Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.

Find the number of all $\text{permutations}$ of $\left \{ 1,2,\cdots ,n \right \}$ like $p$ such that there exists a unique $i \in \left \{ 1,2,\cdots ,n \right \}$ that : $$p(p(i)) \geq i$$

On a board the numbers $1,2,3,\dots,98,99$ are written. One has to mark $50$ of them, such that the sum of two marked numbers is never equal to $99$ or $100$. How many ways one can mark these numbers?

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

There are a total of $n$ boys and girls sitting in a big circle. Now, Dave wants to walk around the circle. For a start point, if at any time, one of the following two conditions holds: 1. he doesn't see any girl 2. the number of boys he saw $\ge$ the number of girls he saw $+k$ Then we say this point is [i]good[/i]. What is the maximum of $r$ with the property that there is at least one good point whenever the number of girls is $r$?

Several circles are drawn on the plane and all points of their meeting or touching are marked. May be that each circle contains exactly four marked points and exactly four marked points lie on each circle?

How many spanning trees does the following graph (with $6$ vertices and $9$ edges) have? (A spanning tree is a subset of edges that spans all of the vertices of the original graph, but does not contain any cycles.) [img]https://cdn.artofproblemsolving.com/attachments/0/4/0e53e0fbb141b66a7b1c08696be2c5dfe68067.png[/img]

Let $m \ge n$ be positive integers. Frieder is given $mn$ posters of Linus with different integer dimensions of $k \times l$ with $1 \ge k \ge m$ and $1 \ge l \ge n$. He must put them all up one by one on his bedroom wall without rotating them. Every time he puts up a poster, he can either put it on an empty spot on the wall or on a spot where it entirely covers a single visible poster and does not overlap any other visible poster. Determine the minimal area of the wall that will be covered by posters.

The $2018$ inhabitants of a city are divided in two groups: the knights(only speak the truth) and the liars(only speak the lie). The inhabitants sat in a circle and everybody spoke "My two neighbours(in the left and in the right) are liars". After this, one inhabitant got off the circle. The $2017$ inhabitants sat again in a circle(not necessarily in the same order), and everybody spoke "None of my two neighbours(in the left and in the right) is of the same group of myself" Can we determine the group of the inhabitant that got off the city?

There are $n$ points in general position in space (no three lie on the same straight line, no four lie in the same plane). A plane is drawn through every three of them. Prove that If you take any whatever $n-3$ points in space, there is a plane from those drawn that does not contain any of these $n - 3$ points.

