$$Problem 4 $$ Straight lines $k,l,m$ intersecting each other in three different points are drawn on a classboard. Bob remembers that in some coordinate system the lines$ k,l,m$ have the equations $y = ax, y = bx$ and $y = c +2\frac{ab}{a+b}x$ (where $ab(a + b)$ is non zero). Misfortunately, both axes are erased. Also, Bob remembers that there is missing a line $n$ ($y = -ax + c$), but he has forgotten $a,b,c$. How can he reconstruct the line $n$?

We have $n$ bags each having $100$ coins. All of the bags have $10$ gram coins except one of them which has $9$ gram coins. We have a balance which can show weights of things that have weight of at most $1$ kilogram. At least how many times shall we use the balance in order to find the different bag? [i]Proposed By Hamidreza Ziarati[/i]

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.

Compute the number of rearrangements $a_1, a_2, \dots, a_{2018}$ of the sequence $1, 2, \dots, 2018$ such that $a_k > k$ for $\textit{exactly}$ one value of $k$.

On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue. At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet. After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls. Let $k(\mathfrak{L})$ be the largest number $k$ such that, no matter how Merlin paints the labyrinth $\mathfrak{L},$ Morgana can always place at least $k$ knights such that no two of them can ever meet. For each $n,$ what are all possible values for $k(\mathfrak{L}),$ where $\mathfrak{L}$ is a labyrinth with $n$ walls?

In a classroom there are $m$ students. During the month of July each of them visited the library at least once but none of them visited the library twice in the same day. It turned out that during the month of July each student visited the library a different number of times, furthermore for any two students $A$ and $B$ there was a day in which $A$ visited the library and $B$ did not and there was also a day when $B$ visited the library and $A$ did not do so. Determine the largest possible value of $m$.

If I roll three fair $4$-sided dice, what is the probability that the sum of the resulting numbers is relatively prime to the product of the resulting numbers?

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

Six thousand points are marked on a circle, and they are colored using 10 colors in such a way that within every group of 100 consecutive points all the colors are used. Determine the least positive integer $ k$ with the following property: In every coloring satisfying the condition above, it is possible to find a group of $ k$ consecutive points in which all the colors are used.

Given an integer $ k > 1.$ We call a $ k \minus{}$digits decimal integer $ a_{1}a_{2}\cdots a_{k}$ is $ p \minus{}$monotonic, if for each of integers $ i$ satisfying $ 1\le i\le k \minus{} 1,$ when $ a_{i}$ is an odd number, $ a_{i} > a_{i \plus{} 1};$ when $ a_{i}$ is an even number, $ a_{i}<a_{i \plus{} 1}.$ Find the number of $ p \minus{}$monotonic $ k \minus{}$digits integers.

Let $m,n,k$ be positive integers, $m> n> k$. An $1 \times m$ strip of paper is divided into the $1 \times 1$ cells. A teacher asks Bill and Pit to place numbers $0$ and $1$ in the cells of the strip so that the sum of the numbers in any $n$ consecutive cells is equal to $k$. After the task was performed it turned out that the sum $S(B)$ of all numbers on the strip of Bill was different from the sum $S(P)$ of Pit. Find the largest possible value of $|S(B) - S(P) |$. (I. Voronovich)

We call a system of non-empty sets $H$ [i]entwined[/i], if for every disjoint pair of sets $A$ and $B$ in $H$ there exists $b\in B$ such that $A\cup\{b\}$ is in $H$ or there exists $a\in A$ such that $B\cup\{a\}$ is in $H.$ Let $H$ be an entwined system of sets containing all of $\{1\},\{2\},\ldots,\{n\}.$ Prove that if $n>k(k+1)/2,$ then $H$ contains a set with at least $k+1$ elements, and this is sharp for every $k,$ i.e. if $n=k(k+1),$ it is possible that every set in $H$ has at most $k$ elements.

At a meeting, there are \( N \) people who do not know each other. Prove that it is possible to introduce them in such a way that no three of them have the same number of acquaintances.

