[b]p1.[/b] Prove that $x = 2$ is the only real number satisfying $3^x + 4^x = 5^x$. [b]p2.[/b] Show that $\sqrt{9 + 4\sqrt5} -\sqrt{9 - 4\sqrt5}$ is an integer. [b]p3.[/b] Two players $A$ and $B$ play a game on a round table. Each time they take turn placing a round coin on the table. The coin has a uniform size, and this size is at least $10$ times smaller than the table size. They cannot place the coin on top of any part of other coins, and the whole coin must be on the table. If a player cannot place a coin, he loses. Suppose $A$ starts first. If both of them plan their moves wisely, there will be one person who will always win. Determine whether $A$ or $B$ will win, and then determine his winning strategy. [b]p4.[/b] Suppose you are given $4$ pegs arranged in a square on a board. A “move” consists of picking up a peg, reflecting it through any other peg, and placing it down on the board. For how many integers $1 \le n \le 2013$ is it possible to arrange the $4$ pegs into a [i]larger [/i] square using exactly $n$ moves? Justify your answers. [b]p5.[/b] Find smallest positive integer that has a remainder of $1$ when divided by $2$, a remainder of $2$ when divided by $3$, a remainder of $3$ when divided by $5$, and a remainder of $5$ when divided by $7$. [b]p6.[/b] Find the value of $$\sum_{m|496,m>0} \frac{1}{m},$$ where $m|496$ means $496$ is divisible by $m$. [b]p7.[/b] What is the value of $${100 \choose 0}+{100 \choose 4}+{100 \choose 8}+ ... +{100 \choose 100}?$$ [b]p8.[/b] An $n$-term sequence $a_0, a_1, ...,a_n$ will be called [i]sweet [/i] if, for each $0 \le i \le n -1$, $a_i$ is the number of times that the number $i$ appears in the sequence. For example, $1, 2, 1,0$ is a sweet sequence with $4$ terms. Given that $a_0$, $a_1$, $...$, $a_{2013}$ is a sweet sequence, find the value of $a^2_0+ a^2_1+ ... + a^2_{2013}.$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a cube $C$ and two planes $\sigma, \tau$, which divide Euclidean space into several regions. Prove that the interior of at least one of these regions meets at least three faces of the cube.

Let $G = (V, E)$ be a simple connected graph. Show that there exists a subset of edges $F \subseteq E$ such that every vertex in $H = (V, F)$ has odd degree if and only if $|V |$ is even. Note: A connected graph is a graph such that any two vertices have a sequence of edges connecting one to the other. Note: A simple graph has no loops (edges of the form $(v, v)$) or duplicate edges.

a) Calculate the product $$\left(1+\frac{1}{2}\right) \left(1+\frac{1}{3}\right) \left(1+\frac{1}{4}\right)... \left(1+\frac{1}{2006}\right) \left(1+\frac{1}{2007}\right)$$ b) Let the set $$A =\left\{\frac{1}{2}, \frac{1}{3},\frac{1}{4}, ...,\frac{1}{2006}, \frac{1}{2007}\right\}$$ Determine the sum of all products of $2$, of $4$, of $6$,... , of $2004$ ¸and of $ 2006$ different elements of the set $A$.

On the Alphamegacentavra planet there are $2023$ cities, some of which are connected by non-directed flights. It turned out that among any $4$ cities one can find two with no flight between them. Find the maximum number of triples of cities such that between any two of them there is a flight.

Find the intersection of all sets of consecutive positive integers having at least four elements and the sum of elements equal to $2001$.

At a point on the real line sits a greyhound. On one of the sides a hare runs, away from the hound. The only thing known is that the (maximal) speed of the hare is strictly less than the (maximal) speed of the greyhound (but not their precise ratio). Does the greyhound have a strategy for catching the hare in a finite amount of time? ([i]Dan Schwarz[/i])

Let $a_1,a_2,..., a_n$ be different integers and let $(b_1,b_2,..., b_n),(c_1,c_2,..., c_n)$ be two of their permutations, different from the identity. Prove that $$(|a_1-b_1|+|a_2-b_2|+...+|a_n-b_n| , |a_1-c_1|+|a_2-c_2|+...+|a_n-c_n| ) \ge 2$$ where $(x,y)$ denotes the greatest common divisor of the numbers $x,y$

Let $n> 2$ be a natural number. We consider $n$ candy bags, each containing exactly one candy. Ali and Omar play the following game in which they move alternately (Ali moves the first): At each move, the player who has to make a move chooses two bags containing $x$, respectively $y$ candy, with $(x,y)=1$, and he puts the $x + y$ candies in one bag (he chooses where). The player who can't make a move loses. Which of the two players has a strategy to win this game?

