Gus is an inhabitant on an $11$ by $11$ grid of squares. He can walk from one square to an adjacent square (vertically or horizontally) in $1$ unit of time. There are also two vents on the grid, one at the top left and one at the bottom right. If Gus is at one vent, he can teleport to the other vent in $0.5$ units of time. Let an ordered pair of squares $(a,b)$ on the grid be [i]sus [/i] if the fastest path from $a$ to $b$ requires Gus to teleport between vents. Walking on top of a vent does not count as teleporting between vents. What is the total number of ordered pairs of squares that are [i]sus[/i]? Note that the pairs $(a_1,b_1)$ and $(a_2,b_2)$ are considered distinct if and only if $a_1 \ne a_2$ or $b_1 \ne b_2$.

In the following image is a beehive lattice of hexagons. Each cell is colored in one of three colors Red, Blue, or Green (denoted by the letters $R, B, G$). The frame is colored according to the instructions in the image, and the rest of the hexagons are colored however one wants. Is there necessarily a point where three hexagons of different colors meet?

Let $ n$, $ k$, $ p$ be positive integers with $ 2 \le k \le \frac {n}{p \plus{} 1}$. Let $ n$ distinct points on a circle be given. These points are colored blue and red so that exactly $ k$ points are blue and, on each arc determined by two consecutive blue points in clockwise direction, there are at least $ p$ red points. How many such colorings are there?

Let $S_n$ be the number of sequences $(a_1, a_2, \ldots, a_n),$ where $a_i \in \{0,1\},$ in which no six consecutive blocks are equal. Prove that $S_n \rightarrow \infty$ when $n \rightarrow \infty.$

Let $n \in \mathbb N$. Prove that for each factor $m \ge n$ of $n(n + 1)/2$, one can partition the set $\{1,2, 3,\cdots , n\}$ into disjoint subsets such that the sum of elements in each subset is equal to $m$.

There three piles of pebbles, containing 5, 49, and 51 pebbles respectively. It is allowed to combine any two piles into a new one or to split any pile consisting of even number of pebbles into two equal piles. Is it possible to have 105 piles with one pebble in each in the end? [i](3 points)[/i]

The game Prongle is played with a special deck of cards: on each card is a nonempty set of distinct colors. No two cards in the deck contain the exact same set of colors. In this game, a “Prongle” is a set of at least $2$ cards such that each color is on an even number of cards in the set. Let k be the maximum possible number of prongles in a set of $2019$ cards. Find $\lfloor \log 2 (k) \rfloor$.

There are three piles of matches on the table: one with $100$ matches, one with $200$, and one with $300$. Two players play the following game. They play alternatively, and a player on turn removes one of the piles and divides one of the remaining piles into two nonempty piles. The player who cannot make a legal move loses. Who has a winning strategy? (K. Kokhas’)

[u]Round 1[/u] [b]p1.[/b] A party is attended by ten people (men and women). Among them is Pat, who always lies to people of the opposite gender and tells the truth to people of the same gender. Pat tells five of the guests: “There are more men than women at the party.” Pat tells four of the guests: “There are more women than men at the party.” Is Pat a man or a woman? [b]p2.[/b] Once upon a time in a land far, far away there lived $100$ knights, $99$ princesses, and $101$ dragons. Over time, knights beheaded dragons, dragons ate princesses, and princesses poisoned knights. But they always obeyed an ancient law that prohibits killing any creature who has killed an odd number of others. Eventually only one creature remained alive. Could it have been a knight? A dragon? A princess? [b]p3.[/b] The numbers $1 \circ 2 \circ 3 \circ 4 \circ 5 \circ 6 \circ 7 \circ 8 \circ 9 \circ 10$ are written in a line. Alex and Vicky play a game, taking turns inserting either an addition or a multiplication symbol between adjacent numbers. The last player to place a symbol wins if the resulting expression is odd and loses if it is even. Alex moves first. Who wins? (Remember that multiplication is performed before addition.) [b]p4.[/b] A chess tournament had $8$ participants. Each participant played each other participant once. The winner of a game got $1$ point, the loser $0$ points, and in the case of a draw each got $1/2$ a point. Each participant scored a different number of points, and the person who got $2$nd prize scored the same number of points as the $5$th, $6$th, $7$th and $8$th place participants combined. Can you determine the result of the game between the $3$rd place player and the $5$th place player? [b]p5.[/b] One hundred gnomes sit in a circle. Each gnome gets a card with a number written on one side and a different number written on the other side. Prove that it is possible for all the gnomes to lay down their cards so that no two neighbors have the same numbers facing up. [u]Round 2[/u] [b]p6.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png[/img] [b]p7.[/b] Each of the $100$ residents of Pleasantville has at least $30$ friends in town. A resident votes in the mayoral election only if one of her friends is a candidate. Prove that it is possible to nominate two candidates for mayor so that at least half of the residents will vote. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements. Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating

Find the smallest natural number $n$ for which it is possible to partition the set $M = \{1,2, ... ,40\}$ into n subsets $M_1, . . . ,M_n$ so that none of the $M_i$ contains elements $a,b,c$ (not necessarily distinct) with $a+b = c$.

A positive integer $n < 2017$ is given. Exactly $n$ vertices of a regular 2017-gon are colored red, and the remaining vertices are colored blue. Prove that the number of isosceles triangles whose vertices are monochromatic does not depend on the chosen coloring (but does depend on $n$.)

Freddy writes down numbers $1, 2,\ldots ,n$ in some order. Then he makes a list of all pairs $(i, j)$ such that $1\le i<j\le n$ and the $i$-th number is bigger than the $j$-th number in his permutation. After that, Freddy repeats the following action while possible: choose a pair $(i, j)$ from the current list, interchange the $i$-th and the $j$-th number in the current permutation, and delete $(i, j)$ from the list. Prove that Freddy can choose pairs in such an order that, after the process finishes, the numbers in the permutation are in ascending order.

The Nationwide Basketball Society (NBS) has $8001$ teams, numbered $2000$ through $10000$. For each $n$, team $n$ has $n+1$ players, and in a sheer coincidence, this year each player attempted $n$ shots and on team $n$, exactly one player made $0$ shots, one player made $1$ shot, . . ., one player made $n$ shots. A player's [i]field goal percentage[/i] is defined as the percentage of shots the player made, rounded to the nearest tenth of a percent (For instance, $32.45\%$ rounds to $32.5\%$). A player in the NBS is randomly selected among those whose field goal percentage is $66.6\%$. If this player plays for team $k$, the probability that $k\geq 6000$ can be expressed as $\tfrac{p}{q}$ for relatively prime positive integers $p$ and $q$. Find $p+q$.

Consider an infinite sequence $a_1,a_2,\cdots$ whose terms all belong to $\left\{1,2\right\}$. A positive integer with $n$ digits is said to be [i]good[/i] if its decimal representation has the form $a_ra_{r+1}\cdots a_{r+(n-1)}$, for some positive integer $r$. Suppose that there are at least $2008$ [i]good[/i] numbers with a million digits. Prove that there are at least $2008$ [i]good[/i] numbers with $2007$ digits.

Consider a set of $n > 3$ points in the plane, no three of which are collinear, and a natural number $k < n$. Prove the following statements: (a) If $k \le \frac{n}{2}$, then each point can be connected with at least k other points by segments so that no three segments form a triangle. (b) If $k \ge \frac{n}{2}$, and each point is connected with at least k other points by segments, then some three segments form a triangle.

Define $f : \mathbb{Z}_+ \to \mathbb{Z}_+$ such that $f(1) = 1$ and $f(n) $ is the greatest prime divisor of $n$ for $n > 1$. Aino and Väinö play a game, where each player has a pile of stones. On each turn the player to turn with $m$ stones in his pile may remove at most $f(m)$ stones from the opponent's pile, but must remove at least one stone. (The own pile stays unchanged.) The first player to clear the opponent's pile wins the game. Prove that there exists a positive integer $n$ such that Aino loses, when both players play optimally, Aino starts, and initially both players have $n$ stones.

