Call a set $S \subseteq \{0,1,\dots,14\}$ $\textit{sparse}$ if $x+1 \pmod{15}$ is not in $S$ whenever $x \in S$. Find the number of sparse sets $T$ such that the sum of the elements of $T$ is a multiple of 15.

The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$. [i]Proposed by Croatia[/i]

Divide a circle into $2n$ equal sections. We call a circle [i]filled[/i] if it is filled with the numbers $0,1,2,\dots,n-1$. We call a filled circle [i] good[/i] if it has the following properties: $i$. Each number $0 \leq a \leq n-1$ is used exactly twice $ii$. For any $a$ we have that there are exactly $a$ sections between the two sections that have the number $a$ in them. Here is an example of a good filling for $n=5$ (View attachment) Prove that there doesn’t exist a good filling for $n=1399$

[b]p1.[/b] Will is distributing his money to three friends. Since he likes some friends more than others, the amount of money he gives each is in the ratio of $5 : 3 : 2$. If the person who received neither the least nor greatest amount of money was given $42$ dollars, how many dollars did Will distribute in all? [b]p2.[/b] Fan, Zhu, and Ming are driving around a circular track. Fan drives $24$ times as fast as Ming and Zhu drives $9$ times as fast as Ming. All three drivers start at the same point on the track and keep driving until Fan and Zhu pass Ming at the same time. During this interval, how many laps have Fan and Zhu driven together? [b]p3.[/b] Mr. DoBa is playing a game with Gunga the Gorilla. They both agree to think of a positive integer from $1$ to $120$, inclusive. Let the sum of their numbers be $n$. Let the remainder of the operation $\frac{n^2}{4}$ be $r$. If $r$ is $0$ or $1$, Mr. DoBa wins. Otherwise, Gunga wins. Let the probability that Mr. DoBa wins a given round of this game be $p$. What is $120p$? [b]p4.[/b] Let S be the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. How many subsets of $S$ are there such that if $a$ is the number of even numbers in the subset and $b$ is the number of odd numbers in the subset, then $a$ and $b$ are either both odd or both even? By definition, subsets of $S$ are unordered and only contain distinct elements that belong to $S$. [b]p5.[/b] Phillips Academy has five clusters, $WQN$, $WQS$, $PKN$, $FLG$ and $ABB$. The Blue Key heads are going to visit all five clusters in some order, except $WQS$ must be visited before $WQN$. How many total ways can they visit the five clusters? [b]p6.[/b] An astronaut is in a spaceship which is a cube of side length $6$. He can go outside but has to be within a distance of $3$ from the spaceship, as that is the length of the rope that tethers him to the ship. Out of all the possible points he can reach, the surface area of the outer surface can be expressed as $m+n\pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? [b]p7.[/b] Let $ABCD$ be a square and $E$ be a point in its interior such that $CDE$ is an equilateral triangle. The circumcircle of $CDE$ intersects sides $AD$ and $BC$ at $D$, $F$ and $C$, $G$, respectively. If $AB = 30$, the area of $AFGB$ can be expressed as $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and c is not divisible by the square of any prime. Find $a + b + c$. [b]p8.[/b] Suppose that $x, y, z$ satisfy the equations $$x + y + z = 3$$ $$x^2 + y^2 + z^2 = 3$$ $$x^3 + y^3 + z^3 = 3$$ Let the sum of all possible values of $x$ be $N$. What is $12000N$? [b]p9.[/b] In circle $O$ inscribe triangle $\vartriangle ABC$ so that $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be the midpoint of arc $BC$, and let $AD$ intersect $BC$ at $E$. Determine the value of $DE \cdot DA$. [b]p10.[/b] How many ways are there to color the vertices of a regular octagon in $3$ colors such that no two adjacent vertices have the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two players play a card game. They have a deck of $n$ distinct cards. About any two cards from the deck know which of them has a different (in this case, if $A$ beats $B$, and $B$ beats $C$, then it may be that $C$ beats $A$). The deck is split between players in an arbitrary manner. In each turn the players over the top card from his deck and one whose card has a card from another player takes both cards and puts them to the bottom of your deck in any order of their discretion. Prove that for any initial distribution of cards, the players can with knowing the location agree and act so that one of the players left without a card. [i]E. Lakshtanov[/i]

