Kevin has a square piece of paper with creases drawn to split the paper in half in both directions, and then each of the four small formed squares diagonal creases drawn, as shown below. [img]https://cdn.artofproblemsolving.com/attachments/2/2/70d6c54e86856af3a977265a8054fd9b0444b0.png[/img] Find the sum of the corresponding numerical values of figures below that Kevin can create by folding the above piece of paper along the creases. (The figures are to scale.) Kevin cannot cut the paper or rip it in any way. [img]https://cdn.artofproblemsolving.com/attachments/a/c/e0e62a743c00d35b9e6e2f702106016b9e7872.png[/img]

On a planet there are $M$ countries and $N$ cities. There are two-way roads between some of the cities. It is given that: (1) In each county there are at least three cities; (2) For each country and each city in the country is connected by roads with at least half of the other cities in the countries; (3) Each city is connceted with exactly one other city ,that is not in its country; (4) There are at most two roads between cities from cities in two different countries; (5) If two countries contain less than $2M$ cities in total then there is a road between them. Prove that there is cycle of lenght at least $M+\frac{N}{2}$.

Prove that the number permutations $ \alpha$ of $ \{1,2,\dots,n\}$ s.t. there does not exist $ i<j<n$ s.t. $ \alpha(i)<\alpha(j\plus{}1)<\alpha(j)$ is equal to the number of partitions of that set.

Chip and Dale play on a $100 \times 100$ table. In the beginning, a chess king stands in the upper left corner of the table. At each move the king is moved one square right, down or right-down diagonally. A player cannot move in the direction used by his opponent in the previous move. The players move in turn, Chip begins. The player that cannot move loses. Which player has a winning strategy?

Consider a regular triangular array of $n(n+1)/2$ points.Let $f(n)$ denote the number of equilateral triangles formed by taking some $3$ points in the array as vertices.Prove that $f(n)=\frac{(n-1)n(n+1)(n+2)}{24}$.

Several pupils with different heights are standing in a row. If they were arranged according to their heights, such that the highest would stand on the right, then each pupil would move for at most 8 positions. Prove that every pupil has no more than 8 pupils lower then him on his right.

[u]Round 9[/u] [b]p25.[/b] For how many nonempty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}$ is the sum of the elements divisble by $32$? [b]p26.[/b] America declared independence in $1776$. Take the sum of the cubes of the digits of $1776$ and let that equal $S_1$. Sum the cubes of the digits of $S_1$ to get $S_2$. Repeat this process $1776$ times. What is $S_{1776}$? [b]p27.[/b] Every Golden Grahams box contains a randomly colored toy car, which is one of four colors. What is the expected number of boxes you have to buy in order to obtain one car of each color? [u]Round 10[/u] [b]p28.[/b] Let $B$ be the answer to Question $29$ and $C$ be the answer to Question $30$. What is the sum of the square roots of $B$ and $C$? [b]p29.[/b] Let $A$ be the answer to Question $28$ and $C$ be the answer to Question $30$. What is the sum of the sums of the digits of $A$ and $C$? [b]p30.[/b] Let $A$ be the answer to Question $28$ and $B$ be the answer to Question $29$. What is $A + B$? [u]Round 11[/u] [b]p31.[/b] If $x + \frac{1}{x} = 4$, find $x^6 + \frac{1}{x^6}$. [b]p32.[/b] Given a positive integer $n$ and a prime $p$, there is are unique nonnegative integers $a$ and $b$ such that $n = p^b \cdot a$ and $gcd (a, p) = 1$. Let $v_p(n)$ denote this uniquely determined $a$. Let $S$ denote the set of the first 20 primes. Find $\sum_{ p \in S} v_p \left(1 + \sum^{100}_{i=0} p^i \right)$. [b]p33. [/b] Find the maximum value of n such that $n+ \sqrt{(n - 1) +\sqrt{(n - 2) + ... +\sqrt{1}}} < 49$ (Note: there would be $n - 1$ square roots and $n$ total terms). [u]Round 12[/u] [b]p34.[/b] Give two numbers $a$ and $b$ such that $2015^a < 2015! < 2015^b$. If you are incorrect you get $-5$ points; if you do not answer you get $0$ points; otherwise you get $\max \{20-0.02(|b - a| - 1), 0\}$ points, rounded down to the nearest integer. [b]p35.[/b] Twin primes are prime numbers whose difference is $2$. Let $(a, b)$ be the $91717$-th pair of twin primes, with $a < b$. Let $k = a^b$, and suppose that $j$ is the number of digits in the base $10$ representation of $k$. What is $j^5$? If the correct answer is $n$ and you say $m$, you will receive $\max \left(20 - | \log \left(| \frac{m}{n} |\right), 0 \right)$ points, rounded down to the nearest integer. [b]p36.[/b] Write down any positive integer. Let the sum of the valid submissions (i.e. positive integer submissions) for all teams be $S$. One team will be chosen randomly, according to the following distribution: if your team's submission is $n$, you will be chosen with probability $\frac{n}{S}$ . The amount of points that the chosen team will win is the greatest integer not exceeding $\min \{K, \frac{ 10000}{S} \}$. $K$ is a predetermined secret value. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3157009p28696627]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3157013p28696685]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many ways are there to arrange three indistinguishable rooks on a $ 6 \times 6$ board such that no two rooks are attacking each other? (Two rooks are attacking each other if and only if they are in the same row or the same column.)

