For which positive integers $n \ge 4$ one can find n points in the plane, no three collinear, such that for each triangle formed with three of the $n$ points which are on the convex hull, exactly one of the $n - 3$ remaining points belongs to its interior.

On a plane are given finitely many red and blue lines, no two parallel, such that any intersection point of two lines of the same color also lies on another line of the other color. Prove that all the lines pass through a single point.

There are $12$ dentists in a clinic near a school. The students of the $5$th year, who are $29$, attend the clinic. Each dentist serves at least $2$ students. Determine the greater number of students that can attend to a single dentist .

A set consisting of $n$ points of a plane is called a [i]bosonti $n$-point[/i] if any three of its points are located in vertices of an isosceles triangle. Find all positive integers $n$ for which there exists a bosonti $n$-point.

In an archery competition of 30 contestants, the target is divided into two zones, zone 1 and zone 2. Each arrow hitting the zone 1 gets 10 points, when hitting zone 2 will get 5 points and no score for miss. Each contestant throws 16 arrows. At the end of the competition, the statistics show that more than 50% of the arrows hit zone 2. The number of arrows that hit zone 1 is equal to the arrows which are missed. Prove than there are two contestants having equal score.

[b]p1.[/b] One chilly morning, $10$ penguins ate a total of $50$ fish. No fish was shared by two or more penguins. Assuming that each penguin ate at least one fish, prove that at least two penguins ate the same number of fish. [b]p2.[/b] A triangle of area $1$ has sides of lengths $a > b > c$. Prove that $b > 2^{1/2}$. [b]p3.[/b] Imagine ducks as points in a plane. Three ducks are said to be in a row if a straight line passes through all three ducks. Three ducks, Huey, Dewey, and Louie, each waddle along a different straight line in the plane, each at his own constant speed. Although their paths may cross, the ducks never bump into each other. Prove: If at three separate times the ducks are in a row, then they are always in a row. [b]p4.[/b] Two computers and a number of humans participated in a large round-robin chess tournament (i.e., every participant played every other participant exactly once). In every game, the winner of the game received one point, the loser zero. If a game ended in a draw, each player received half a point. At the end of the tournament, the sum of the two computers' scores was $38$ points, and all of the human participants finished with the same total score. Describe (with proof) ALL POSSIBLE numbers of humans that could have participated in such a tournament. [b]p5.[/b] One thousand cows labeled $000$, $001$,$...$, $998$, $999$ are requested to enter $100$ empty barns labeled $00$, $01$,$...$,$98$, $99$. One hundred Dalmatians - one at the door of each barn - enforce the following rule: In order for a cow to enter a barn, the label of the barn must be obtainable from the label of the cow by deleting one of the digits. For example, the cow labeled $357$ would be admitted into any of the barns labeled $35$, $37$ or $57$, but would not admitted into any other barns. a) Demonstrate that there is a way for all $1000$ cows to enter the barns so that at least $50$ of the barns remain empty. b) Prove that no matter how they distribute themselves, after all $1000$ cows enter the barns, at most $50$ of the barns will remain empty. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We call the figure shown in the picture consisting of five unit squares a $\emph{plus}$, and each rectangle consisting of two such squares a $\emph{minus}$. Does there exist an odd integer $n$ with the property that a square with side length $n$ can be dissected into pluses and minuses? Justify your answer. [img] https://wiki-images.artofproblemsolving.com//6/6a/18-1-6.png [/img]

Given a positive integer $n \ge 2$, let $\mathcal{P}$ and $\mathcal{Q}$ be two sets, each consisting of $n$ points in three-dimensional space. Suppose that these $2n$ points are distinct. Show that it is possible to label the points of $\mathcal{P}$ as $P_1,P_2,\dots,P_n$ and the points of $\mathcal{Q}$ as $Q_1,Q_2,\dots,Q_n$ such that for any indices $i$ and $j$, the balls of diameters $P_iQ_i$ and $P_jQ_j$ have at least one common point.