[b]p1.[/b] Find $20 \cdot 18 + 20 + 18 + 1$. [b]p2.[/b] Suzie’s Ice Cream has $10$ flavors of ice cream, $5$ types of cones, and $5$ toppings to choose from. An ice cream cone consists of one flavor, one cone, and one topping. How many ways are there for Sebastian to order an ice cream cone from Suzie’s? [b]p3.[/b] Let $a = 7$ and $b = 77$. Find $\frac{(2ab)^2}{(a+b)^2-(a-b)^2}$ . [b]p4.[/b] Sebastian invests $100,000$ dollars. On the first day, the value of his investment falls by $20$ percent. On the second day, it increases by $25$ percent. On the third day, it falls by $25$ percent. On the fourth day, it increases by $60$ percent. How many dollars is his investment worth by the end of the fourth day? [b]p5.[/b] Square $ABCD$ has side length $5$. Points $K,L,M,N$ are on segments $AB$,$BC$,$CD$,$DA$ respectively,such that $MC = CL = 2$ and $NA = AK = 1$. The area of trapezoid $KLMN$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [b]p6.[/b] Suppose that $p$ and $q$ are prime numbers. If $p + q = 30$, find the sum of all possible values of $pq$. [b]p7.[/b] Tori receives a $15 - 20 - 25$ right triangle. She cuts the triangle into two pieces along the altitude to the side of length $25$. What is the difference between the areas of the two pieces? [b]p8.[/b] The factorial of a positive integer $n$, denoted $n!$, is the product of all the positive integers less than or equal to $n$. For example, $1! = 1$ and $5! = 120$. Let $m!$ and $n!$ be the smallest and largest factorial ending in exactly $3$ zeroes, respectively. Find $m + n$. [b]p9.[/b] Sam is late to class, which is located at point $B$. He begins his walk at point $A$ and is only allowed to walk on the grid lines. He wants to get to his destination quickly; how many paths are there that minimize his walking distance? [img]https://cdn.artofproblemsolving.com/attachments/a/5/764e64ac315c950367357a1a8658b08abd635b.png[/img] [b]p10.[/b] Mr. Iyer owns a set of $6$ antique marbles, where $1$ is red, $2$ are yellow, and $3$ are blue. Unfortunately, he has randomly lost two of the marbles. His granddaughter starts drawing the remaining $4$ out of a bag without replacement. She draws a yellow marble, then the red marble. Suppose that the probability that the next marble she draws is blue is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positiveintegers. What is $m + n$? [b]p11.[/b] If $a$ is a positive integer, what is the largest integer that will always be a factor of $(a^3+1)(a^3+2)(a^3+3)$? [b]p12.[/b] What is the largest prime number that is a factor of $160,401$? [b]p13.[/b] For how many integers $m$ does the equation $x^2 + mx + 2018 = 0$ have no real solutions in $x$? [b]p14.[/b] What is the largest palindrome that can be expressed as the product of two two-digit numbers? A palindrome is a positive integer that has the same value when its digits are reversed. An example of a palindrome is $7887887$. [b]p15.[/b] In circle $\omega$ inscribe quadrilateral $ADBC$ such that $AB \perp CD$. Let $E$ be the intersection of diagonals $AB$ and $CD$, and suppose that $EC = 3$, $ED = 4$, and $EB = 2$. If the radius of $\omega$ is $r$, then $r^2 =\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine $m + n$. [b]p16.[/b] Suppose that $a, b, c$ are nonzero real numbers such that $2a^2 + 5b^2 + 45c^2 = 4ab + 6bc + 12ca$. Find the value of $\frac{9(a + b + c)^3}{5abc}$ . [b]p17.[/b] Call a positive integer n spicy if there exist n distinct integers $k_1, k_2, ... , k_n$ such that the following two conditions hold: $\bullet$ $|k_1| + |k_2| +... + |k_n| = n2$, $\bullet$ $k_1 + k_2 + ...+ k_n = 0$. Determine the number of spicy integers less than $10^6$. [b]p18.[/b] Consider the system of equations $$|x^2 - y^2 - 4x + 4y| = 4$$ $$|x^2 + y^2 - 4x - 4y| = 4.$$ Find the sum of all $x$ and $y$ that satisfy the system. [b]p19.[/b] Determine the number of $8$ letter sequences, consisting only of the letters $W,Q,N$, in which none of the sequences $WW$, $QQQ$, or $NNNN$ appear. For example, $WQQNNNQQ$ is a valid sequence, while $WWWQNQNQ$ is not. [b]p20.[/b] Triangle $\vartriangle ABC$ has $AB = 7$, $CA = 8$, and $BC = 9$. Let the reflections of $A,B,C$ over the orthocenter H be $A'$,$B'$,$C'$. The area of the intersection of triangles $ABC$ and $A'B'C'$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ , where $b$ is squarefree and $a$ and $c$ are relatively prime. determine $a+b+c$. (The orthocenter of a triangle is the intersection of its three altitudes.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

At a mathematical olympiad, eight problems were given to 30 contestants. In order to take the difficulty of each problem into account, the jury decided to assign weights to the problems as follows: a problem is worth $n$ points if it was not solved by exactly $n$ contestants. For example, if a problem was solved by all contestants, then it is worth no points. (It is assumed that there are no partial marks for a problem.) Ivan got less points than any other contestant. Find the greatest score he can have.

Given a finite set $X$, let $f$ be a rule such that $f$ maps every [i]even-element-subset[/i] $E$ of $X$ (i.e. $E \subseteq X$, $|E|$ is even) into a real number $f(E)$. Suppose that $f$ satisfies the following conditions: (I) there exists an [i]even-element-subset[/i] $D$ of $X$ such that $f(D)>1990$; (II) for any two disjoint [i]even-element-subsets [/i]$A,B$ of $X$, equation $f(A\cup B)=f(A)+f(B)-1990$ holds. Prove that there exist two subsets $P,Q$ of $X$ satisfying: (1) $P\cap Q=\emptyset$, $P\cup Q=X$; (2) for any [i]non-even-element-subset [/i]$S$ of $P$ (i.e. $S\subseteq P$, $|S|$ is odd), we have $f(S)>1990$; (3) for any [i]even-element-subset[/i] $T$ of $Q$, we have $f(T)\le 1990$.

Given an integer $n\geq 3$, determine the smallest positive number $k$ such that any two points in any $n$-gon (or at its boundary) in the plane can be connected by a polygonal path consisting of $k$ line segments contained in the $n$-gon (including its boundary).

