For what natural number $n> 100$ can $n$ pairwise distinct numbers be arranged on a circle such that each number is either greater than $100$ numbers following it clockwise or less than all of them? and would any property be violated when deleting any of those numbers?

Prove that the number of ways of choosing $6$ among the first $49$ positive integers, at least two of which are consecutive, is equal to $\binom{49}6-\binom{44}6$.

[b]p7[/b] Cindy cuts regular hexagon $ABCDEF$ out of a sheet of paper. She folds $B$ over $AC$, resulting in a pentagon. Then, she folds $A$ over $CF$, resulting in a quadrilateral. The area of $ABCDEF$ is $k$ times the area of the resulting folded shape. Find $k$. [b]p8[/b] Call a sequence $\{a_n\} = a_1, a_2, a_3, . . .$ of positive integers [i]Fib-o’nacci[/i] if it satisfies $a_n = a_{n-1}+a_{n-2}$ for all $n \ge 3$. Suppose that $m$ is the largest even positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = m$, and suppose that $n$ is the largest odd positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = n$. Find $mn$. [b]p9[/b] Compute the number of ways there are to pick three non-empty subsets $A$, $B$, and $C$ of $\{1, 2, 3, 4, 5, 6\}$, such that $|A| = |B| = |C|$ and the following property holds: $$A \cap B \cap C = A \cap B = B \cap C = C \cap A.$$

In the Rainbow Nation there are two airways: Red Rockets and Blue Boeings. For any two cities in the Rainbow Nation it is possible to travel from the one to the other using either or both of the airways. It is known, however, that it is impossible to travel from Beanville to Mieliestad using only Red Rockets - not directly nor by travelling via other cities. Show that, using only Blue Boeings, one can travel from any city to any other city by stopping at at most one city along the way.

[b]p1. [/b] Evaluate $S$. $$S =\frac{10000^2 - 1}{\sqrt{10000^2 - 19999}}$$ [b]p2. [/b] Starting on a triangular face of a right triangular prism and allowing moves to only adjacent faces, how many ways can you pass through each of the other four faces and return to the first face in five moves? [b]p3.[/b] Given that $$(a + b) + (b + c) + (c + a) = 18$$ $$\frac{1}{a + b}+\frac{1}{b + c}+ \frac{1}{c + a}=\frac59,$$ determine $$\frac{c}{a + b}+\frac{a}{b + c}+\frac{b}{c + a}.$$ [b]p4.[/b] Find all primes $p$ such that $2^{p+1} + p^3 - p^2 - p$ is prime. [b]p5.[/b] In right triangle $ABC$ with the right angle at $A$, $AF$ is the median, $AH$ is the altitude, and $AE$ is the angle bisector. If $\angle EAF = 30^o$ , find $\angle BAH$ in degrees. [b]p6.[/b] For which integers $a$ does the equation $(1 - a)(a - x)(x- 1) = ax$ not have two distinct real roots of $x$? [b]p7. [/b]Given that $a^2 + b^2 - ab - b +\frac13 = 0$, solve for all $(a, b)$. [b]p8. [/b] Point $E$ is on side $\overline{AB}$ of the unit square $ABCD$. $F$ is chosen on $\overline{BC}$ so that $AE = BF$, and $G$ is the intersection of $\overline{DE}$ and $\overline{AF}$. As the location of $E$ varies along side $\overline{AB}$, what is the minimum length of $\overline{BG}$? [b]p9.[/b] Sam and Susan are taking turns shooting a basketball. Sam goes first and has probability $P$ of missing any shot, while Susan has probability $P$ of making any shot. What must $P$ be so that Susan has a $50\%$ chance of making the first shot? [b]p10.[/b] Quadrilateral $ABCD$ has $AB = BC = CD = 7$, $AD = 13$, $\angle BCD = 2\angle DAB$, and $\angle ABC = 2\angle CDA$. Find its area. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

suppose that $\mathcal F\subseteq p(X)$ and $|X|=n$. prove that if $|\mathcal F|>\sum_{i=0}^{k-1}\dbinom{n}{i}$ then there exist $Y\subseteq X$ with $|Y|=k$ such that $p(Y)=\mathcal F\cap Y$ that $\mathcal F\cap Y=\{F\cap Y:F\in \mathcal F\}$(20 points) you can see this problem also here: COMBINATORIAL PROBLEMS AND EXERCISES-SECOND EDITION-by LASZLO LOVASZ-AMS CHELSEA PUBLISHING- chapter 13- problem 10(c)!!!