[b]p1.[/b] What is the maximum amount of a $12\%$ acid solution that can be obtained from $1$ liter of $5\%$, $10\%$ and $15\%$ solutions? [b]p2.[/b] Which number is greater: $199,719,971,997^2$ or $199,719,971,996 * 19,9719,971,998$ ? [b]p3.[/b] Is there a convex $1998$-gon whose angles are all integer degrees? [b]p4.[/b] Is there a ten-digit number divisible by $11$ that uses all the digits from$ 0$ to $9$? [b]p5.[/b] There are $20$ numbers written in a circle, each of which is equal to the sum of its two neighbors. Prove that the sum of all numbers is $0$. [b]p6.[/b] Is there a convex polygon that has neither an axis of symmetry nor a center of symmetry, but which transforms into itself when rotated around some point through some angle less than $180$ degrees? [b]p7.[/b] In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon. [b]p8.[/b] Give an example of a natural number that is divisible by $30$ and has exactly $105$ different natural factors, including $1$ and the number itself. [b]p9.[/b] In the writing of the antipodes, numbers are also written with the digits $0, ..., 9$, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes $5 * 8 + 7 + 1 = 48$ $2 * 2 * 6 = 24$ $5* 6 = 30$ a) How will the equality $2^3 = ...$ in the writing of the antipodes be continued? b) What does the number$ 9$ mean among the Antipodes? Clarifications: a) It asks to convert $2^3$ in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems. b) What does the digit $9$ mean among the antipodes, i.e. with which digit is it equal in our number system? [b]p10.[/b] Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts? PS.1. There was typo in problem $9$, it asks for $2^3$ and not $23$. PS.2. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Given a simple, connected graph with $n$ vertices and $m$ edges. Prove that one can find at least $m$ ways separating the set of vertices into two parts, such that the induced subgraphs on both parts are connected.

Prove that there exist $0 < \alpha< \beta<1$ numbers have the following properties. (i) for any sufficiently large n, n points can be specified in $\Bbb R^3$ , so that each point is equidistant from at least $n^\alpha$ other points. (ii) the above statement is no longer true with $n^\beta$ instead of $n^\alpha$

There are $14$ players participating at a chess tournament, each playing one game with every other player. After the end of the tournament, the players were ranked in descending order based on their points. The sum of the points of the first three players is equal with the sum of the points of the last nine players. What is the highest possible number of draws in the tournament.(For a victory the player gets $1$ point, for a loss $0$ points, in a draw both players get $0,5$ points.)

Let $X$ be a finite set of real numbers. For any $x,x' \in X$ with $x<x'$, define a function $f(x,x')$, then $f$ is called an ordered pair function on $X$. For any given ordered pair function $f$ on $X$, if there exist elements $x_1 <x_2 <\cdots<x_k$ in $X$ such that $f(x_1 ,x_2 ) \le f(x_2 ,x_3 ) \le \cdots \le f(x_{k-1} ,x_k )$, then $x_1 ,x_2 ,\cdots,x_k$ is called an $f$-ascending sequence of length $k$ in $X$. Similarly, define an $f$-descending sequence of length $l$ in $X$. For integers $k,l \ge 3$, let $h(k,l)$ denote the smallest positive integer such that for any set $X$ of $s$ real numbers and any ordered pair function $f$ on $X$, there either exists an $f$-ascending sequence of length $k$ in $X$ or an $f$-descending sequence of length $l$ in $X$ if $s \ge h(k,l)$. Prove: 1.For $k,l>3,h(k,l) \le h(k-1,l)+h(k,l-1)-1$; 2.$h(k,l) \le \binom{l-2}{k+l-4} +1$.

A meeting is held at a round table. It is known that 7 women have a woman on their right side, and 12 women have a man on their right side. It is also known that 75% of the men have a woman on their right side. How many people are sitting at the round table?

Amir enters Fine Hall and sees the number $2$ written on the blackboard. Amir can perform the following operation: he flips a coin, and if it is heads, he replaces the number $x$ on the blackboard with $3x+1;$ otherwise, he replaces $x$ with $\lfloor x/3\rfloor.$ If Amir performs the operation four times, let $\tfrac{m}{n}$ denote the expected number of times that he writes the digit $1$ on the blackboard, where $m,n$ are relatively prime positive integers. Find $m+n.$

We have a complete graph with $n$ vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?

For $S_n=\{1, 2, ..., n\}$, find the maximum value of $m$ that makes the following proposition true. [b]Proposition[/b] There exists $m$ different subsets of $S$, say $A_1, A_2, ..., A_m$, such that for every $i, j=1, 2, ..., m$, the set $A_i \cup A_j$ is not $S$.

NASA has proposed populating Mars with $2,004$ settlements. The only way to get from one settlement to another will be by a connecting tunnel. A bored bureaucrat draws on a map of Mars, randomly placing $N$ tunnels connecting the settlements in such a way that no two settlements have more than one tunnel connecting them. What is the smallest value of $N$ that guarantees that, no matter how the tunnels are drawn, it will be possible to travel between any two settlements?

suppose that $\mathcal F\subseteq p(X)$ and $|X|=n$. we know that for every $A_i,A_j\in \mathcal F$ that $A_i\supseteq A_j$ we have $3\le |A_i|-|A_j|$. prove that: $|\mathcal F|\le \lfloor\frac{2^n}{3}+\frac{1}{2}\dbinom{n}{\lfloor\frac{n}{2}\rfloor}\rfloor$ (20 points)

If $C$ is a set of $n$ points in the plane that has the following property: For each point $P$ of $C$, there are four points of $C$, each one distinct from $P$ , which are the vertices of a square. Find the smallest possible value of $n$.

