In the real axis, there is bug standing at coordinate $x=1$. Each step, from the position $x=a$, the bug can jump to either $x=a+2$ or $x=\frac{a}{2}$. Show that there are precisely $F_{n+4}-(n+4)$ positions (including the initial position) that the bug can jump to by at most $n$ steps. Recall that $F_n$ is the $n^{th}$ element of the Fibonacci sequence, defined by $F_0=F_1=1$, $F_{n+1}=F_n+F_{n-1}$ for all $n\geq 1$.

Yuri is looking at the great Mayan table. The table has $200$ columns and $2^{200}$ rows. Yuri knows that each cell of the table depicts the sun or the moon, and any two rows are different (i.e. differ in at least one column). Each cell of the table is covered with a sheet. The wind has blown aways exactly two sheets from each row. Could it happen that now Yuri can find out for at least $10000$ rows what is depicted in each of them (in each of the columns)? [i]Proposed by I. Bogdanov, K. Knop[/i]

What is the smallest number of sides that an polygon can have (not necessarily convex), which can be cut into parallelograms?

How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

A checkered table consists of $2012$ rows and $k > 2$ columns. A marker is placed in a cell of the left-most column. Two players move the marker in turns. During each move, the player moves the marker by $1$ cell to the right, up or down to a cell that had never been occupied by the marker before. The game is over when any of the players moves the marker to the right-most column. However, whether this player is to win or to lose, the players are advised only when the marker reaches the second column from the right. Can any player secure his win?

Two persons are playing the following game. They have a pile of stones and take turns removing stones from it, with the first player taking the first turn. At each turn, the first player removes either 1 or 10 stones from the pile, and the second player removes either m or n stones. The player who can not make his move loses. It is known that for any number of stones in the pile, the first player can always win (regardless of the second player's moves). What are the possible values of m and n?

For finite sets $A,M$ such that $A \subseteq M \subset \mathbb{Z}^+$, we define $$f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}.$$ Given a positive integer $k$, we call $M$ [i]k-colorable[/i] if it is possible to color the subsets of $M$ with $k$ colors so that for any $A \subseteq M$, if $f_M(A)\neq A$ then $f_M(A)$ and $A$ have different colors. Determine the least positive integer $k$ such that every finite set $M \subset\mathbb{Z}^+$ is k-colorable.

Given natural $n$. We shall call "universal" such a sequence of natural number $a_1, a_2, ... , a_k, k\ge n$, if we can obtain every transposition of the first $n$ natural numbers (i.e such a sequence of $n$ numbers, that every one is encountered only once) by deleting some its members. (Examples: ($1,2,3,1,2,1,3$) is universal for $n=3$, and ($1,2,3,2,1,3,1$) -- not, because you can't obtain ($3,1,2$) from it.) The goal is to estimate the length of the shortest universal sequence for given $n$. a) Give an example of the universal sequence of $n2$ members. b) Give an example of the universal sequence of $(n^2 - n + 1)$ members. c) Prove that every universal sequence contains not less than $n(n + 1)/2$ members d) Prove that the shortest universal sequence for $n=4$ contains 12 members e) Find as short universal sequence, as you can. The Organising Committee knows the method for $(n^2 - 2n +4) $ members.

[b]p1.[/b] Find the largest k such that the equation $x^2 - 2x + k = 0$ has at least one real root. [b]p2.[/b] Indiana Jones needs to cross a flimsy rope bridge over a mile long gorge. It is so dark that it is impossible to cross the bridge without a flashlight. Furthermore, the bridge is so weak that it can only support the weight of two people. The party has only one flashlight, which has a weak beam so whenever two people cross, they are constrained to walk together, at the speed of the slower person. Indiana Jones can cross the bridge in $5$ minutes. His girlfriend can cross in $10$ minutes. His father needs $20$ minutes, and his father’s side kick needs $25$ minutes. They need to get everyone across safely in on hour to escape the bay guys. Can they do it? [b]p3.[/b] There are ten big bags with coins. Nine of them contain fare coins weighing $10$ g. each, and one contains counterfeit coins weighing $9$ g. each. By one weighing on a digital scale find the bag with counterfeit coins. [b]p4.[/b] Solve the equation: $\sqrt{x^2 + 4x + 4} = x^2 + 5x + 5$. [b]p5.[/b] (a) In the $x - y$ plane, analytically determine the length of the path $P \to A \to C \to B \to P$ around the circle $(x - 6)^2 + (y - 8)^2 = 25$ from the point $P(12, 16)$ to itself. [img]https://cdn.artofproblemsolving.com/attachments/f/b/24888b5b478fa6576a54d0424ce3d3c6be2855.png[/img] (b) Determine coordinates of the points $A$ and $B$. [b]p6.[/b] (a) Let $ABCD$ be a convex quadrilateral (it means that diagonals are inside the quadrilateral). Prove that $$Area\,\, (ABCD) \le \frac{|AB| \cdot |AD| + |BC| \cdot |CD|}{2}$$ (b) Let $ABCD$ be an arbitrary quadrilateral (not necessary convex). Prove the same inequality as in part (a). (c) For an arbitrary quadrilateral $ABCD$ prove that $Area\,\, (ABCD) \le \frac{|AB| \cdot |CD| + |BC| \cdot |AD|}{2}$ PS. You should use hide for answers.