The set $N$ is partitioned into three subsets $A_1,A_2,A_3$. Prove that at least one of them has the following property: There exists a positive number $m$ such that for any $k$ one can find numbers $a_1 < a_2 < ... < a_k$ in that subset satisfying $a_{j+1} -a_j \le m$ for $j = 1,...,k -1$.

Given a paper on which the numbers $1,2,3\dots ,14,15$ are written. Andy and Bobby are bored and perform the following operations, Andy chooses any two numbers (say $x$ and $y$) on the paper, erases them, and writes the sum of the numbers on the initial paper. Meanwhile, Bobby writes the value of $xy(x+y)$ in his book. They were so bored that they both performed the operation until only $1$ number remained. Then Bobby adds up all the numbers he wrote in his book, let’s call $k$ as the sum. $a$. Prove that $k$ is constant which means it does not matter how they perform the operation, $b$. Find the value of $k$.

* A shipment of $13.5$ tons is packed in a number of weightless containers. Each loaded container weighs not more than $350$ kg. Prove that $11$ trucks each of which is capable of carrying · $1.5$ ton can carry this load.

The country Dreamland consists of $2016$ cities. The airline Starways wants to establish some one-way flights between pairs of cities in such a way that each city has exactly one flight out of it. Find the smallest positive integer $k$ such that no matter how Starways establishes its flights, the cities can always be partitioned into $k$ groups so that from any city it is not possible to reach another city in the same group by using at most $28$ flights. [i]Warut Suksompong, Thailand[/i]

Let $a$ and $b$ be positive integers. The cells of an $(a+b+1)\times (a+b+1)$ grid are colored amber and bronze such that there are at least $a^2+ab-b$ amber cells and at least $b^2+ab-a$ bronze cells. Prove that it is possible to choose $a$ amber cells and $b$ bronze cells such that no two of the $a+b$ chosen cells lie in the same row or column.

How many finite sequences $x_1,x_2,\ldots,x_m$ are there such that $x_i=1$ or $2$ and $\sum_{i=1}^mx_i=10$? $\textbf{(A)}~89$ $\textbf{(B)}~73$ $\textbf{(C)}~107$ $\textbf{(D)}~119$

There are $m \ge 2$ blue points and $n \ge 2$ red points in three-dimensional space, and no four points are coplanar. Geoff and Nazar take turns, picking one blue point and one red point and connecting the two with a straight-line segment. Assume that Geoff starts first and the one who first makes a cycle wins. Who has the winning strategy?

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

Let $n$ be a positive integer, and Megavan has a $(3n+1)\times (3n+1)$ board. All squares, except one, are tiled by non-overlapping $1\times 3$ triominoes. In each step, he can choose a triomino that is untouched in the step right before it, and then shift this triomino horizontally or vertically by one square, as long as the triominoes remain non-overlapping after this move. Show that there exist some $k$, such that after $k$ moves Megavan can no longer make any valid moves irregardless of the initial configuration, and find the smallest possible $k$ for each $n$. [i](Note: While he cannot undo a move immediately before the current step, he may still choose to move a triomino that has already been moved at least two steps before.)[/i] [i]Proposed by Ivan Chan Kai Chin[/i]

Determine the number of arrangements $ a_1,a_2,...,a_{10}$ of the numbers $ 1,2,...,10$ such that $ a_i>a_{2i}$ for $ 1 \le i \le 5$ and $ a_i>a_{2i\plus{}1}$ for $ 1 \le i \le 4$.

Jerry writes down all binary strings of length $10$ without any two consecutive $1$s. How many $1$s does Jerry write?

Let $n$ be an even positive integer. We consider rectangles with integer side lengths $k$ and $k +1$, where $k$ is greater than $\frac{n}{2}$ and at most equal to $n$. Show that for all even positive integers $ n$ the sum of the areas of these rectangles equals $$\frac{n(n + 2)(7n + 4)}{24}.$$