We define $I(n)$ as the result when the digits of $n$ are reversed. For example, $I(123)=321$, $I(2008)=8002$. Find all integers $n$, $1\leq{n}\leq10000$ for which $I(n)=\lceil{\frac{n}{2}}\rceil$. Note: $\lceil{x}\rceil$ denotes the smallest integer greater than or equal to $x$. For example, $\lceil{2.1}\rceil=3$, $\lceil{3.9}\rceil=4$, $\lceil{7}\rceil=7$.

In quadrilateral $ABCD$ holds $AB=6$, $AD=4$, $\angle DAB=\angle ABC = 60^{\circ}$ and $\angle ADC = 90^{\circ}$. Find length of diagonals and area of the quadrilateral

Let $P_1$ and $P_2$ be regular polygons of 1985 sides and perimeters $x$ and $y$ respectively. Each side of $P_1$ is tangent to a given circle of circumference $c$ and this circle passes through each vertex of $P_2$. Prove $x + y \ge 2c$. (You may assume that $\tan \theta \ge \theta$ for $0 \le \theta < \frac{\pi}{2}$.)

1. A strip for playing "hopscotch" consists of ten squares numbered consecutively $1,2, \ldots, 10$. Clarissa and Marissa start from the center of the first square, jump 9 times to the centers of the other squares so that they visit each square once, and end up at the tenth square. (Jumps forward and backward are allowed.) Each jump of Clarissa was for the same distance as the corresponding jump of Marissa. Does this mean that they both visited the squares in the same order? Alexey Tolpygo

Determine for which natural numbers $n$ it is possible to completely cover a board of $ n \times n$, divided into $1 \times 1$ squares, with pieces like the one in the figure, without gaps or overlays and without leaving the board. Each of the pieces covers exactly six boxes. Note: Parts can be rotated. [img]https://cdn.artofproblemsolving.com/attachments/c/2/d87d234b7f9799da873bebec845c721e4567f9.png[/img]

The captain and his three sailors get $2009$ golden coins with the same value . The four people decided to divide these coins by the following rules : sailor $1$,sailor $2$,sailor $3$ everyone write down an integer $b_1,b_2,b_3$ , satisfies $b_1\ge b_2\ge b_3$ , and ${b_1+b_2+b_3=2009}$; the captain dosen't know what the numbers the sailors have written . He divides $2009$ coins into $3$ piles , with number of coins: $a_1,a_2,a_3$ , and $a_1\ge a_2\ge a_3$ . For sailor $k$ ($k=1,2,3$) , if $b_k<a_k$ , then he can take $b_k$ coins from the $k$th pile ; if $b_k\ge a_k$ , then he can't take any coins away . At last , the captain own the rest of the coins .If no matter what the numbers the sailors write , the captain can make sure that he always gets $n$ coins . Find the largest possible value of $n$ and prove your conclusion .

$n$ consecutive positive integers are put down in a row (not necessarily in order) so that the sum of any three successive integers in the row is divisible by the leftmost number in the triple. What is the largest possible value of $n$ if the last number in the row is odd? (A Shapovalov)

Albrecht fills in each cell of an $8 \times 8$ table with a $0$ or a $1$. Then at the end of each row and column he writes down the sum of the $8$ digits in that row or column, and then he erases the original digits in the table. Afterwards, he claims to Berthold that given only the sums, it is possible to restore the $64$ digits in the table uniquely. Show that the $8 \times 8$ table contained either a row full of $0$’s or a column full of $1$’s

Prove that $2^{n - 1}$ divides $n!$ if and only if $n = 2^{k - 1}$ for some positive integer $k$.

Define the operation "$\circ$" by $x \circ y=4x-3y+xy$, for all real numbers $x$ and $y$. For how many real numbers $y$ does $3 \circ y=12$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{more than}\ 4 $

Suppose that $ f: \mathbb R^2\rightarrow \mathbb R$ is a non-negative and continuous function that $ \iint_{\mathbb R^2}f(x,y)dxdy\equal{}1$. Prove that there is a closed disc $ D$ with the least radius possible such that $ \iint_D f(x,y)dxdy\equal{}\frac12$.

There are $2008$ trinomials $x^2-a_kx+b_k$ where $a_k$ and $b_k$ are all different numbers from set $(1,2,...,4016)$. These trinomials has not common real roots. We mark all real roots on the $Ox$-axis. Prove, that distance between some two marked points is $\leq \frac{1}{250}$

The circles $\mathcal{Y}_0$ and $\mathcal{Y}_1$ lies outside each other. Let $O_0$ be the center of $\mathcal{Y}_0$ and $O_1$ be the center of $\mathcal{Y}_1$. From $O_0$, draw the rays which are tangents to $\mathcal{Y}_1$ and similarty from $O_1$, draw the rays which are tangents to $\mathcal{Y}_0$. Let the intersection points of rays and circle $\mathcal{Y}_i$ be $A_i$ and $B_i$. Show that the line segments $A_0B_0$ and $A_1B_1$ have equal lengths.

$\frac{a}{c}=\frac{b}{d}=\frac{3}{4}$, $\sqrt{a^2+c^2}-\sqrt{b^2+d^2}=15$. Find $ac+bd-ad-bc$.

We say a ring $(A,+,\cdot)$ has property $(P)$ if : \[ \begin{cases} \text{the set } A \text{ has at least } 4 \text{ elements} \\ \text{the element } 1+1 \text{ is invertible}\\ x+x^4=x^2+x^3 \text{ holds for all } x \in A \end{cases} \] a) Prove that if a ring $(A,+,\cdot)$ has property $(P)$, and $a,b \in A$ are distinct elements, such that $a$ and $a+b$ are units, then $1+ab$ is also a unit, but $b$ is not a unit. b) Provide an example of a ring with property $(P)$.

Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ ,$2$ ,$3$ ,$4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$. Prove that $p$ divides all the coefficients of $Q(x)$.