[b]p1.[/b] Let $A$ be the point $(-1, 0)$, $B$ be the point $(0, 1)$ and $C$ be the point $(1, 0)$ on the $xy$-plane. Assume that $P(x, y)$ is a point on the $xy$-plane that satisfies the following condition $$d_1 \cdot d_2 = (d_3)^2,$$ where $d_1$ is the distance from $P$ to the line $AB$, $d_2$ is the distance from $P$ to the line $BC$, and $d_3$ is the distance from $P$ to the line $AC$. Find the equation(s) that must be satisfied by the point $P(x, y)$. [b]p2.[/b] On Day $1$, Peter sends an email to a female friend and a male friend with the following instructions: $\bullet$ If you’re a male, send this email to $2$ female friends tomorrow, including the instructions. $\bullet$ If you’re a female, send this email to $1$ male friend tomorrow, including the instructions. Assuming that everyone checks their email daily and follows the instructions, how many emails will be sent from Day $1$ to Day $365$ (inclusive)? [b]p3.[/b] For every rational number $\frac{a}{b}$ in the interval $(0, 1]$, consider the interval of length $\frac{1}{2b^2}$ with $\frac{a}{b}$ as the center, that is, the interval $\left( \frac{a}{b}- \frac{1}{2b^2}, \frac{a}{b}+\frac{1}{2b^2}\right)$ . Show that $\frac{\sqrt2}{2}$ is not contained in any of these intervals. [b]p4.[/b] Let $a$ and $b$ be real numbers such that $0 < b < a < 1$ with the property that $$\log_a x + \log_b x = 4 \log_{ab} x - \left(\log_b (ab^{-1} - 1)\right)\left(\log_a (ab^{-1} - 1) + 2 log_a ab^{-1} \right)$$ for some positive real number $x \ne 1$. Find the value of $\frac{a}{b}$. [b]p5.[/b] Find the largest positive constant $\lambda$ such that $$\lambda a^2 b^2 (a - b)^2 \le (a^2 - ab + b^2)^3$$ is true for all real numbers $a$ and $b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The target is a triangle divided by three families of parallel lines into $100$ equal regular triangles with single sides. A sniper shoots at a target. He aims at triangle and hits either it or one of the sides adjacent to it. He sees the results of his shooting and can choose when stop shooting. What is the greatest number of triangles he can with a guarantee of hitting five times?

Maisy and Jeff are playing a game with a deck of cards with $4$ $0$’s, $4$ $1$’s, $4$ $2$’s, all the way up to $4$ $9$’s. You cannot tell apart cards of the same number. After shuffling the deck, Maisy and Jeff each take $4$ cards, make the largest $4$-digit integer they can, and then compare. The person with the larger $4$-digit integer wins. Jeff goes first and draws the cards $2,0,2,1$ from the deck. Find the number of hands Maisy can draw to beat that, if the order in which she draws the cards matters. [i]Proposed by Richard Chen[/i]

Let $n \ge 3$ be an integer. A frog is to jump along the real axis, starting at the point $0$ and making $n$ jumps: one of length $1$, one of length $2$, $\dots$ , one of length $n$. It may perform these $n$ jumps in any order. If at some point the frog is sitting on a number $a \le 0$, its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number $a > 0$, its next jump must be to the left (towards the negative numbers). Find the largest positive integer $k$ for which the frog can perform its jumps in such an order that it never lands on any of the numbers $1, 2, \dots , k$.

$\definecolor{A}{RGB}{70,255,50}\color{A}\fbox{C6.}$ There are $n$ $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ and $n$ $\definecolor{A}{RGB}{255,0,255}\color{A}\text{girls}$ in a club. Some of them are friends with each other. The $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ want to get into a [i]relationship[/i], so some subset of them wants to ask some $\definecolor{A}{RGB}{255,0,255}\color{A}\text{girls}$ out for a trip. Because the $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ are shy, for a nonempty set $B$ of $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$, they want to make sure that each of the girl they ask out is friend with one of the $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ in $B$. If the number of $\definecolor{A}{RGB}{255,0,255}\color{A}\text{girls}$ they are able to ask out is smaller than the number of the $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ in $B$, then the nonempty set $B$ of those $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ is called a group of complete losers. Show that for any $0 \le k < 2n$, there exists an arrangement of the [i]friendships[/i] among those $2n$ people so that there are exactly $k$ groups of complete losers. [i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b]. [color=#3D9186]#1737[/color]

In a square field of side length $12$ there is a source that contains a system of straight irrigation ditches. This is laid out in such a way that for every point of the field the distance to the next ditch is at most $1$. Here, the source is as a point and are the ditches to be regarded as stretches. It must be verified that the total length of the irrigation ditches is greater than $70$ m. The sketch shows an example of a trench system of the type indicated. [img]https://cdn.artofproblemsolving.com/attachments/6/5/5b51511da468cf14b5823c6acda3c4d2fe8280.png[/img]

Find the number of ways to distribute $11$ balls into $5$ boxes with different sizes, so that each box receives at least one ball, and the total number of balls in the largest and smallest boxes is more than the total number of balls in the remaining boxes.

a) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$, one can find $100$ members for which $a_k > k$. b) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$ there are infinitely many such numbers $a_k$ such that $a_k > k$. (A Andjans, Riga) PS. (a) for juniors (b) for seniors

A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the [b]interior[/b] of the polygon, in such a way that all the resulting triangles have vertices of all three colours.

