At a strange party, each person knew exactly $22$ others. For any pair of people $X$ and $Y$ who knew each other, there was no other person at the party that they both knew. For any pair of people $X$ and $Y$ who did not know one another, there were exactly $6$ other people that they both knew. How many people were at the party?

Let $n$ and $k$ be positive integers and $G$ be a complete graph on $n$ vertices. Each edge of $G$ is colored one of $k$ colors such that every triangle consists of either three edges of the same color or three edges of three different colors. Furthermore, there exist two different-colored edges. Prove that $n\le(k-1)^2$. [i]Linus Tang[/i]

[b]p1.[/b] Sarah needed a ride home to the farm from town. She telephoned for her father to come and get her with the pickup truck. Being eager to get home, she began walking toward the farm as soon as she hung up the phone. However, her father had to finish milking the cows, so could not leave to get her until fifteen minutes after she called. He drove rapidly to make up for lost time. They met on the road, turned right around and drove back to the farm at two-thirds of the speed her father drove coming. They got to the farm two hours after she had called. She walked and he drove both ways at constant rates of speed. How many minutes did she spend walking? [b]p2.[/b] Let $A = (a,b)$ be any point in a coordinate plane distinct from the origin $O$. Let $M$ be the midpoint of $OA$, and let $P$ be a point such that $MP$ is perpendicular to $OA$ and the lengths $\overline{MP}$ and $\overline{OM}$ are equal. Determine the coordinates $(x,y)$ of $P$ in terms of $a$ and $b$. Give all possible solutions. [b]p3.[/b] Determine the exact sum of the series $$\frac{1}{1 \cdot 2\cdot 3} + \frac{1}{2\cdot 3\cdot 4} + \frac{1}{3\cdot 4\cdot 5} + ... + \frac{1}{98\cdot 99\cdot 100}$$ [b]p4.[/b] A six pound weight is attached to a four foot nylon cord that is looped over two pegs in the manner shown in the drawing. At $B$ the cord passes through a small loop in its end. The two pegs $A$ and $C$ are one foot apart and are on the same level. When the weight is released the system obtains an equilibrium position. Determine angle $ABC$ for this equilibrium position, and verify your answer. (Your verification should assume that friction and the weight of the cord are both negligible, and that the tension throughout the cord is a constant six pounds.) [img]https://cdn.artofproblemsolving.com/attachments/a/1/620c59e678185f01ca8743c39423234d5ba04d.png[/img] [b]p5.[/b] The four corners of a rectangle have the property that when they are taken three at a time, they determine triangles all of which have the same perimeter. We will consider whether a set of five points can have this property. Let $S = \{p_1, p_2, p_3, p_4, p_5\}$ be a set of five points. For each $i$ and $j$, let $d_{ij}$ denote the distance from $p_i$ to $p_j$. Suppose that $S$ has the property that all triangles with vertices in $S$ have the same perimeter. (a) Prove that $d$ must be the same for every pair $(i,j)$ with $i \ne j$. (b) Can such a five-element set be found in three dimensional space? Justify your answer. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $n$ MOPpers $p_1,...,p_n$ designing a carpool system to attend their morning class. Each $p_i$'s car fits $\chi (p_i)$ people ($\chi : \{p_1,...,p_n\} \to \{1,2,...,n\}$). A $c$-fair carpool system is an assignment of one or more drivers on each of several days, such that each MOPper drives $c$ times, and all cars are full on each day. (More precisely, it is a sequence of sets $(S_1, ...,S_m)$ such that $|\{k: p_i\in S_k\}|=c$ and $\sum_{x\in S_j} \chi(x) = n$ for all $i,j$. ) Suppose it turns out that a $2$-fair carpool system is possible but not a $1$-fair carpool system. Must $n$ be even? [i]Proposed by Nathan Ramesh and Palmer Mebane

In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.

Let $n$ be a positive integer. There are $n$ islands with $n-1$ bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let $X$ be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of $X$ over all possible configurations of bridges and island where the fire starts at.

In a social network with a fixed finite setback of users, each user had a fixed set of [i]followers[/i] among the other users. Each user has an initial positive integer rating (not necessarily the same for all users). Every midnight, the rating of every user increases by the sum of the ratings that his followers had just before midnight. Let $m$ be a positive integer. A hacker, who is not a user of the social network, wants all the users to have ratings divisible by $m$. Every day, he can either choose a user and increase his rating by 1, or do nothing. Prove that the hacker can achieve his goal after some number of days. [i]Vladislav Novikov[/i]

