Given an integer $n\ge 2$, given $n+1$ distinct points $X_0,X_1,\ldots,X_n$ in the plane, and a positive real number $A$, show that the number of triangles $X_0X_iX_j$ of area $A$ does not exceed $4n\sqrt n$.

On a straight line lie $100$ points and another point outside the line. Which is the biggest the number of isosceles triangles can be formed from the vertices of these $101$ points?

A jeweller makes a chain consisting of $N>3$ numbered links. A querulous customer then asks him to change the order of the links, in such a way that the number of links the jeweller must open is maximized. What is the maximum number?

Let $n \geq 2$ be a natural number, and let $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ be a permutation of $\left(1;\;2;\;...;\;n\right)$. For any integer $k$ with $1 \leq k \leq n$, we place $a_k$ raisins on the position $k$ of the real number axis. [The real number axis is the $x$-axis of a Cartesian coordinate system.] Now, we place three children A, B, C on the positions $x_A$, $x_B$, $x_C$, each of the numbers $x_A$, $x_B$, $x_C$ being an element of $\left\{1;\;2;\;...;\;n\right\}$. [It is not forbidden to place different children on the same place!] For any $k$, the $a_k$ raisins placed on the position $k$ are equally handed out to those children whose positions are next to $k$. [So, if there is only one child lying next to $k$, then he gets the raisin. If there are two children lying next to $k$ (either both on the same position or symmetric with respect to $k$), then each of them gets one half of the raisin. Etc..] After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places). For which $n$ does there exist a configuration $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ and numbers $x_A$, $x_B$, $x_C$, such that all three children are happy?

On a circle $4n$ points are chosen ($n \ge 1$). The points are alternately colored yellow and blue. The yellow points are divided into $n$ pairs and the points in each pair are connected with a yellow line segment. In the same manner the blue points are divided into $n$ pairs and the points in each pair are connected with a blue segment. Assume that no three of the segments pass through a single point. Show that there are at least $n$ intersection points of blue and yellow segments.

In the multiplication magic square below, $l, m, n, p, q, r, s, t, u$ are positive integers. The product of any three numbers in any row, column or diagonal is equal to a constant $k$, where $k$ is a number between $11, 000$ and $12, 500$. Find the value of $k$. \begin{tabular}{|l|l|l|} \hline $l$ & $m$ & $n$ \\ \hline $p$ & $q$ & $r$ \\ \hline $s$ & $t$ & $u$ \\ \hline \end{tabular}

Find the minimum value of $c$ such that for any positive integer $n\ge 4$ and any set $A\subseteq \{1,2,\cdots,n\}$, if $|A| >cn$, there exists a function $f:A\to\{1,-1\}$ satisfying $$\left| \sum_{a\in A}a\cdot f(a)\right| \le 1.$$

Let $n$ and $k$ be relatively prime positive integers with $k<n$. Each number in the set $M=\{1,2,3,\ldots,n-1\}$ is colored either blue or white. For each $i$ in $M$, both $i$ and $n-i$ have the same color. For each $i\ne k$ in $M$ both $i$ and $|i-k|$ have the same color. Prove that all numbers in $M$ must have the same color.

We consider $S$ a set of odd positive interger numbers with $n\geq 3$ elements such that no element divides another element. We say that a set $S$ is $beautiful$ if for any 3 elements from $S$, there is one the divides the sum of the other 2. We call a beautiful set $S$ $maximal$ if we can't add another number to the set such that $S$ will still be beautiful. Find the values of $n$ for which there exists a $maximal$ set.

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?

Show that the number of $3-$element subsets $\{ a , b, c \}$ of $\{ 1 , 2, 3, \ldots, 63 \}$ with $a+b +c < 95$ is less than the number of those with $a + b +c \geq 95.$

For a given set of points in space it is allowed to mirror a point from the set with respect to another point from the set, and to include the image in the set. Starting with a set of seven vertices of a cube, is it possible to include the eight vertex in the set after finitely many such steps?

Several square-shaped papers are situated on a table such that every side of the paper is positioned parallel to the sides of the table. Each paper has a colour, and there are $n$ different coloured papers. It is known that for every $n$ papers with distinct colors, we can always find an overlapping pair of papers. Prove that, using $2n- 2$ nails, it is possible to hammer all the squares of a certain colour to the table.