At Durant University, an A grade corresponds to raw scores between $90$ and $100$, and a B grade corresponds to raw scores between $80$ and $90$. Travis has $3$ equally weighted exams in his math class. Given that Travis earned an A on his first exam and a B on his second (but doesn't know his raw score for either), what is the minimum score he needs to have a $90\%$ chance of getting an A in the class? Note that scores on exams do not necessarily have to be integers.

$G$ is a graph with $n$ vertices $A_1,A_2,\ldots,A_n,$ such that for each pair of non adjacent vertices $A_i$ and $A_j$ , there exist another vertex $A_k$ that is adjacent to both $A_i$ and $A_j .$ [b](a) [/b]Find the minimum number of edges in such a graph. [b](b) [/b]If $n = 6$ and $A_1,A_2,A_3,A_4,A_5,$ and $A_6$ form a cycle of length $6,$ find the number of edges that must be added to this cycle such that the above condition holds.

Suppose $n$ houses are to be assigned to $n$ people. Each person ranks the houses in the order of preference, with no ties. After the assignment is made, it is observed that every other assignment would assign to at least one person a less preferred house. Prove that there is at least one person who received the house he/she preferred most under this assignment.

During a school year 44 competitions were held. Exactly 7 students won in each of the competition. For any two competitions, there exists exactly 1 student who won both competitions. Is it true that there exists a student who won all the competitions?

$ m$ and $ n$ are positive integers. In a $ 8 \times 8$ chessboard, $ (m,n)$ denotes the number of grids a Horse can jump in a chessboard ($ m$ horizontal $ n$ vertical or $ n$ horizontal $ m$ vertical ). If a $ (m,n) \textbf{Horse}$ starts from one grid, passes every grid once and only once, then we call this kind of Horse jump route a $ \textbf{H Route}$. For example, the $ (1,2) \textbf{Horse}$ has its $ \textbf{H Route}$. Find the smallest positive integer $ t$, such that from any grid of the chessboard, the $ (t,t\plus{}1) \textbf{Horse}$ does not has any $ \textbf{H Route}$.

A V-tromino is a diagram formed by three unit squares.(As attachment.) (a)Is it possible to cover a $3\times 2013$ table by $3\times 671$ V-trominoes? (b)Is it possible to cover a $5\times 2013$ table by $5\times 671$ V-trominoes?

At the beginning of a two-player game, the number $2004!$ is written on the blackboard. The players move alternately. In each move, a positive integer smaller than the number on the blackboard and divisible by at most $20$ different prime numbers is chosen. This is subtracted from the number on the blackboard, which is erased and replaced by the difference. The winner is the player who obtains $0$. Does the player who goes first or the one who goes second have a guaranteed win, and how should that be achieved?

$p>2$ persons participate at a chess tournament, two players play at most one game against each other. After $n$ games were made, no more game is running and in every subset of three players, we can find at least two that havem't played against each other. Show that $n \leq \frac{p^{2}}4$.

We say that a set of points $M$ in the plane is convex if for any two points $A, B \in M$, all the points from the segment $(AB)$ also belong to $M$. Let $n \ge 2$ be an integer and let $F$ be a family of $n$ disjoint convex sets in the plane. Prove that there exists a line $\ell$ in the plane, a set $M \in F$ and a subset $S \subset F$ with at least $\lceil \frac{n}{12} \rceil $ elements such that $M$ is contained in one closed half-plane determined by $\ell$, and all the sets $N \in S$ are contained in the complementary closed half-plane determined by $\ell$.

Pedro has $111$ blue chips and $88$ white chips. There is a machine that for every $14$ blue chips , it gives $11$ white pieces and for every $7$ white chips, it gives $13$ blue pieces. Decide if Pedro can achieve, through successive operations with the machine, increase the total number of chips by $33$, so that the number of blue chips equals $\frac53$ of the amount of white chips. If possible, indicate how to do it. If not, indicate why.

A designer took a wooden cube $5 \times 5 \times 5$, divided each face into unit squares and painted each square black, white or red so that any two squares with a common side have different colours. What is the least possible number of black squares? (Squares with a common side may belong to the same face of the cube or to two different faces.) [i](8 points)[/i] [i]Mikhail Evdokimov[/i]