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Ben was trying to solve for $x$ in the equation $6 + x = 1$. Unfortunately, he was reading upside-down and misread the equation as $1 = x + 9$. What is the positive difference between Ben's answer and the correct answer? [b]p2.[/b] Anjali and Meili each have a chocolate bar shaped like a rectangular box. Meili's bar is four times as long as Anjali's, while Anjali's is three times as wide and twice as thick as Meili's. What is the ratio of the volume of Anjali's chocolate to the volume of Meili's chocolate? [b]p3.[/b] For any two nonnegative integers $m, n$, not both zero, define $m?n = m^n + n^m$. Compute the value of $((2?0)?1)?7$. [b]p4.[/b] Eliza is making an in-scale model of the Phillips Exeter Academy library, and her prototype is a cube with side length $6$ inches. The real library is shaped like a cube with side length $120$ feet, and it contains an entrance chamber in the front. If the chamber in Eliza's model is $0.8$ inches wide, how wide is the real chamber, in feet? [b]p5.[/b] One day, Isaac begins sailing from Marseille to New York City. On the exact same day, Evan begins sailing from New York City to Marseille along the exact same route as Isaac. If Marseille and New York are exactly $3000$ miles apart, and Evan sails exactly 40 miles per day, how many miles must Isaac sail each day to meet Evan's ship in $30$ days? [b]p6.[/b] The conversion from Celsius temperature C to Fahrenheit temperature F is: $$F = 1.8C + 32.$$ If the lowest temperature at Exeter one day was $20^o$ F, and the next day the lowest temperature was $5^o$ C higher, what would be the lowest temperature that day, in degrees Fahrenheit? [b]p7.[/b] In a school, $60\%$ of the students are boys and $40\%$ are girls. Given that $40\%$ of the boys like math and $50\%$ of the people who like math are girls, what percentage of girls like math? [b]p8.[/b] Adam and Victor go to an ice cream shop. There are four sizes available (kiddie, small, medium, large) and seventeen different flavors, including three that contain chocolate. If Victor insists on getting a size at least as large as Adam's, and Adam refuses to eat anything with chocolate, how many different ways are there for the two of them to order ice cream? [b]p9.[/b] There are $10$ (not necessarily distinct) positive integers with arithmetic mean $10$. Determine the maximum possible range of the integers. (The range is defined to be the nonnegative difference between the largest and smallest number within a list of numbers.) [b]p10.[/b] Find the sum of all distinct prime factors of $11! - 10! + 9!$. [b]p11.[/b] Inside regular hexagon $ZUMING$, construct square $FENG$. What fraction of the area of the hexagon is occupied by rectangle $FUME$? [b]p12.[/b] How many ordered pairs $(x, y)$ of nonnegative integers satisfy the equation $4^x \cdot 8^y = 16^{10}$? [b]p13.[/b] In triangle $ABC$ with $BC = 5$, $CA = 13$, and $AB = 12$, Points $E$ and $F$ are chosen on sides $AC$ and $AB$, respectively, such that $EF \parallel BC$. Given that triangle $AEF$ and trapezoid $EFBC$ have the same perimeter, find the length of $EF$. [b]p14.[/b] Find the number of two-digit positive integers with exactly $6$ positive divisors. (Note that $1$ and $n$ are both counted among the divisors of a number $n$.) [b]p15.[/b] How many ways are there to put two identical red marbles, two identical green marbles, and two identical blue marbles in a row such that no red marble is next to a green marble? [b]p16.[/b] Every day, Yannick submits $8$ more problems to the EMCC problem database than he did the previous day. Every day, Vinjai submits twice as many problems to the EMCC problem database as he did the previous day. If Yannick and Vinjai initially both submit one problem to the database on a Monday, on what day of the week will the total number of Vinjai's problems first exceed the total number of Yannick's problems? [b]p17.[/b] The tiny island nation of Konistan is a cone with height twelve meters and base radius nine meters, with the base of the cone at sea level. If the sea level rises four meters, what is the surface area of Konistan that is still above water, in square meters? [b]p18.[/b] Nicky likes to doodle. On a convex octagon, he starts from a random vertex and doodles a path, which consists of seven line segments between vertices. At each step, he chooses a vertex randomly among all unvisited vertices to visit, such that the path goes through all eight vertices and does not visit the same vertex twice. What is the probability that this path does not cross itself? [b]p19.[/b] In a right-angled trapezoid $ABCD$, $\angle B = \angle C = 90^o$, $AB = 20$, $CD = 17$, and $BC = 37$. A line perpendicular to $DA$ intersects segment $BC$ and $DA$ at $P$ and $Q$ respectively and separates the trapezoid into two quadrilaterals with equal area. Determine the length of $BP$. [b]p20.[/b] A sequence of integers $a_i$ is defined by $a_1 = 1$ and $a_{i+1} = 3i - 2a_i$ for all integers $i \ge 1$. Given that $a_{15} = 5476$, compute the sum $a_1 + a_2 + a_3 + ...+ a_{15}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider $111$ distinct points which lie on or in the internal of a circle with radius 1.Prove that there are at least $1998$ segments formed by these points with length $\leq \sqrt{3}$

Let $n$ be a positive integer. $n$ letters are written around a circle, each $A$, $B$, or $C$. When the letters are read in clockwise order, the sequence $AB$ appears $100$ times, the sequence $BA$ appears $99$ times, and the sequence $BC$ appears $17$ times. How many times does the sequence $CB$ appear?

If $a$ and $b$ are randomly selected real numbers between $0$ and $1$, find the probability that the nearest integer to $\frac{a-b}{a+b}$ is odd.