Alice and Bob want to play a game. In the beginning of the game, they are teleported to two random position on a train, whose length is $ 1 $ km. This train is closed and dark. So they dont know where they are. Fortunately, both of them have iPhone 133, it displays some information: 1. Your facing direction 2. Your total walking distance 3. whether you are at the front of the train 4. whether you are at the end of the train Moreover, one may see the information of the other one. Once Alice and Bob meet, the game ends. Alice and Bob can only discuss their strategy before the game starts. Find the least value $ x $ so that they are guarantee to end the game with total walking distance $ \le x $ km.

How many permutations $s$ does the set $\{1,2,..., 15\}$ have with the following properties: for every $1 \le k \le 13$ we have $s(k) < s(k+2)$ and for every $1 \le k \le 12$ we have $s(k) < s(k+3)$?

p1. Let $A = \{(x, y)|3x + 5y\ge 15, x + y^2\le 25, x\ge 0, x, y$ integer numbers $\}$. Find all pairs of $(x, zx)\in A$ provided that $z$ is non-zero integer. p2. A shop owner wants to be able to weigh various kinds of weight objects (in natural numbers) with only $4$ different weights. (For example, if he has weights $ 1$, $2$, $5$ and $10$. He can weighing $ 1$ kg, $2$ kg, $3$ kg $(1 + 2)$, $44$ kg $(5 - 1)$, $5$ kg, $6$ kg, $7$ kg, $ 8$ kg, $9$ kg $(10 - 1)$, $10$ kg, $11$ kg, $12$ kg, $13$ kg $(10 + 1 + 2)$, $14$ kg $(10 + 5 -1)$, $15$ kg, $16$ kg, $17$ kg and $18$ kg). If he wants to be able to weigh all the weight from $ 1$ kg to $40$ kg, determine the four weights that he must have. Explain that your answer is correct. p3. Given the following table. [img]https://cdn.artofproblemsolving.com/attachments/d/8/4622407a72656efe77ccaf02cf353ef1bcfa28.png[/img] Table $4\times 4$ ​​is a combination of four smaller table sections of size $2\times 2$. This table will be filled with four consecutive integers such that: $\bullet$ The horizontal sum of the numbers in each row is $10$ . $\bullet$ The vertical sum of the numbers in each column is $10$ $\bullet$ The sum of the four numbers in each part of $2\times 2$ which is delimited by the line thickness is also equal to $10$. Determine how many arrangements are possible. p4. A sequence of real numbers is defined as following: $U_n=ar^{n-1}$, if $n = 4m -3$ or $n = 4m - 2$ $U_n=- ar^{n-1}$, if $n = 4m - 1$ or $n = 4m$, where $a > 0$, $r > 0$, and $m$ is a positive integer. Prove that the sum of all the $ 1$st to $2009$th terms is $\frac{a(1+r-r^{2009}+r^{2010})}{1+r^2}$ 5. Cube $ABCD.EFGH$ is cut into four parts by two planes. The first plane is parallel to side $ABCD$ and passes through the midpoint of edge $BF$. The sceond plane passes through the midpoints $AB$, $AD$, $GH$, and $FG$. Determine the ratio of the volumes of the smallest part to the largest part.

Michael and James are playing a game where they alternate throwing darts at a simplified dartboard. Each dart throw is worth either 25 points or 50 points. They track the sequence of scores per throw (which is shared between them), and on the first time the three most recent scores sum to 125, the person who threw the last dart wins. On each throw, a given player has a $2/3$ chance of getting the score they aim for, and a $1/3$ chance of getting the other score. Suppose Michael goes first, and the first two throws are both 25. If both players use an optimal strategy, what is the probability Michael wins? [i]Proposed by Michael Duncan[/i]

A sequence $a_0, a_1, \dots , a_n, \dots$ of positive integers is constructed as follows: [list] [*] If the last digit of $a_n$ is less than or equal to $5$, then this digit is deleted and $a_{n+1}$ is the number consisting of the remaining digits. (If $a_{n+1}$ contains no digits, the process stops.) [*] Otherwise, $a_{n+1}= 9a_n$. [/list] Can one choose $a_0$ so that this sequence is infinite?