[b]p1.[/b] How many primes $p < 100$ satisfy $p = a^2 + b^2$ for some positive integers $a$ and $b$? [b]p2. [/b] For $a < b < c$, there exists exactly one Pythagorean triple such that $a + b + c = 2000$. Find $a + c - b$. [b]p3.[/b] Five points lie on the surface of a sphere of radius $ 1$ such that the distance between any two points is at least $\sqrt2$. Find the maximum volume enclosed by these five points. [b]p4.[/b] $ABCDEF$ is a convex hexagon with $AB = BC = CD = DE = EF = FA = 5$ and $AC = CE = EA = 6$. Find the area of $ABCDEF$. [b]p5.[/b] Joe and Wanda are playing a game of chance. Each player rolls a fair $11$-sided die, whose sides are labeled with numbers $1, 2, ... , 11$. Let the result of the Joe’s roll be $X$, and the result of Wanda’s roll be $Y$ . Joe wins if $XY$ has remainder $ 1$ when divided by $11$, and Wanda wins otherwise. What is the probability that Joe wins? [b]p6.[/b] Vivek picks a number and then plays a game. At each step of the game, he takes the current number and replaces it with a new number according to the following rule: if the current number $n$ is divisible by $3$, he replaces $n$ with $\frac{n}{3} + 2$, and otherwise he replaces $n$ with $\lfloor 3 \log_3 n \rfloor$. If he starts with the number $3^{2011}$, what number will he have after $2011$ steps? Note that $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. [b]p7.[/b] Define a sequence an of positive real numbers with a$_1 = 1$, and $$a_{n+1} =\frac{4a^2_n - 1}{-2 + \frac{4a^2_n -1}{-2+ \frac{4a^2_n -1}{-2+...}}}.$$ What is $a_{2011}$? [b]p8.[/b] A set $S$ of positive integers is called good if for any $x, y \in S$ either $x = y$ or $|x - y| \ge 3$. How many subsets of $\{1, 2, 3, ..., 13\}$ are good? Include the empty set in your count. [b]p9.[/b] Find all pairs of positive integers $(a, b)$ with $a \le b$ such that $10 \cdot lcm \, (a, b) = a^2 + b^2$. Note that $lcm \,(m, n)$ denotes the least common multiple of $m$ and $n$. [b]p10.[/b] For a natural number $n$, $g(n)$ denotes the largest odd divisor of $n$. Find $$g(1) + g(2) + g(3) + ... + g(2^{2011})$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Denmark has played an international football match against Georgia. the fight ended $5-5$, and between the first and the last goal the game has justnever stood . No country has scored three goals in a row, and Denmark scored the sixth goal. Can you use this information to determine which country scored the fifth goal?

A kingdom has $2021$ towns. All of the towns lie on a circle, and there is a one-way road going from every town to the next $101$ towns in a clockwise order. Each road is colored in one color. Additionally, it is known that for any ordered pair of towns $A$ and $B$ it is possible to find a path from $A$ to $B$ so that no two roads of the path would have the same color. Find the minimal number of road colors in the kingdom.

(a) Two identical cogwheels with $14$ teeth each are given . One is laid horizontally on top of the other in such a way that their teeth coincide (thus the projections of the teeth on the horizontal plane are identical ) . Four pairs of coinciding teeth are cut off. Is it always possible to rotate the two cogwheels with respect to each other so that their common projection looks like that of an entire cogwheel? (The cogwheels may be rotated about their common axis, but not turned over.) (b) Answer the same question , but with two $13$-tooth cogwheels and four pairs of cut-off teeth.

In the Republic of Unistan there are $n$ national roads, each of them links two cities exactly. You can travel from one city to another of your choice using a sequence of roads. The President of Unistan ordered to label the national roads with the integers from $1$ to $n$ by an old law: if a city is adjacent to more than one road, the greatest common divisor of the numbers of that roads must be one. Show that you can label the national roads without breaking the law.

An arbitrary number of lines divide the plane into regions. Show that the regions can be colored red and blue so that neighboring regions have different colors.

