Determine the number of paths from $A$ to $B$ on the picture that go along gridlines only, do not pass through any point twice, and never go upwards? [img]https://cdn.artofproblemsolving.com/attachments/0/2/87868e24a48a2e130fb5039daeb85af42f4b9d.png[/img]

Let $n$ be a given positive integer. Prove that there are infinitely many pairs of positive integers $(a, b)$ with $a, b > n$ such that $$\prod_{i=1}^{2015} (a + i) | b(b + 2016), \prod_{i=1}^{2015}(a + i) \nmid b, \prod_{i=1}^{2015} (a + i)\mid (b + 2016)$$.

Call a positive integer [i]monotonous[/i] if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. How many monotonous positive integers are there? $\textbf{(A)} \text{ 1024} \qquad \textbf{(B)} \text{ 1524} \qquad \textbf{(C)} \text{ 1533} \qquad \textbf{(D)} \text{ 1536} \qquad \textbf{(E)} \text{ 2048}$

A boy write three times the natural number $n$ in a blackboard. He then performed an operation of the following type several times: He erased one of the numbers and wrote in its place the sum of the two others minus $1$. After several moves, one of the three numbers in the blackboard is $900$. Find all the posible values of $n$.

Let $a=3^{1/223}+1$ and for all $n \ge 3$ let \[f(n)= \dbinom{n}{0} a^{n-1} - \dbinom{n}{1} a^{n-2} + \dbinom{n}{2} a^{n-3}- ... +(-1)^{n-1} \dbinom{n}{n-1} a^0.\] Find $f(2007)+f(2008).$

There is a list of $n$ football matches. Determine how many possible columns, with an even number of draws, there are.

In a class, there are $n \geqslant 3$ students and a teacher with $M$ marbles. The teacher then play a [i]Marble distribution[/i] according to the following rules. At the start, the teacher distributed all her marbles to students, so that each student receives at least $1$ marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of $2(n-1)$. That chosen student then equally distribute half of his marbles to $n-1$ other students. The same goes on until the teacher is not able to choose anymore student. Find all integer $M$, such that for some initial numbers of marbles that the students receive, the teacher can choose all the student(according to the rule above), so that each student receiving equal amount of marbles at the end.

Consider an acute-angled triangle $ABC$ and its circumcircle. Let $D$ be a point on the arc $AB$ which does not include point $C$ and let $A_1$ and $B_1$ be points on the lines $DA$ and $DB$, respectively, such that $CA_1 \perp DA$ and $CB_1 \perp DB$. Prove that $|AB| \ge |A_1B_1|$.

Let $\vartriangle ABC$ be a triangle such that angle $\angle ABC$ is less than angle $\angle ACB$. The bisector of angle $\angle BAC$ cuts side $BC$ at $D$. Let $E$ be on side $AB$ such that $\angle EDB = 90^o$ and $F$ on side $AC$ such that $\angle BED = \angle DEF$. Prove that $\angle BAD = \angle FDC$.

Let $ABC$ be a triangle with $AB = 13$, $AC = 14$, and $BC = 15$, and let $\Gamma$ be its incircle with incenter $ I$. Let $D$ and $E$ be the points of tangency between $\Gamma$ and $BC$ and $AC$ respectively, and let $\omega$ be the circle inscribed in $CDIE$. If $Q$ is the intersection point between $\Gamma$ and $\omega$ and $P$ is the intersection point between $CQ$ and $\omega$, compute the length of $P Q$.

Among the points corresponding to number $1,2,...,2n$ on the real line, $n$ are colored in blue and $n$ in red. Let $a_1,a_2,...,a_n$ be the blue points and $b_1,b_2,...,b_n$ be the red points. Prove that the sum $\mid a_1-b_1\mid+...+\mid a_n-b_n\mid$ does not depend on coloring , and compute its value. :roll:

We are given fixed circles $\Omega$ and $\omega$ such that there exists a hexagon $ABCDEF$ inscribed in $\Omega$ and circumscribed around $\omega$. (Note that then, by virtue of Poncelet's theorem, there is an infinite family of such hexagons.) Prove that the value of $\dfrac{S_{ABCDEF}}{AD+BE+CF}$ it does not depend on the choice of the hexagon $ABCDEF$. [i]A. Zaslavsky and Tran Quang Hung[/i]

Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.

Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.

Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.

Turbo the snail plays a game on a board with $2024$ rows and $2023$ columns. There are hidden monsters in $2022$ of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at most one monster. Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an adjacent cell sharing a common side. (He is allowed to return to a previously visited cell.) If he reaches a cell with a monster, his attempt ends and he is transported back to the first row to start a new attempt. The monsters do not move, and Turbo remembers whether or not each cell he has visited contains a monster. If he reaches any cell in the last row, his attempt ends and the game is over. Determine the minimum value of $n$ for which Turbo has a strategy that guarantees reaching the last row on the $n$-th attempt or earlier, regardless of the locations of the monsters. [i]Proposed by Cheuk Hei Chu, Hong Kong[/i]

Three lights are placed horizontally on a line on the ceiling. All the lights are initially off. Every second, Neil picks one of the three lights uniformly at random to switch: if it is off, he switches it on; if it is on, he switches it off. When a light is switched, any lights directly to the left or right of that light also get turned on (if they were off) or off (if they were on). The expected number of lights that are on after Neil has flipped switches three times can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

suppose that $p$ is a prime number. find that smallest $n$ such that there exists a non-abelian group $G$ with $|G|=p^n$. SL is an acronym for Special Lesson. this year our special lesson was Groups and Symmetries. the exam time was 5 hours.

Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b}- \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.

The alphabet of language $BAU$ consists of letters $B, A$, and $U$. Independently of the choice of the $BAU$ word of length n from which to start, one can construct all the $BAU$ words with length n using iteratively the following rules: (1) invert the order of the letters in the word; (2) replace two consecutive letters: $BA \to UU, AU \to BB, UB \to AA, UU \to BA, BB \to AU$ or $AA \to UB$. Given that $BBAUABAUUABAUUUABAUUUUABB$ is a $BAU$ word, does $BAU$ have a) the word $BUABUABUABUABAUBAUBAUBAUB$ ? b) the word $ABUABUABUABUAUBAUBAUBAUBA$ ?

Let $ABC$ be a triangle with points $E, F$ on $CA$, $AB$, respectively. Circle $C_1$ passes through $E, F$ and is tangent to segment $BC$ at $D$. Suppose that $AE = AF = EF = 3$, $BF = 1$, and $CE = 2$. What is $\frac{ED^2}{F D^2}$ ?

A [i]short-sighted[/i] rook is a rook that beats all squares in the same column and in the same row for which he can not go more than $60$-steps. What is the maximal amount of short-sighted rooks that don't beat each other that can be put on a $100\times 100$ chessboard.

Let $\triangle ABC$ a triangle, and $D$ the intersection of the angle bisector of $\angle BAC$ and the perpendicular bisector of $AC$. the line parallel to $AC$ passing by the point $B$, intersect the line $AD$ at $X$. the line parallel to $CX$ passing by the point $B$, intersect $AC$ at $Y$. $E = (AYB) \cap BX$ . prove that $C$ , $D$ and $E$ collinear.

Let $M$ be the set of all lattice points in the plane (i.e. points with integer coordinates, in a fixed Cartesian coordinate system). For any point $P=(x,y)\in M$ we call the points $(x-1,y)$, $(x+1,y)$, $(x,y-1)$, $(x,y+1)$ neighbors of $P$. Let $S$ be a finite subset of $M$. A one-to-one mapping $f$ of $S$ onto $S$ is called perfect if $f(P)$ is a neighbor of $P$, for any $P\in S$. Prove that if such a mapping exists, then there exists also a perfect mapping $g:S\to S$ with the additional property $g(g(P))=P$ for $P\in S$.

Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.