Given any $22$ points in the plane, no three collinear. Show that the points can be divided into $11$ pairs, so that the $11$ line segments defined by the pairs have at least five different intersections

In a $2 \times 3$ rectangle there is a polyline of length $36$, which can have self-intersections. Show that there exists a line parallel to two sides of the rectangle, which intersects the other two sides in their interior points and intersects the polyline in fewer than $10$ points. [i]Proposed by Josef Tkadlec, Czechia and Vojtech Bálint, Slovakia[/i]

The following process is applied to each positive integer: the sum of its digits is subtracted from the number, and the result is divided by $9$. For example, the result of the process applied to $938$ is $102$, since $\frac{938-(9 + 3 + 8)}{9} = 102.$ Applying the process twice to $938$ the result is $11$, applied three times the result is $1$, and applying it four times the result is $0$. When the process is applied one or more times to an integer $n$, the result is eventually $0$. The number obtained before obtaining $0$ is called the [i]house[/i] of $n$. How many integers less than $26000$ share the same [i]house[/i] as $2012$?

Let $n, k \geq 3$ be integers, and let $S$ be a circle. Let $n$ blue points and $k$ red points be chosen uniformly and independently at random on the circle $S$. Denote by $F$ the intersection of the convex hull of the red points and the convex hull of the blue points. Let $m$ be the number of vertices of the convex polygon $F$ (in particular, $m=0$ when $F$ is empty). Find the expected value of $m$.

p1. Two integers $m$ and $n$ are said to be [i]coprime [/i] if there are integers $a$ and $ b$ such that $am + bn = 1$. Show that for each integer $p$, the pair of numbers formed by $21p + 4$ and $14p + 3$ are always coprime. p2. Two farmers, Person $A$ and Person $B$ intend to change the boundaries of their land so that it becomes like a straight line, not curvy as in image below. They do not want the area of ​​their origin to be reduced. Try define the boundary line they should agree on, and explain why the new boundary does not reduce the area of ​​their respective origins. [img]https://cdn.artofproblemsolving.com/attachments/4/d/ec771d15716365991487f3705f62e4566d0e41.png[/img] p3. The system of equations of four variables is given: $\left\{\begin{array}{l} 23x + 47y - 3z = 434 \\ 47x - 23y - 4w = 183 \\ 19z + 17w = 91 \end{array} \right. $ where $x, y, z$, and $w$ are positive integers. Determine the value of $(13x - 14y)^3 - (15z + 16w)^3$ p4. A person drives a motorized vehicle so that the material used fuel is obtained at the following graph. [img]https://cdn.artofproblemsolving.com/attachments/6/f/58e9f210fafe18bfb2d9a3f78d90ff50a847b2.png[/img] Initially the vehicle contains $ 3$ liters of fuel. After two hours, in the journey of fuel remains $ 1$ liter. a. If in $ 1$ liter he can cover a distance of $32$ km, what is the distance taken as a whole? Explain why you answered like that? b. After two hours of travel, is there any acceleration or deceleration? Explain your answer. c. Determine what the average speed of the vehicle is. p5. Amir will make a painting of the circles, each circle to be filled with numbers. The circle's painting is arrangement follows the pattern below. [img]https://cdn.artofproblemsolving.com/attachments/8/2/533bed783440ea8621ef21d88a56cdcb337f30.png[/img] He made a rule that the bottom four circles would be filled with positive numbers less than $10$ that can be taken from the numbers on the date of his birth, i.e. $26 \,\, - \,\, 12 \,\, - \,\,1961$ without recurrence. Meanwhile, the circles above will be filled with numbers which is the product of the two numbers on the circles in underneath. a. In how many ways can he place the numbers from left to right, right on the bottom circles in order to get the largest value on the top circle? Explain. b. On another occasion, he planned to put all the numbers on the date of birth so that the number of the lowest circle now, should be as many as $8$ circles. He no longer cares whether the numbers are repeated or not . i. In order to get the smallest value in the top circle, how should the numbers be arranged? ii. How many arrays are worth considering to produce the smallest value?

Arnaldo and Bernaldo train for a marathon along a circular track, which has in its center a mast with a flag raised. Arnaldo runs faster than Bernaldo, so that every $30$ minutes of running, while Arnaldo gives $15$ laps on the track, Bernaldo can only give $10$ complete laps. Arnaldo and Bernaldo left at the same moment of the line and ran with constant velocities, both in the same direction. Between minute $1$ and minute $61$ of the race, how many times did Arnaldo, Bernaldo and the mast become collinear?

Let $N$ be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of $N$ which form a triangle of area smaller than 1.

Let $n$ be an even positive integer. We fill in a number on each cell of a rectangle table of $n$ columns and multiple rows as following: i. Each row is assigned to some positive integer $a$ and its cells are filled by $0$ or $a$ (in any order); ii. The sum of all numbers in each row is $n$. Note that we cannot add any more row to the table such that the conditions (i) and (ii) still hold. Prove that if the number of $0$’s on the table is odd then the maximum odd number on the table is a perfect square.

We say that a finite set $S$ of red and green points in the plane is [i]separable[/i] if there exists a triangle $\delta$ such that all points of one colour lie strictly inside $\delta$ and all points of the other colour lie strictly outside of $\delta$. Let $A$ be a finite set of red and green points in the plane, in general position. Is it always true that if every $1000$ points in $A$ form a separable set then $A$ is also separable?