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

In a television series about incidents in a conspicuous town there are $n$ citizens staging in it, where $n$ is an integer greater than $3$. Each two citizens plan together a conspiracy against one of the other citizens. Prove that there exists a citizen, against whom at least $\sqrt{n}$ other citizens are involved in the conspiracy.

Find all triples $(a, b, c)$ of positive integers with $a+b+c = 10$ such that there are $a$ red, $b$ blue and $c$ green points (all different) in the plane satisfying the following properties: $\bullet$ for each red point and each blue point we consider the distance between these two points, the sum of these distances is $37$, $\bullet$ for each green point and each red point we consider the distance between these two points, the sum of these distances is $30$, $\bullet$ for each blue point and each green point we consider the distance between these two points, the sum of these distances is $1$.

There are $2^{2n+1}$ towns with $2n+1$ companies and each two towns are connected with airlines from one of the companies. What’s the greatest number $k$ with the following property: We can close $k$ of the companies and their airlines in such way that we can still reach each town from any other (connected graph).

On each vertex of a regular $ n\minus{}$gon there was a crow. Call this as initial configuration. At a signal, they all flew by and after a while, those $ n$ crows came back to the $ n\minus{}$gon, one crow for each vertex. Call this as final configuration. Determine all $ n$ such that: there are always three crows such that the triangle they formed in the initial configuration and the triangle they formed in the final configuration are both right-angled triangle.

A grasshopper rests on the point $(1,1)$ on the plane. Denote by $O,$ the origin of coordinates. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point $A$ to $B,$ then the area of $\triangle AOB$ is equal to $\frac 12.$ $(a)$ Find all the positive integral poijnts $(m,n)$ which can be covered by the grasshopper after a finite number of steps, starting from $(1,1).$ $(b)$ If a point $(m,n)$ satisfies the above condition, then show that there exists a certain path for the grasshopper to reach $(m,n)$ from $(1,1)$ such that the number of jumps does not exceed $|m-n|.$

A baker with access to a number of different spices bakes ten cakes. He uses more than half of the different kinds of spices in each cake, but no two of the combinations of spices are exactly the same. Show that there exist three spices $a,b,c$ such that every cake contains at least one of these.

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

consider $n\geq 6$ points $x_1,x_2,\dots,x_n$ on the plane such that no three of them are colinear. We call graph with vertices $x_1,x_2,\dots,x_n$ a "road network" if it is connected, each edge is a line segment, and no two edges intersect each other at points other than the vertices. Prove that there are three road networks $G_1,G_2,G_3$ such that $G_i$ and $G_j$ don't have a common edge for $1\leq i,j\leq 3$. Proposed by Morteza Saghafian

A circle is divided by $2019$ points into equal parts. Two players delete these points in turns. A player loses, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?