In an $8$-square board -like the one in the figure- there is initially one checker in each square. $ \begin{tabular}{ | l | c | c |c | c| c | c | c | r| } \hline & & & & & & & \\ \hline \end{tabular} $ A move consists of choosing two tokens and moving one of them one square to the right and the other one one square to the left. If after $4$ moves the $8$ checkers are distributed in only $2$ boxes, determine what those boxes can be and how many checkers are in each one.

A total of $a \cdot b \cdot c$ cubical boxes are joined together in a $a \times b \times c$ rectangular stack, where $a, b, c \ge 2$. A bee is found inside one of the boxes. It can fly from one box to another through a hole in the wall, but not through edges or corners. Also, it cannot fly outside the stack. For which triples $(a, b, c)$ is it possible for the bee to fly through all of the boxes exactly once, and end up in the same box where it started?

We visit all squares exactly once on a $n\times n$ chessboard (colored in the usual way) with a king. Find the smallest number of times we had to switch colors during our walk. [i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

Determine whether it's possible to cover a $K_{2012}$ with a) 1000 $K_{1006}$'s; b) 1000 $K_{1006,1006}$'s. [i]David Yang.[/i]

[u]Round 1[/u] [b]p1. [/b] What is the sum of the digits in the binary representation of $2023$? [b]p2.[/b] Jack is buying fruits at the EMCCmart. Three apples and two bananas cost $\$11.00$. Five apples and four bananas cost $\$19.00$. In cents, how much more does an apple cost than a banana? [b]p3.[/b] Define $a \sim b$ as $a! - ab$. What is $(4 \sim 5) \sim (5 \sim (3 \sim 1))$? [u] Round 2[/u] [b]p4.[/b] Alan has $24$ socks in his drawer. Of these socks, $4$ are red, $8$ are blue, and $12$ are green. Alan takes out socks one at a time from his drawer at random. What is the minimum number of socks he must pull out to guarantee that the number of green socks is at least twice the number of red socks? [b]p5.[/b] What is the remainder when the square of the $24$th smallest prime number is divided by $24$? [b]p6.[/b] A cube and a sphere have the same volume. If $k$ is the ratio of the length of the longest diagonal of the cube to the diameter of the sphere, find $k^6$. [u]Round 3[/u] [b]p7.[/b] Equilateral triangle $ABC$ has side length $3\sqrt3$. Point $D$ is drawn such that $BD$ is tangent to the circumcircle of triangle $ABC$ and $BD = 4$. Find the distance from the circumcenter of triangle $ABC$ to $D$. [b]p8.[/b] If $\frac{2023!}{2^k}$ is an odd integer for an integer $k$, what is the value of $k$? [b]p9.[/b] Let $S$ be a set of 6 distinct positive integers. If the sum of the three smallest elements of $S$ is $8$, and the sum of the three largest elements of $S$ is $19$, find the product of the elements in $S$. [u]Round 4[/u] [b]p10.[/b] For some integers $b$, the number $1 + 2b + 3b^2 + 4b^3 + 5b^4$ is divisible by $b + 1$. Find the largest possible value of $b$. [b]p11.[/b] Let $a, b, c$ be the roots of cubic equation $x^3 + 7x^2 + 8x + 1$. Find $a^2 + b^2 + c^2 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ [b]p12.[/b] Let $C$ be the set of real numbers $c$ such that there are exactly two integers n satisfying $2c < n < 3c$. Find the expected value of a number chosen uniformly at random from $C$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3131590p28370327]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In how many ways can $n$ rooks be placed on a $2n$ x $2n$ chessboard, so that they cover all the white fields?

Pieces are placed in some squares of an $8 \times 8$ board sothat: a) There is at least one token in any rectangle with sides $2 \times 1$ or $1\times 2$. b) There are at least two neighboring pieces in any rectangle with sides $7\times 1$ or $1\times 7$. Find the smallest number of tokens that can be taken to fulfill with both conditions.

What is the largest possible value of $|a_1 - 1| + |a_2-2|+...+ |a_n- n|$ where $a_1, a_2,..., a_n$ is a permutation of $1,2,..., n$?