Let $n$ be a positive integer. An $n \times n$ matrix (a rectangular array of numbers with $n$ rows and $n$ columns) is said to be a platinum matrix if: [list=i] [*] the $n^2$ entries are integers from $1$ to $n$; [*] each row, each column, and the main diagonal (from the upper left corner to the lower right corner) contains each integer from $1$ to $n$ exactly once; and [*] there exists a collection of $n$ entries containing each of the numbers from $1$ to $n$, such that no two entries lie on the same row or column, and none of which lie on the main diagonal of the matrix. [/list] Determine all values of $n$ for which there exists an $n \times n$ platinum matrix.

All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.

On the whiteboard, the numbers are written sequentially: $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9$. Andi has to paste a $+$ (plus) sign or $-$ (minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process.

[b]p1[/b] How many two-digit positive integers with distinct digits satisfy the conditions that 1) neither digit is $0$, and 2) the units digit is a multiple of the tens digit? [b]p2[/b] Mirabel has $47$ candies to pass out to a class with $n$ students, where $10\le n < 20$. After distributing the candy as evenly as possible, she has some candies left over. Find the smallest integer $k$ such that Mirabel could have had $k$ leftover candies. [b]p3[/b] Callie picks two distinct numbers from $\{1, 2, 3, 4, 5\}$ at random. The probability that the sum of the numbers she picked is greater than the sum of the numbers she didn’t pick is $p$. $p$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd (a, b) = 1$. Find $a + b$.

For each positive integer $m$, we define $L_m$ as the figure that is obtained by overlapping two $1 \times m$ and $m \times 1$ rectangles in such a way that they coincide at the $1 \times 1$ square at their ends, as shown in the figure. [asy] pair h = (1, 0), v = (0, 1), o = (0, 0); for(int i = 1; i < 5; ++i) { o = (i*i/2 + i, 0); draw(o -- o + i*v -- o + i*v + h -- o + h + v -- o + i*h + v -- o + i*h -- cycle); string s = "$L_" + (string)(i) + "$"; label(s, o + ((i / 2), -1)); for(int j = 1; j < i; ++j) { draw(o + j*v -- o + j*v + h); draw(o + j*h -- o + j*h + v); } } label("...", (18, 0.5)); [/asy] Using some figures $L_{m_1}, L_{m_2}, \dots, L_{m_k}$, we cover an $n \times n$ board completely, in such a way that the edges of the figure coincide with lines in the board. Among all possible coverings of the board, find the minimal possible value of $m_1 + m_2 + \dots + m_k$. Note: In covering the board, the figures may be rotated or reflected, and they may overlap or not be completely contained within the board.

Let $n$ be a fixed positive integer and fix a point $O$ in the plane. There are $n$ lines drawn passing through the point $O$. Determine the largest $k$ (depending on $n$) such that we can always color $k$ of the $n$ lines red in such a way that no two red lines are perpendicular to each other. [i]Proposed by Nikola Velov[/i]

Suppose that a professor has $n \ge 4$ students. Let $P$ denote the set of all ordered pairs $(n, k)$ such that the number of ways for the professor to choose one pair of students equals the number of ways for the professor to choose $k > 1$ pairs of students. For each such ordered pair $(n, k) \in P$, consider the sum $n+k=s$. Find the sum of all $s$ over all ordered pairs $(n, k)$ in $P$. [i]If the same value of $s$ appears in multiple distinct elements $(n, k)$ in $P$, count this value multiple times.[/i]

A group of 6 students decided to make [i]study groups[/i] and [i]service activity groups[/i] according to the following principle: Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members. Supposing there are at least one group and no three students belong to the same study group and service activity group, find the minimum number of groups.

There are eight rooks on a chessboard, no two attacking each other. Prove that some two of the pairwise distances between the rooks are equal. (The distance between two rooks is the distance between the centers of their cell.)

In a certain language there are $n$ letters. A sequence of letters is a word, if there are no two equal letters between two other equal letters. Find the number of words of the maximum length.

The streets of the city of Duzhinsk are simple polygonal lines not intersecting each other in internal points. Each street connects two crossings and is colored in one of three colors: white, red, or blue. At each crossing exactly three streets meet, one of each color. A crossing is called positive if the streets meeting at it are white, blue and red in counterclockwise direction, and negative otherwise. Prove that the difference between the numbers of positive and negative crossings is a multiple of 4.

A positive integer $n$ is called nice if it has at least $3$ proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is nice because its largest proper divisors are $3,2,1$ and $6=3+2+1$. Find the number of nice integers not greater than $3000$.