We say that a digit is [i]high[/i] if it is placed between two other digits and it is bigger than both of them. The digits $0$,$1$,$2$,$\dots$,$9$ are used exactly once to form a 10-digit number. How many numbers can be formed with the property such that they don’t have any high digits?

There are $6$ computers(power off) and $3$ printers. Between a printer and a computer, they are connected with a wire or not. Printer can be only activated if and only if at least one of the connected computer's power is on. Your goal is to connect wires in such a way that, no matter how you choose three computers to turn on among the six, you can activate all $3$ printers. What is the minimum number of wires required to make this possible?

Let $ A \equal{} (a_{ij})$, where $ i,j \equal{} 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} \equal{} 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.

a) 11 kayakers row on the Danube from Szentendre to Kopaszi-gát. They do not necessarily start at the same time, but we know that they all take the same route and that each kayaker rows with a constant speed. Whenever a kayaker passes another one, they do a high five. After they all arrive, everybody claims to have done precisely $10$ high fives in total. Show that it is possible for the kayakers to have rowed in such a way that this is true. b) At a different occasion $13$ kayakers rowed in the same manner; now after arrival everybody claims to have done precisely$ 6$ high fives. Prove that at least one kayaker has miscounted.

We say that distinct positive integers $n, m$ are $friends$ if $\vert n-m \vert$ is a divisor of both ${}n$ and $m$. Prove that, for any positive integer $k{}$, there exist $k{}$ distinct positive integers such that any two of these integers are friends.

Kira has $3$ blocks with the letter $A$, $3$ blocks with the letter $B$, and $3$ blocks with the letter $C$. She puts these $9$ blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence $ABCAABCBC$ the blocks with the letter A have distances $1, 3$, and $4$ between one another, the blocks with the letter $B$ have distances $2, 4$, and $6$ between one another, and the blocks with the letter $C$ have distances $2, 4$, and $6$ between one another. Altogether, we got distances of $1, 2, 3, 4$, and $6$; these are $5$ distinct distances. What is the maximum number of distinct distances that can occur?

Let $S$ be the set of all 3-tuples $(a, b, c)$ of positive integers such that $a + b + c = 2013$. Find $$\sum_{(a,b,c)\in S} abc.$$

Kevin and Yang are playing a game. Yang has $2017 + \tbinom{2017}{2}$ cards with their front sides face down on the table. The cards are constructed as follows: [list] [*] For each $1 \le n \le 2017$, there is a blue card with $n$ written on the back, and a fraction $\tfrac{a_n}{b_n}$ written on the front, where $\gcd(a_n, b_n) = 1$ and $a_n, b_n > 0$. [*] For each $1 \le i < j \le 2017$, there is a red card with $(i, j)$ written on the back, and a fraction $\tfrac{a_i+a_j}{b_i+b_j}$ written on the front. [/list] It is given no two cards have equal fractions. In a turn Kevin can pick any two cards and Yang tells Kevin which card has the larger fraction on the front. Show that, in fewer than $10000$ turns, Kevin can determine which red card has the largest fraction out of all of the red cards.

Michael and Andrew are playing the game Bust, which is played as follows: Michael chooses a positive integer less than or equal to $99$, and writes it on the board. Andrew then makes a move, which consists of him choosing a positive integer less than or equal to $ 8$ and increasing the integer on the board by the integer he chose. Play then alternates in this manner, with each person making exactly one move, until the integer on the board becomes greater than or equal to $100$. The person who made the last move loses. Let S be the sum of all numbers for which Michael could choose initially and win with both people playing optimally. Find S.

The participants to an international conference are native and foreign scientist. Each native scientist sends a message to a foreign scientist and each foreign scientist sends a message to a native scientist. There are native scientists who did not receive a message. Prove that there exists a set $S$ of native scientists such that the outer $S$ scientists are exactly those who received messages from those foreign scientists who did not receive messages from scientists belonging to $S$. [i]Radu Niculescu[/i]

Among a group of 120 people, some pairs are friends. A [i]weak quartet[/i] is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ?

The enemy ship has landed on a $9\times 9$ board that covers exactly $5$ squares of the board, like this: [img]https://cdn.artofproblemsolving.com/attachments/2/4/ae5aa95f5bb5e113fd5e25931a2bf8eb872dbe.png[/img] The ship is invisible. Each defensive missile covers exactly one square, and destroys the ship if it hits one of the $5$ squares that it occupies. Determine the minimum number of missiles needed to destroy the enemy ship with certainty .