How many diagonals can you draw in a convex $2009$-gon if in the finished drawing, every drawn diagonal inside the $2009$-gon may cut at most another drawn diagonal?

Does there exist a set $S$ of positive integers satisfying the following conditions? $\text{(i)}$ $S$ contains $2020$ distinct elements; $\text{(ii)}$ the number of distinct primes in the set $\{\gcd(a, b) : a, b \in S, a \neq b\}$ is exactly $2019$; and $\text{(iii)}$ for any subset $A$ of $S$ containing at least two elements, $\sum\limits_{a,b\in A; a<b} ab$ is not a prime power.

We are given $1978$ sets of size $40$ each. The size of the intersection of any two sets is exactly $1$. Prove that all the sets have a common element.

[u]Set 1[/u] [b]p1.[/b] Farmer John has $4000$ gallons of milk in a bucket. On the first day, he withdraws $10\%$ of the milk in the bucket for his cows. On each following day, he withdraws a percentage of the remaining milk that is $10\%$ more than the percentage he withdrew on the previous day. For example, he withdraws $20\%$ of the remaining milk on the second day. How much milk, in gallons, is left after the tenth day? [b]p2.[/b] Will multiplies the first four positive composite numbers to get an answer of $w$. Jeremy multiplies the first four positive prime numbers to get an answer of $j$. What is the positive difference between $w$ and $j$? [b]p3.[/b] In Nathan’s math class of $60$ students, $75\%$ of the students like dogs and $60\%$ of the students like cats. What is the positive difference between the maximum possible and minimum possible number of students who like both dogs and cats? [u]Set 2[/u] [b]p4.[/b] For how many integers $x$ is $x^4 - 1$ prime? [b]p5.[/b] Right triangle $\vartriangle ABC$ satisfies $\angle BAC = 90^o$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 60$ and $AB = 65$, find the area of $\vartriangle ABC$. [b]p6.[/b] Define $n! = n \times (n - 1) \times ... \times 1$. Given that $3! + 4! + 5! = a^2 + b^2 + c^2$ for distinct positive integers $a, b, c$, find $a + b + c$. [u]Set 3[/u] [b]p7.[/b] Max nails a unit square to the plane. Let M be the number of ways to place a regular hexagon (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Vincent, on the other hand, nails a regular unit hexagon to the plane. Let $V$ be the number of ways to place a square (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Find the nonnegative difference between $M$ and $V$ . [b]p8.[/b] Let a be the answer to this question, and suppose $a > 0$. Find $\sqrt{a +\sqrt{a +\sqrt{a +...}}}$ . [b]p9.[/b] How many ordered pairs of integers $(x, y)$ are there such that $x^2 - y^2 = 2019$? [u]Set 4[/u] [b]p10.[/b] Compute $\frac{p^3 + q^3 + r^3 - 3pqr}{p + q + r}$ where $p = 17$, $q = 7$, and $r = 8$. [b]p11.[/b] The unit squares of a $3 \times 3$ grid are colored black and white. Call a coloring good if in each of the four $2 \times 2$ squares in the $3 \times 3$ grid, there is either exactly one black square or exactly one white square. How many good colorings are there? Consider rotations and reflections of the same pattern distinct colorings. [b]p12.[/b] Define a $k$-[i]respecting [/i]string as a sequence of $k$ consecutive positive integers $a_1$, $a_2$, $...$ , $a_k$ such that $a_i$ is divisible by $i$ for each $1 \le i \le k$. For example, $7$, $8$, $9$ is a $3$-respecting string because $7$ is divisible by $1$, $8$ is divisible by $2$, and $9$ is divisible by $3$. Let $S_7$ be the set of the first terms of all $7$-respecting strings. Find the sum of the three smallest elements in $S_7$. [u]Set 5[/u] [b]p13.[/b] A triangle and a quadrilateral are situated in the plane such that they have a finite number of intersection points $I$. Find the sum of all possible values of $I$. [b]p14.[/b] Mr. DoBa continuously chooses a positive integer at random such that he picks the positive integer $N$ with probability $2^{-N}$ , and he wins when he picks a multiple of 10. What is the expected number of times Mr. DoBa will pick a number in this game until he wins? [b]p15.[/b] If $a, b, c, d$ are all positive integers less than $5$, not necessarily distinct, find the number of ordered quadruples $(a, b, c, d)$ such that $a^b - c^d$ is divisible by $5$. PS. You had better use hide for answers. Last 4 sets have been posted [url=https://artofproblemsolving.com/community/c4h2777362p24370554]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Randy has a deck of $29$ distinct cards. He chooses one of the $29!$ permutations of the deck and then repeatedly rearranges the deck using that permutation until the deck returns to its original order for the first time. What is the maximum number of times Randy may need to rearrange the deck?