[b]p1.[/b] Find all integer solutions of the equation $a^2 - b^2 = 2002$. [b]p2.[/b] Prove that the disks drawn on the sides of a convex quadrilateral as on diameters cover this quadrilateral. [b]p3.[/b] $30$ students from one school came to Mathematical Olympiad. In how many different ways is it possible to place them in four rooms? [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Integers from 1 to 100 are placed in a row in some order. Let us call a number [i]large-right[/i], if it is greater than each number to the right of it; let us call a number [i]large-left[/i], is it is greater than each number to the left of it. It appears that in the row there are exactly $k$ large-right numbers and exactly $k$ large-left numbers. Find the maximal possible value of $k$.

Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]

[u]Set 3[/u] [b]p7.[/b] On unit square $ABCD$, a point $P$ is selected on segment $CD$ such that $DP =\frac14$ . The segment $BP$ is drawn and its intersection with diagonal $AC$ is marked as $E$. What is the area of triangle $AEP$? [b]p8.[/b] Five distinct points are arranged on a plane, creating ten pairs of distinct points. Seven pairs of points are distance $1$ apart, two pairs of points are distance $\sqrt3$ apart, and one pair of points is distance $2$ apart. Draw a line segment from one of these points to the midpoint of a pair of these points. What is the longest this line segment can be? [b]p9.[/b] The inhabitants of Mars use a base $8$ system. Mandrew Mellon is competing in the annual Martian College Interesting Competition of Math (MCICM). The first question asks to compute the product of the base $8$ numerals $1245415_8$, $7563265_8$, and $ 6321473_8$. Mandrew correctly computed the product in his scratch work, but when he looked back he realized he smudged the middle digit. He knows that the product is $1014133027\blacksquare 27662041138$. What is the missing digit? PS. You should use hide for answers.

We are given an infinite set of rectangles in the plane, each with vertices of the form $(0, 0)$, $(0,m)$, $(n, 0)$ and $ (n,m)$, where $m$ and $n$ are positive integers. Prove that there exist two rectangles in the set such that one contains the other.

A finite set $\mathcal{L}$ of coplanar lines, no three of which are concurrent, is called [i]odd[/i] if, for every line $\ell$ in $\mathcal{L}$ the total number of lines in $\mathcal{L}$ crossed by $\ell$ is odd. [list=a] [*]Prove that every finite set of coplanar lines, no three of which are concurrent, extends to an odd set of coplanar lines. [*]Given a positive integer $n$ determine the smallest nonnegative integer $k$ satisfying the following condition: Every set of $n$ coplanar lines, no three of which are concurrent, extends to an odd set of $n+k$ coplanar lines. [/list]

On a table there are $1999$ counters, red on one side and black on the other side, arranged arbitrarily. Two people alternately make moves, where each move is of one of the following two types: (1) Remove several counters which all have the same color up; (2) Reverse several counters which all have the same color up. The player who takes the last counter wins. Decide which of the two players (the one playing first or the other one) has a wining strategy.

Let $S= \{ a_1, \ldots, a_n\}$ be a set of $n\geq 1$ positive real numbers. For each nonempty subset of $S$ the sum of its elements is written down. Show that all written numbers can be divided into $n$ classes such that in each class the ratio of the greatest number to the smallest number is not greater than $2$.

$512$ persons meet at a meeting[ Under every six of these people there is always at least two who know each other. Prove that there must be six people at this gathering, all mutual know.

For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: \[ o(Q_{i_{1}}+Q_{i_{2}}+\ldots+Q_{i_{n}})\ge o(Q_{i_{1}}). \]

On the table, there're $1000$ cards arranged on a circle. On each card, a positive integer was written so that all $1000$ numbers are distinct. First, Vasya selects one of the card, remove it from the circle, and do the following operation: If on the last card taken out was written positive integer $k$, count the $k^{th}$ clockwise card not removed, from that position, then remove it and repeat the operation. This continues until only one card left on the table. Is it possible that, initially, there's a card $A$ such that, no matter what other card Vasya selects as first card, the one that left is always card $A$?

A number of objects, each coloured in one of two given colours, are arranged in a line (there is at least one object having each of the given colours). It is known that each two objects, between which there are exactly $10$ or $15$ other objects, are of the same colour. What is the greatest possible number of such pieces?

Initially, on the lower left and right corner of a $2018\times 2018$ board, there're two horses, red and blue, respectively. $A$ and $B$ alternatively play their turn, $A$ start first. Each turn consist of moving their horse ($A$-red, and $B$-blue) by, simultaneously, $20$ cells respect to one coordinate, and $17$ cells respect to the other; while preserving the rule that the horse can't occupied the cell that ever occupied by any horses in the game. The player who can't make the move loss, who has the winning strategy?

Let $r,n$ be positive integers. For a set $A$, let ${A \choose r}$ denote the family of all $r$-element subsets of $A$. Prove that if $A$ is infinite and $f : {A \choose r} \to {1,2,...,n}$ is any function, then there exists an infinite subset $B$ of $A$ such that $f(X) = f(Y)$ for all $X,Y \in {B \choose r}$.