[b]p1.[/b] Evaluate $(2-0)^2 \cdot 3+ \frac{20}{2+3}$ . [b]p2.[/b] Let $x = 11 \cdot 99$ and $y = 9 \cdot 101$. Find the sumof the digits of $x \cdot y$. [b]p3.[/b] A rectangle is cut into two pieces. The ratio between the areas of the two pieces is$ 3 : 1$ and the positive difference between those areas is $20$. What’s the area of the rectangle? [b]p4.[/b] Edgeworth is scared of elevators. He is currently on floor $50$ of a building, and he wants to go down to floor $1$. Edgeworth can go down at most $4$ floors each time he uses the elevator. What’s the minimum number of times he needs to use the elevator to get to floor $1$? [b]p5.[/b] There are $20$ people at a party. Fifteen of those people are normal and $5$ are crazy. A normal person will shake hands once with every other normal person, while a crazy person will shake hands twice with every other crazy person. How many total handshakes occur at the party? [b]p6.[/b] Wam and Sang are chewing gum. Gum comes in packages, each package consisting of $14$ sticks of gum. Wam eats $6$ packs and $9$ individual sticks of gum. Sang wants to eat twice as much gum as Wam. How many packs of gum must Sang buy? [b]p7.[/b] At Lakeside Health School (LHS), $40\%$ of students are male and $60\%$ of the students are female. If half of the students at the school take biology, and the same number ofmale and female students take biology, to the nearest percent, what percent of female students take biology? [b]p8.[/b] Evin is bringing diluted raspberry iced tea to the annual LexingtonMath Team party. He has a cup with $10$ mL of iced tea and a $2000$ mL cup of water with $10\%$ raspberry iced tea. If he fills up the cup with $20$ more mL of $10\%$ raspberry iced tea water, what percent of the solution will be iced tea? [b]p9.[/b] Tree $1$ starts at height $220$ m and grows continuously at $3$ m per year. Tree $2$ starts at height $20$ m and grows at $5$ m during the first year, $7$ m per during the second year, $9$ m during the third year, and in general $(3+2n)$ m in the nth year. After which year is Tree $2$ taller than Tree $1$? [b]p10.[/b] Leo and Chris are playing a game in which Chris flips a coin. The coin lands on heads with probability $\frac{499}{999}$ , tails with probability $\frac{499}{999}$ , and it lands on its side with probability $\frac{1}{999}$ . For each flip of the coin, Leo agrees to give Chris $4$ dollars if it lands on heads, nothing if it lands on tails, and $2$ dollars if it lands on its side. What’s the expected value of the number of dollars Chris gets after flipping the coin $17$ times? [b]p11.[/b] Ephram has a pile of balls, which he tries to divide into piles. If he divides the balls into piles of $7$, there are $5$ balls that don’t get divided into any pile. If he divides the balls into piles of $11$, there are $9$ balls that aren’t in any pile. If he divides the balls into piles of $13$, there are $11$ balls that aren’t in any pile. What is the minimumnumber of balls Ephram has? [b]p12.[/b] Let $\vartriangle ABC$ be a triangle such that $AB = 3$, $BC = 4$, and $C A = 5$. Let $F$ be the midpoint of $AB$. Let $E$ be the point on $AC$ such that $EF \parallel BC$. Let CF and $BE$ intersect at $D$. Find $AD$. [b]p13.[/b] Compute the sum of all even positive integers $n \le 1000$ such that: $$lcm(1,2, 3, ..., (n -1)) \ne lcm(1,2, 3,, ...,n)$$. [b]p14.[/b] Find the sum of all palindromes with $6$ digits in binary, including those written with leading zeroes. [b]p15.[/b] What is the side length of the smallest square that can entirely contain $3$ non-overlapping unit circles? [b]p16.[/b] Find the sum of the digits in the base $7$ representation of $6250000$. Express your answer in base $10$. [b]p17.[/b] A number $n$ is called sus if $n^4$ is one more than a multiple of $59$. Compute the largest sus number less than $2023$. [b]p18.[/b] Michael chooses real numbers $a$ and $b$ independently and randomly from $(0, 1)$. Given that $a$ and $b$ differ by at most $\frac14$, what is the probability $a$ and $b$ are both greater than $\frac12$ ? [b]p19.[/b] In quadrilateral $ABCD$, $AB = 7$ and $DA = 5$, $BC =CD$, $\angle BAD = 135^o$ and $\angle BCD = 45^o$. Find the area of $ABCD$. [b]p20.[/b] Find the value of $$\sum_{i |210} \sum_{j |i} \left \lfloor \frac{i +1}{j} \right \rfloor$$ [b]p21.[/b] Let $a_n$ be the number of words of length $n$ with letters $\{A,B,C,D\}$ that contain an odd number of $A$s. Evaluate $a_6$. [b]p22.[/b] Detective Hooa is investigating a case where a criminal stole someone’s pizza. There are $69$ people involved in the case, among whom one is the criminal and another is a witness of the crime. Every day, Hooa is allowed to invite any of the people involved in the case to his rather large house for questioning. If on some given day, the witness is present and the criminal is not, the witness will reveal who the criminal is. What is the minimum number of days of questioning required such that Hooa is guaranteed to learn who the criminal is? [b]p23.[/b] Find $$\sum^{\infty}_{n=2} \frac{2n +10}{n^3 +4n^2 +n -6}.$$ [b]p24.[/b] Let $\vartriangle ABC$ be a triangle with circumcircle $\omega$ such that $AB = 1$, $\angle B = 75^o$, and $BC =\sqrt2$. Let lines $\ell_1$ and $\ell_2$ be tangent to $\omega$ at $A$ and $C$ respectively. Let $D$ be the intersection of $\ell_1$ and $\ell_2$. Find $\angle ABD$ (in degrees). [b]p25.[/b] Find the sum of the prime factors of $14^6 +27$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Thirty people sit at a round table. Each of them is either smart or dumb. Each of them is asked: "Is your neighbor to the right smart or dumb?" A smart person always answers correctly, while a dumb person can answer both correctly and incorrectly. It is known that the number of dumb people does not exceed $F$. What is the largest possible value of $F$ such that knowing what the answers of the people are, you can point at at least one person, knowing he is smart?