[b]p1.[/b] Let $U = \{-2, 0, 1\}$ and $N = \{1, 2, 3, 4, 5\}$. Let $f$ be a function that maps $U$ to $N$. For any $x \in U$, $x + f(x) + xf(x)$ is an odd number. How many $f$ satisfy the above statement? [b]p2.[/b] Around a circle are written all of the positive integers from $ 1$ to $n$, $n \ge 2$ in such a way that any two adjacent integers have at least one digit in common in their decimal expressions. Find the smallest $n$ for which this is possible. [b]p3.[/b] Michael loses things, especially his room key. If in a day of the week he has $n$ classes he loses his key with probability $n/5$. After he loses his key during the day he replaces it before he goes to sleep so the next day he will have a key. During the weekend(Saturday and Sunday) Michael studies all day and does not leave his room, therefore he does not lose his key. Given that on Monday he has 1 class, on Tuesday and Thursday he has $2$ classes and that on Wednesday and Friday he has $3$ classes, what is the probability that loses his key at least once during a week? [b]p4.[/b] Given two concentric circles one with radius $8$ and the other $5$. What is the probability that the distance between two randomly chosen points on the circles, one from each circle, is greater than $7$ ? [b]p5.[/b] We say that a positive integer $n$ is lucky if $n^2$ can be written as the sum of $n$ consecutive positive integers. Find the number of lucky numbers strictly less than $2015$. [b]p6.[/b] Let $A = \{3^x + 3^y + 3^z|x, y, z \ge 0, x, y, z \in Z, x < y < z\}$. Arrange the set $A$ in increasing order. Then what is the $50$th number? (Express the answer in the form $3^x + 3^y + 3^z$). [b]p7.[/b] Justin and Oscar found $2015$ sticks on the table. I know what you are thinking, that is very curious. They decided to play a game with them. The game is, each player in turn must remove from the table some sticks, provided that the player removes at least one stick and at most half of the sticks on the table. The player who leaves just one stick on the table loses the game. Justin goes first and he realizes he has a winning strategy. How many sticks does he have to take off to guarantee that he will win? [b]p8.[/b] Let $(x, y, z)$ with $x \ge y \ge z \ge 0$ be integers such that $\frac{x^3+y^3+z^3}{3} = xyz + 21$. Find $x$. [b]p9.[/b] Let $p < q < r < s$ be prime numbers such that $$1 - \frac{1}{p} -\frac{1}{q} -\frac{1}{r}- \frac{1}{s}= \frac{1}{pqrs}.$$ Find $p + q + r + s$. [b]p10.[/b] In ”island-land”, there are $10$ islands. Alex falls out of a plane onto one of the islands, with equal probability of landing on any island. That night, the Chocolate King visits Alex in his sleep and tells him that there is a mountain of chocolate on one of the islands, with equal probability of being on each island. However, Alex has become very fat from eating chocolate his whole life, so he can’t swim to any of the other islands. Luckily, there is a teleporter on each island. Each teleporter will teleport Alex to exactly one other teleporter (possibly itself) and each teleporter gets teleported to by exactly one teleporter. The configuration of the teleporters is chosen uniformly at random from all possible configurations of teleporters satisfying these criteria. What is the probability that Alex can get his chocolate? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are n boys and m girls at Daehan Mathematical High School. Let $d(B)$ a number of girls who know Boy $B$ each other, and let $d(G)$ a number of boys who know Girl $G$ each other. Each girl knows at least one boy each other. Prove that there exist Boy $B$ and Girl $G$ who knows each other in condition that $\frac{d(B)}{d(G)}\ge\frac{m}{n}$.

p1. A fraction is called Toba-$n$ if the fraction has a numerator of $1$ and the denominator of $n$. If $A$ is the sum of all the fractions of Toba-$101$, Toba-$102$, Toba-$103$, to Toba-$200$, show that $\frac{7}{12} <A <\frac56$. p2. If $a, b$, and $c$ satisfy the system of equations $$ \frac{ab}{a+b}=\frac12$$ $$\frac{bc}{b+c}=\frac13 $$ $$ \frac{ac}{a+c}=\frac17 $$ Determine the value of $(a- c)^b$. p3. Given triangle $ABC$. If point $M$ is located at the midpoint of $AC$, point $N$ is located at the midpoint of $BC$, and the point $P$ is any point on $AB$. Determine the area of ​​the quadrilateral $PMCN$. [img]https://cdn.artofproblemsolving.com/attachments/4/d/175e2d55f889b9dd2d8f89b8bae6c986d87911.png[/img] p4. Given the rule of motion of a particle on a flat plane $xy$ as following: $N: (m, n)\to (m + 1, n + 1)$ $T: (m, n)\to (m + 1, n - 1)$, where $m$ and $n$ are integers. How many different tracks are there from $(0, 3)$ to $(7, 2)$ by using the above rules ? p5. Andra and Dedi played “SUPER-AS”. The rules of this game as following. Players take turns picking marbles from a can containing $30$ marbles. For each take, the player can take the least a minimum of $ 1$ and a maximum of $6$ marbles. The player who picks up the the last marbels is declared the winner. If Andra starts the game by taking $3$ marbles first, determine how many marbles should be taken by Dedi and what is the next strategy to take so that Dedi can be the winner.

What is the greatest number of parts that $5$ spheres can divide the space into?