Given a collection of sets $X = \{A_1, A_2, ..., A_n\}$. A set $\{a_1, a_2, ..., a_n\}$ is called a single representation of $X$ if $a_i \in A_i$ for all i. Let $|S| = mn$, $S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n$ with $|A_i| = |B_i| = m$ for all $i$. Prove that $S = C_1 \cup C_2 \cup ... \cup C_n$ where for every $i, C_i $ is a single represenation for $\{A_j\}_{j=1}^n $and $\{B_j\}_{j=1}^n$.

Let $n \ge 4$ be a positive integer and there exist $n$ positive integers that are arranged on a circle such that: $\bullet$ The product of each pair of two non-adjacent numbers is divisible by $2015 \cdot 2016$. $\bullet$ The product of each pair of two adjacent numbers is not divisible by $2015 \cdot 2016$. Find the maximum value of $n$

A teacher wishes to separate her $12$ students into groups. Yesterday, the teacher put the students into $4$ groups of $3$. Today, the teacher decides to put the students into $4$ groups of $3$ again. However, she doesn’t want any pair of students to be in the same group on both days. Find how many ways she could formthe groups today.

On the screen of a computer there is an $2^n\times 2^n$ board. On each cell of the main diagonal there is a file. At each step, we may select some files and move them to the left, on their respective rows, by the same distance. What is the minimum number of necessary moves in order to put all files on the first column? [i]Proposed by Vlad Spătaru[/i]

A floor is tiled with equilateral triangles of side length $1$, as shown. If you drop a needle of length $2$ somewhere on the floor , what is the largest number of triangles it could end up intersecting? (Only count the triangles whose interiors are met by the needle --- touching along edges or at corners doesn't qualify.) [img]https://cdn.artofproblemsolving.com/attachments/5/6/e7555c22ffe890b46a3ebdbda2169d23e43700.png[/img]

Is it possible to color each square on a $9\times 9$ board so that each $2\times 3$ or $3\times 2$ block contains exactly $2$ black squares? If so, what is/are the possible total number(s) of black squares?

Hi everybody! I've an interesting problem! Can you solve it? Prove [b]Erdös-Ginzburg-Ziv Theorem[/b]: [i]"Among any $2n-1$ integers, there are some $n$ whose sum is divisible by $n$."[/i]

Given a set $P$ of $n>100$ points on the plane such that no three of them are collinear, and a set $S$ of $20n$ distinct segments, each joining a pair of points from $P$. Prove that there exists a line not passing through a point from $P$ and intersecting at least $200$ segments from $S$.

There are $n \ge 4$ cities on a round lake, between which $n -4$ people travel and one green drivers operate. Each ferry connects two non-adjacent cities, and itself do not cross two driving routes so that collisions can be avoided. In order to better adapt the transport routes to the needs of the passengers, the following change can be done: A new route can be assigned to any driver. The routes of the remaining drives must not cross and also must not be changed at the same time. Let Santa Marta and Cape Town be two non-adjacent cities. Show that you have finitely many route changes so that the Green Driver will operate between Santa Marta and Cape Town after these changes. Note: At no time may two trips between the same cities or one drive between two neighboring cities. [hide=original wording]An einem runden See liegen $n >= 4$ Stadte, zwischen denen $n - 4$ Personenfahren und eine grune Autofahre verkehren. Jede Fahre verbindet zwei nicht benachbarte Stadte, wobei sich keine zwei Fahrenrouten uberkreuzen, damit Kollisionen vermieden werden konnen. Um die Transportrouten besser den Bedurfnissen der Passagiere anzupassen, kann folgende Anderung vorgenommen werden: Einer beliebigen Fahre kann eine neue Route zugeordnet werden. Dabei durfen die Routen der restlichen Fahren nicht uberkreuzt und auch nicht gleichzeitig verandert werden. Seien Santa Marta und Kapstadt zwei nicht benachbarte Stadte. Zeige, dass man endlich viele Routenanderungen vornehmen kann, sodass die grune Autofahre nach diesen Anderungen zwischen Santa Marta und Kapstadt verkehrt. Bemerkung: Zu keinem Zeitpunkt durfen zwei Fahren zwischen denselben Stadten oder eine Fahre zwischen zwei benachbarten Stadten verkehren.[/hide]