In a certain big city, all the streets go in one of two perpendicular directions. During a drive in the city, a car does not pass through any place twice, and returns to the parking place along a street from which it started. If it has made 100 left turns, how many right turns must it have made? [i](5 points)[/i]

A [i]snake of length $k$[/i] is an animal which occupies an ordered $k$-tuple $(s_1, \dots, s_k)$ of cells in a $n \times n$ grid of square unit cells. These cells must be pairwise distinct, and $s_i$ and $s_{i+1}$ must share a side for $i = 1, \dots, k-1$. If the snake is currently occupying $(s_1, \dots, s_k)$ and $s$ is an unoccupied cell sharing a side with $s_1$, the snake can [i]move[/i] to occupy $(s, s_1, \dots, s_{k-1})$ instead. The snake has [i]turned around[/i] if it occupied $(s_1, s_2, \dots, s_k)$ at the beginning, but after a finite number of moves occupies $(s_k, s_{k-1}, \dots, s_1)$ instead. Determine whether there exists an integer $n > 1$ such that: one can place some snake of length $0.9n^2$ in an $n \times n$ grid which can turn around. [i]Nikolai Beluhov[/i]

The $4$ code words $$\square * \otimes \,\,\,\, \oplus \rhd \bullet \,\,\,\, * \square \bullet \,\,\,\, \otimes \oslash \oplus$$ they are in some order $$AMO \,\,\,\, SUR \,\,\,\, REO \,\,\,\, MAS$$ Decrypt $$\otimes \oslash \square * \oplus \rhd \square \bullet \otimes $$

Boeotia is comprised of $3$ islands which are home to $2019$ towns in total. Each flight route connects three towns, each on a different island, providing connections between any two of them in both directions. Any two towns in the country are connected by at most one flight route. Find the maximal number of flight routes in the country

In a group of $90$ children each has at least $30$ friends (friendship is mutual). Prove that they can be divided into three $30$-member groups so that each child has its own a group of at least one friend.

The Facebook group Olympiad training has at least five members. There is a certain integer $k$ with following property: [i]for each $k$-tuple of members there is at least one member of this $k$-tuple friends with each of the other $k - 1$.[/i] (Friendship is mutual: if $A$ is friends with $B$, then also $B$ is friends with $A$.) (a) Suppose $k = 4$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members? (b) Suppose $k = 5$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members?

Ann and Max play a game on a $100 \times 100$ board. First, Ann writes an integer from 1 to 10 000 in each square of the board so that each number is used exactly once. Then Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves in order to reach the rightmost column. In each move the token is moved to a square adjacent by side or vertex. For each visited square (including the starting one) Max pays Ann the number of coins equal to the number written in that square. Max wants to pay as little as possible, whereas Ann wants to write the numbers in such a way to maximise the amount she will receive. How much money will Max pay Ann if both players follow their best strategies? [i]Lev Shabanov[/i]