Three parallel lines $\ell_1, \ell_2$ and $\ell_3$ of a plane are given such that the line $\ell_2$ has the same distance $a$ from $\ell_1$ and $\ell_3$. We put $5$ points $M_1, M_2, M_3,M_4$ and $M_5$ on the lines $\ell_1, \ell_2$ and $\ell_3$ in such a way that each line contains at least one point. Detennine the maximal number of isosceles triangles that are possible to be formed with vertices three of the points $M_1, M_2, M_3, M_4$ and $M_5$ in the following cases: (i) $M_1,M_2,M_3 \in \ell_2, M_4 \in \ell_1$ and $M_5 \in \ell_3$. (ii) $M_1,M_2 \in \ell_1, M_3,M_4 \in \ell_3$ and $M_5 \in \ell_2$.

(a) Divide the line segment $[0,1]$ into smaller black and white segments so that, for any polynomial $p(x)$ of degree no greater than two, the sum of increments of $p(x)$ along all the black segments is equal to that along the white ones. (The increment of $p(x)$ along the segment $[a, n]$ is the number $p(b) - p(a)$.) (b) Can this be done for all polynomials of degree no greater than $1995$? (Burkov)

A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied: (i) All the squares are congruent. (ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares. (iii) Each square touches exactly three other squares. How many positive integers $n$ are there with $2018\leq n \leq 3018$, such that there exists a collection of $n$ squares that is tri-connected?

Let $n$ be positive integer and $S$= {$0,1,…,n$}, Define set of point in the plane. $$A = \{(x,y) \in S \times S \mid -1 \leq x-y \leq 1 \} $$, We want to place a electricity post on a point in $A$ such that each electricity post can shine in radius 1.01 unit. Define minimum number of electricity post such that every point in $A$ is in shine area

There are $42$ students taking part in the Team Selection Test. It is known that every student knows exactly $20$ other students. Show that we can divide the students into $2$ groups or $21$ groups such that the number of students in each group is equal and every two students in the same group know each other.

$600$ integer numbers from $[1,1000]$ colored in red. Natural segment $[n,k]$ is called yummy if for every natural $t$ from $[1,k-n]$ there are two red numbers $a,b$ from $[n,k]$ and $b-a=t$ . Prove that there is yummy segment with $[a,b]$ with $b-a \geq 199$

There is an even number of cards on a table; a positive integer is written on each card. Let $a_k$ be the number of cards having $k$ written on them. It is known that \[a_n-a_{n-1}+a_{n-2}- \cdots \ge 0 \] for each positive integer $n$. Prove that the cards can be partitioned into pairs so that the numbers in each pair differ by $1$. [i](A. Golovanov)[/i]

On the edges of a convex polyhedra we draw arrows such that from each vertex at least an arrow is pointing in and at least one is pointing out. Prove that there exists a face of the polyhedra such that the arrows on its edges form a circuit. [i]Dan Schwartz[/i]

Let $S$ be a set of $10$ positive integers. Prove that one can find two disjoint subsets $A =\{a_1, ..., a_k\}$ and $B = \{b_1, ... , b_k\}$ of $S$ with $|A| = |B|$ such that the sums $x =\frac{1}{a_1}+ ... +\frac{1}{a_k}$ and $y =\frac{1}{b_1}+ ... +\frac{1}{b_k}$ differ by less than $0.01$, i.e., $|x - y| < 1/100$.

Let $A$ be a set $A=\{1,2,3,...,2017\}$. Subset $S$ of set $A$ is [i]good [/i] if for all $x\in A$ sum of remaining elements of set $S$ has same last digit as $x$. Prove that [i]good[/i] subset with $405$ elements is not possible.

A "Hishgad" lottery ticket contains the numbers $1$ to $mn$, arranged in some order in a table with $n$ rows and $m$ columns. It is known that the numbers in each row increase from left to right and the numbers in each column increase from top to bottom. An example for $n=3$ and $m=4$: [asy] size(3cm); Label[][] numbers = {{"$1$", "$2$", "$3$", "$9$"}, {"$4$", "$6$", "$7$", "$10$"}, {"$5$", "$8$", "$11$", "$12$"}}; for (int i=0; i<5;++i) { draw((i,0)--(i,3)); } for (int i=0; i<4;++i) { draw((0,i)--(4,i)); } for (int i=0; i<4;++i){ for (int j=0; j<3;++j){ label(numbers[2-j][i], (i+0.5, j+0.5)); }} [/asy] When the ticket is bought the numbers are hidden, and one must "scratch" the ticket to reveal them. How many cells does it always suffice to reveal in order to determine the whole table with certainty?

A set of $1985$ points is distributed around the circumference of a circle and each of the points is marked with $1$ or $-1$. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of points marked with $-1$ is less than $662$, there must be at least one good point.

For each positive integer $k$, let $n_k$ be the smallest positive integer such that there exists a finite set $A$ of integers satisfy the following properties: [list] [*]For every $a\in A$, there exists $x,y\in A$ (not necessary distinct) that $$n_k\mid a-x-y$$[/*] [*]There's no subset $B$ of $A$ that $|B|\leq k$ and $$n_k\mid \sum_{b\in B}{b}.$$ [/list] Show that for all positive integers $k\geq 3$, we've $$n_k<\Big( \frac{13}{8}\Big)^{k+2}.$$

Consider the following array: \[ 3, 5\\3, 8, 5\\3, 11, 13, 5\\3, 14, 24, 18, 5\\3, 17, 38, 42, 23, 5\\ \ldots \] Find the 5-th number on the $ n$-th row with $ n>5$.

Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. At least how many problems are needed?

The numbers $1- \sqrt{2}$, $\sqrt{2}$ and $1+\sqrt{2}$ are written on a blackboard. Every minute, if $x, y, z$ are the numbers written, then they are erased and the numbers, $x^2 + xy + y^2$, $y^2 + yz + z^2$ and $z^2 + zx + x^2$ are written. Determine whether it is possible for all written numbers to be rational numbers after a finite number of minutes.