Two teams $A$ and $B$ play a school ping pong tournament. The team $A$ consists of $m$ students, and the team $B$ consists of $n$ students where $m \ne n$. There is only one ping pong table to play and the tournament is organized as follows: Two students from different teams start to play while other players form a line waiting for their turn to play. After each game the first player in the line replaces the member of the same team at the table and plays with the remaining player. The replaced player then goes to the end of the line. Prove that every two players from the opposite teams will eventually play against each other.

An approximate construction of a regular pentagon goes as follows. Inscribe an arbitrary convex pentagon $P_1P_2P_3P_4P_5$ in a circle. Now choose an arror bound $\epsilon > 0$ and apply the following procedure. (a) Denote $P_0 = P_5$ and $P_6 = P_1$ and construct the midpoint $Q_i$ of the circular arc $P_{i-1}P_{i+1}$ containing $P_i$. (b) Rename the vertices $Q_1,...,Q_5$ as $P_1,...,P_5$. (c) Repeat this procedure until the difference between the lengths of the longest and the shortest among the arcs $P_iP_{i+1}$ is less than $\epsilon$. Prove this procedure must end in a finite time for any choice of $\epsilon$ and the points $P_i$.

In a class of $ n\geq 4$ some students are friends. In this class any $ n \minus{} 1$ students can be seated in a round table such that every student is sitting next to a friend of him in both sides, but $ n$ students can not be seated in that way. Prove that the minimum value of $ n$ is $ 10$.

Consider a permutation $(a_1,a_2,a_3,a_4,a_5)$ of $\{1,2,3,4,5\}$. We say the $5$-tuple $(a_1,a_2,a_3,a_4,a_5)$ is dlawless if for all $1 \le i<j<k \le 5$, the sequence $(a_i,a_j,a_k)$ is [b]not [/b] an arithmetic progression (in that order). Find the number of flawless $5$-tuples.

A square grid is composed of $ n^2\equiv 1\pmod 4 $ unit cells that contained each a locust that jumped the same amount of cells in the direccion of columns or lines, without leaving the grid. Prove that, as a result of this, at least two locusts landed on the same cell. [i]Marius Cavachi[/i]

Let $F$ be the family of all sets of positive integers with $2010$ elements that satisfy the following condition: The difference between any two of its elements is never the same as the difference of any other two of its elements. Let $f$ be a function defined from $F$ to the positive integers such that $f(K)$ is the biggest element of $K \in F$. Determine the least value of $f(K)$.

Let be given $r_1,r_2,\ldots,r_n \in \mathbb R$. Show that there exists a subset $I$ of $\{1,2,\ldots,n \}$ which which has one or two elements in common with the sets $\{i,i + 1,i + 2\} , (1 \leq i \leq n- 2)$ such that \[\left| {\mathop \sum \limits_{i \in I} {r_i}} \right| \geqslant \frac{1}{6}\mathop \sum \limits_{i = 1}^n \left| {{r_i}} \right|.\]

Let $n\geq3$ be an integer. Find the smallest positive integer $k$ with the property that if in a group of $n$ boys for each boy there are at least $k$ other boys that are born in the same year with him, then all the boys are born in the same year.

There are two piles of cards, one with $n$ cards and the other with $m$ cards. $A$ and $B$ play alternately, performing one of the following actions in each turn. following operations: a) Remove a card from a pile. b) Remove one card from each pile. c) Move a card from one pile to the other. Player $A$ always starts the game and whoever takes the last one letter wins . Determine if there is a winning strategy based on $m$ and $n$, so that one of the players following her can win always.

An integer $N \ge 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N - 1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions hold: ($1$) no one stands between the two tallest players, ($2$) no one stands between the third and fourth tallest players, $\;\;\vdots$ ($N$) no one stands between the two shortest players. Show that this is always possible. [i]Proposed by Grigory Chelnokov, Russia[/i]

Let $n, k \ge 1$ be natural numbers. Find the number $A(n, k)$ of solutions in integers of the equation \[|x_1| + |x_2| +\cdots + |x_k| = n\]