6}} \put(56,0){\circle{6}} \put(7,14){\circle{6}} \put(21,14){\circle{6}} \put(35,14){\circle{6}} \put(49,14){\circle{6}} \put(14,28){\circle{6}} \put(28,28){\circle{6}} \put(42,28){\circle{6}} \put(21,42){\circle{6}} \put(35,42){\circle{6}} \put(28,56){\circle{6}}} \end{picture}\]

[b]a)[/b] Let $ \bold {u,v,w,} $ be three coplanar vectors of absolute value $ 1. $ Show that there exist $ \varepsilon_1 ,\varepsilon_2, \varepsilon_3\in \{ \pm 1\} $ such that $$ \big| \varepsilon_1\bold u +\varepsilon_2\bold v +\varepsilon_3\bold w \big|\le 1. $$ [b]b)[/b] Give an example of three vectors such that the inequality above does not work for any sclaras from $ \{ \pm 1\} . $

For $n \ge 2$, consider $n$ boxes aligned from left to right. In each box, one puts a ball that can be red, blue or white such that the following condition is fullled: [i]Each box is neighboring at least one box containing a ball of the same color.[/i] We denote by $I_n$ the number of such congurations. a) Determine $I_{11}$. Justify your answer. b) Find, with proof, the general formula for $I_n$.

The incircle of triangle $ABC$ is tangent to the sides $AB,AC$ and $BC$ at the points $M,N$ and $K$ respectively. The median $AD$ of the triangle $ABC$ intersects $MN$ at the point $L$. Prove that $K,I$ and $L$ are collinear, where $I$ is the incenter of the triangle $ABC$.

Let real numbers $a,b,c$ such that $\left| a-b \right|\ge \left| c \right|,\left| b-c \right|\ge \left| a \right|,\left| c-a \right|\ge \left| b \right|.$ Prove that $a=b+c$ or $b=c+a$ or $c=a+b.$

A rook is allowed to move one cell either horizontally or vertically. After $64$ moves the rook visited all cells of the $8 \times 8$ chessboard and returned back to the initial cell. Prove that the number of moves in the vertical direction and the number of moves in the horizontal direction cannot be equal. (A Shapovalov, R Sadykov)

A parallelogram of paper with sides $25$ and $10$ is given. The distance between the longer sides is $6$. The paper should be cut into exactly two parts in such a way that one can stick both the pieces together and fold it in a suitable manner to form a cube of suitable edge length without any further cuts and overlaps. Show that it is really possible and describe such a fragmentation.

I give you a function $\textbf{rand}$ that returns a number chosen uniformly at random from $[0,T]$ for some number $T$ that you don't know. Your task is to approximate $T$. You do this by calling $\textbf{rand}$ $100$ times, recording the results as $X_1,X_2,\dots,X_{100}$, and guessing \[\hat{T}=\alpha\cdot\max\{X_1,X_2,\dots,X_{100}\}\] for some $\alpha$. Which value of $\alpha$ ensures that $\mathbb{E}[\hat{T}]=T$?

Find the largest $a$ for which there exists a polynomial $$P(x) =a x^4 +bx^3 +cx^2 +dx +e$$ with real coefficients which satisfies $0\leq P(x) \leq 1$ for $-1 \leq x \leq 1.$

Let $P(x, y)$ be a non-constant homogeneous polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for every real number $t$. Prove that there exists a positive integer $k$ such that $P(x, y) = (x^2 + y^2)^k$.

The product $ (1.8)(40.3\plus{}.07)$ is closest to \[ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 42 \qquad \textbf{(C)}\ 74 \qquad \textbf{(D)}\ 84 \qquad \textbf{(E)}\ 737 \]

$BC$ is a diameter of a circle and the points $X,Y$ are on the circle such that $XY\perp BC$. The points $P,M$ are on $XY,CY$ (or their stretches), respectively, such that $CY||PB$ and $CX||PM$. Let $K$ be the meet point of the lines $XC,BP$. Prove that $PB\perp MK$.

Let $n$ be an integer greater than 2. A positive integer is said to be [i]attainable [/i]if it is 1 or can be obtained from 1 by a sequence of operations with the following properties: 1.) The first operation is either addition or multiplication. 2.) Thereafter, additions and multiplications are used alternately. 3.) In each addition, one can choose independently whether to add 2 or $n$ 4.) In each multiplication, one can choose independently whether to multiply by 2 or by $n$. A positive integer which cannot be so obtained is said to be [i]unattainable[/i]. [b]a.)[/b] Prove that if $n\geq 9$, there are infinitely many unattainable positive integers. [b]b.)[/b] Prove that if $n=3$, all positive integers except 7 are attainable.