[b]p1.[/b] The length of the side $AB$ of the trapezoid with bases $AD$ and $BC$ is equal to the sum of lengths $|AD|+|BC|$. Prove that bisectors of angles $A$ and $B$ do intersect at a point of the side $CD$. [b]p2.[/b] Polynomials $P(x) = x^4 + ax^3 + bx^2 + cx + 1$ and $Q(x) = x^4 + cx^3 + bx^2 + ax + 1$ have two common roots. Find these common roots of both polynomials. [b]p3.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace: $\bullet$ two candies in the box with one chocolate bar, $\bullet$ two muffins in the box with one chocolate bar, $\bullet$ two chocolate bars in the box with one candy and one muffin, $\bullet$ one candy and one chocolate bar in the box with one muffin, $\bullet$ one muffin and one chocolate bar in the box with one candy. Is it possible that after some time it remains only one object in the box? [b]p4.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points? [b]p5.[/b] It is known that $x$ is a real number such that $x+\frac{1}{x}$ is an integer. Prove that $x^n+\frac{1}{x^n}$ is an integer for any positive integer $n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

A $1 \times 5n$ rectangle is partitioned into tiles, each of the tile being either a separate $1 \times 1$ square or a broken domino consisting of two such squares separated by four squares (not belonging to the domino). Prove that the number of such partitions is a perfect fifth power. [i](K. Kokhas)[/i]

Problem 4. A chessboard with $2n \times 2n$ tiles is coloured such that every tile is coloured with one out of $n$ colours. Prove that there exists 2 tiles in either the same column or row such that if the colours of both tiles are swapped, then there exists a rectangle where all its four corner tiles have the same colour.

We have two $20 \times 13$ rectangular grids with $260$ unit cells. each one. We insert in the boxes of each of the grids the numbers $1, 2, ..., 260$ as follows: $\bullet$ For the first grid, we start by inserting the numbers $1, 2, ..., 13$ in the boxes in the top row from left to right. We continue inserting numbers $14$, $ 15$, $...$, $26$ in the second row from left to right. We maintain the same procedure until in the last row, $20$, the numbers are placed $248$, $249$, $...$, $260$ from left to right. $\bullet$ For the second grid we start by inserting the numbers $1$, $2$,$ ..$., $20$ from top to bottom in the farthest column right. We continue inserting the numbers $21$, $22$,$ ...$, $40$ in the second column from the right also from top to bottom. We maintain that same procedure until we reach the column on the left where we place the numbers from top to bottom $241$, $242$, $ ...$, $260$. Determines the integers inserted in the boxes located in the same position in both grids.

There are a) $26$; b) $30$ identical-looking coins in a circle. It is known that exactly two of them are fake. Real coins weigh the same, fake ones too, but they are lighter than the real ones. How can you determine in three weighings on a cup scale without weights whether there are fake coins lying nearby or not?? [i]Proposed by A. Gribalko[/i]

A Knight always tells the truth. A Knave always lies. A Normal may either lie or tell the truth. You are allowed to ask questions that can be answered with ''yes" or ''no", such as ''is this person a Normal?" (a) There are three people in front of you. One is a Knight, another one is a Knave, and the third one is a Normal. They all know the identities of one another. How can you too learn the identity of each? (1) (b) There are four people in front of you. One is a Knight, another one is a Knave, and the other two are Normals. They all know the identities of one another. Prove that the Normals may agree in advance to answer your questions in such a way that you will not be able to learn the identity of any of the four people. (3)

[b]p1.[/b] Anne purchased yesterday at WalMart in Puerto Rico $6$ identical notebooks, $8$ identical pens and $7$ identical erasers. Anne remembers that each eraser costs $73$ cents. She did not buy anything else. Anne told her mother that she spent $12$ dollars and $76$ cents at Walmart. Can she be right? Note that in Puerto Rico there is no sales tax. [b]p2.[/b] Two men ski one after the other first in a flat field and then uphill. In the field the men run with the same velocity $12$ kilometers/hour. Uphill their velocity drops to $8$ kilometers/hour. When both skiers enter the uphill trail segment the distance between them is $300$ meters less than the initial distance in the field. What was the initial distance between skiers? (There are $1000$ meters in 1 kilometer.) [b]p3.[/b] In the equality $** + **** = ****$ all the digits are replaced by $*$. Restore the equality if it is known that any numbers in the equality does not change if we write all its digits in the opposite order. [b]p4.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies. Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!” The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!” A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!” Then a stripped polyleg started: ”-None of you has $8$ legs. Only I have 8 legs!” Which polyleg has exactly $8$ legs? [b]p5.[/b] Cut the figure shown below in two equal pieces. (Both the area and the form of the pieces must be the same.) [img]https://cdn.artofproblemsolving.com/attachments/e/4/778678c1e8748e213ffc94ba71b1f3cc26c028.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].