Sir Alex is coaching a soccer team of $n$ players of distinct heights. He wants to line them up so that for each player $P$, the total number of players that are either to the left of $P$ and taller than $P$ or to the right of $P$ and shorter than $P$ is even. In terms of $n$, how many possible orders are there? [i]Michael Ren[/i]

[b]p1.[/b] $ABCD$ is a convex quadrilateral such that $AB = 20$, $BC = 24$, $CD = 7$, $DA = 15$, and $\angle DAB$ is a right angle. What is the area of $ABCD$? [b]p2.[/b] A triangular number is one that can be written in the form $1 + 2 +...·+n$ for some positive number $n$. $ 1$ is clearly both triangular and square. What is the next largest number that is both triangular and square? [b]p3.[/b] Find the last (i.e. rightmost) three digits of $9^{2008}$. [b]p4.[/b] When expressing numbers in a base $b \ge 11$, you use letters to represent digits greater than $9$. For example, $A$ represents $10$ and $B$ represents $11$, so that the number $110$ in base $10$ is $A0$ in base $11$. What is the smallest positive integer that has four digits when written in base $10$, has at least one letter in its base $12$ representation, and no letters in its base $16$ representation? [b]p5.[/b] A fly starts from the point $(0, 16)$, then flies straight to the point $(8, 0)$, then straight to the point $(0, -4)$, then straight to the point $(-2, 0)$, and so on, spiraling to the origin, each time intersecting the coordinate axes at a point half as far from the origin as its previous intercept. If the fly flies at a constant speed of $2$ units per second, how many seconds will it take the fly to reach the origin? [b]p6.[/b] A line segment is divided into two unequal lengths so that the ratio of the length of the short part to the length of the long part is the same as the ratio of the length of the long part to the length of the whole line segment. Let $D$ be this ratio. Compute $$D^{-1} + D^{[D^{-1}+D^{(D^{-1}+D^2)}]}.$$ [b]p7.[/b] Let $f(x) = 4x + 2$. Find the ordered pair of integers $(P, Q)$ such that their greatest common divisor is $1, P$ is positive, and for any two real numbers $a$ and $b$, the sentence: “$P a + Qb \ge 0$” is true if and only if the following sentence is true: “For all real numbers x, if $|f(x) - 6| < b$, then $|x - 1| < a$.” [b]p8.[/b] Call a rectangle “simple” if all four of its vertices have integers as both of their coordinates and has one vertex at the origin. How many simple rectangles are there whose area is less than or equal to $6$? [b]p9.[/b] A square is divided into eight congruent triangles by the diagonals and the perpendicular bisectors of its sides. How many ways are there to color the triangles red and blue if two ways that are reflections or rotations of each other are considered the same? [b]p10.[/b] In chess, a knight can move by jumping to any square whose center is $\sqrt5$ units away from the center of the square that it is currently on. For example, a knight on the square marked by the horse in the diagram below can move to any of the squares marked with an “X” and to no other squares. How many ways can a knight on the square marked by the horse in the diagram move to the square with a circle in exactly four moves? [img]https://cdn.artofproblemsolving.com/attachments/d/9/2ef9939642362182af12089f95836d4e294725.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There is an $n\times n$ chessboard where $n\geq 4$ is a positive even number. The cells of the chessboard are coloured black and white such that adjacent cells sharing a common side have different colours. Let $A$ and $B$ be two interior cells (which means cells not lying on an edge of the chessboard) of distinct colours. Prove that a chess piece can move from $A$ to $B$ by moving across adjacent cells such that every cell of the chessboard is passed through exactly once.

Define the function $ g(\cdot): \mathbb{Z} \to \{0,1\}$ such that $ g(n) \equal{} 0$ if $ n < 0$, and $ g(n) \equal{} 1$ otherwise. Define the function $ f(\cdot): \mathbb{Z} \to \mathbb{Z}$ such that $ f(n) \equal{} n \minus{} 1024g(n \minus{} 1024)$ for all $ n \in \mathbb{Z}$. Define also the sequence of integers $ \{a_i\}_{i \in \mathbb{N}}$ such that $ a_0 \equal{} 1$ e $ a_{n \plus{} 1} \equal{} 2f(a_n) \plus{} \ell$, where $ \ell \equal{} 0$ if $ \displaystyle \prod_{i \equal{} 0}^n{\left(2f(a_n) \plus{} 1 \minus{} a_i\right)} \equal{} 0$, and $ \ell \equal{} 1$ otherwise. How many distinct elements are in the set $ S: \equal{} \{a_0,a_1,\ldots,a_{2009}\}$? [i](Paolo Leonetti)[/i]

Suppose that we have $2n$ non-empty subset of $ \big\{0,1,2,...,2n-1\big\} $ that sum of the elements of these subsets is $ \binom{2n+1}{2}$ . Prove that we can choose one element from every subset that some of them is $ \binom{2n}{2}$ [i]Proposed by Morteza Saghafian and Afrouz Jabalameli [/i]