[b]p1.[/b] Ben and his dog are walking on a path around a lake. The path is a loop $500$ meters around. Suddenly the dog runs away with velocity $10$ km/hour. Ben runs after it with velocity $8$ km/hour. At the moment when the dog is $250$ meters ahead of him, Ben turns around and runs at the same speed in the opposite direction until he meets the dog. For how many minutes does Ben run? [b]p2.[/b] The six interior angles in two triangles are measured. One triangle is obtuse (i.e. has an angle larger than $90^o$) and the other is acute (all angles less than $90^o$). Four angles measure $120^o$, $80^o$, $55^o$ and $10^o$. What is the measure of the smallest angle of the acute triangle? [b]p3.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p4.[/b] Two three-digit whole numbers are called relatives if they are not the same, but are written using the same triple of digits. For instance, $244$ and $424$ are relatives. What is the minimal number of relatives that a three-digit whole number can have if the sum of its digits is $10$? [b]p5.[/b] Three girls, Ann, Kelly, and Kathy came to a birthday party. One of the girls wore a red dress, another wore a blue dress, and the last wore a white dress. When asked the next day, one girl said that Kelly wore a red dress, another said that Ann did not wear a red dress, the last said that Kathy did not wear a blue dress. One of the girls was truthful, while the other two lied. Which statement was true? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The seats in the Parliament of some country are arranged in a rectangle of $10$ rows of $10$ seats each. All the $100$ $MP$s have different salaries. Each of them asks all his neighbours (sitting next to, in front of, or behind him, i.e. $4$ members at most) how much they earn. They feel a lot of envy towards each other: an $MP$ is content with his salary only if he has at most one neighbour who earns more than himself. What is the maximum possible number of $MP$s who are satisfied with their salaries?

Let $A$ be a finite set of positive numbers , $B=\{\frac{a+b}{c+d} |a,b,c,d \in A \}$. Show that: $\left | B \right | \ge 2\left | A \right |^2-1 $, where $|X| $ be the number of elements of the finite set $X$. (High School Affiliated to Nanjing Normal University )

Let $a,t,n$ be positive integers such that $a\le n$. Consider the subsets of $\{1,2,\dots,n\}$ whose any two elements differ by at least $t$. Prove that the number of such subsets not containing $a$ is at most $t^2$ times the number of those that do contain $a$.

An equilateral triangle with side $3$ is divided into $9$ small equal equilateral triangles with sides of length $1$. Each vertex of a triangle small (bold dots) is numbered with a different number than the $1$ to $10$. Inside each small triangle, write the sum of the numbers corresponding to its three vertices. Prove that there are three small triangles for which it is verified that the sum of the numbers written inside is at least $48$. [img]https://cdn.artofproblemsolving.com/attachments/2/1/b2f58b6d59cb26e2fe29d0df59c1a42639a496.png[/img]

2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations: [list] [*] Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip. [*] Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.[/list] Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it. [i]Brian Hamrick.[/i]

Let $n>1$ be an integer. Find the number of permutations $(a_1,a_2,\ldots,a_n)$ of the numbers $1,2,\ldots,n$ such that $a_i>a_{i+1}$ holds for exactly one $i\in\{1,2,\ldots,n-1\}$.

There are 5 points on plane. Prove that you can chose some of them and shift them such that distances between shifted points won't change and as a result there will be symetric by some line set of 5 points.

Hossna is playing with a $m*n$ grid of points.In each turn she draws segments between points with the following conditions. **1.** No two segments intersect. **2.** Each segment is drawn between two consecutive rows. **3.** There is at most one segment between any two points. Find the maximum number of regions Hossna can create.

Let the vertex set \( V \) of a graph be partitioned into \( h \) parts \( (V = V_1 \cup V_2 \cup \cdots \cup V_h) \), with \(|V_1| = n_1, |V_2| = n_2, \ldots, |V_h| = n_h \). If there is an edge between any two vertices only when they belong to different parts, the graph is called a complete \( h \)-partite graph, denoted as \( k(n_1, n_2, \ldots, n_h) \). Let \( n \) and \( r \) be positive integers, \( n \geq 6 \), \( r \leq \frac{2}{3}n \). Consider the complete \( r + 1 \)-partite graph \( k\left(\underbrace{1, 1, \ldots, 1}_{r}, n - r\right) \). Answer the following questions: 1. Find the maximum number of disjoint circles (i.e., circles with no common vertices) in this complete \( r + 1 \)-partite graph. 2. Given \( n \), for all \( r \leq \frac{2}{3}n \), find the maximum number of edges in a complete \( r + 1 \)-partite graph \( k(1, 1, \ldots, 1, n - r) \) where no more than one circle is disjoint.

A sequence of positive integers with $n$ terms satisfies $\sum_{i=1}^{n} a_i=2007$. Find the least positive integer $n$ such that there exist some consecutive terms in the sequence with their sum equal to $30$.

Julian and Johan are playing a game with an even number of cards, say $2n$ cards, ($n \in Z_{>0}$). Every card is marked with a positive integer. The cards are shuffled and are arranged in a row, in such a way that the numbers are visible. The two players take turns picking cards. During a turn, a player can pick either the rightmost or the leftmost card. Johan is the first player to pick a card (meaning Julian will have to take the last card). Now, a player’s score is the sum of the numbers on the cards that player acquired during the game. Prove that Johan can always get a score that is at least as high as Julian’s.

Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is \[ \frac{x_{i-1}+x_{i+1}}{x_i} = k_i \] is an integer for each $i$, where $x_0 = x_n$, $x_{n+1} = x_1$. Prove that \[ 2n \le k_1 + k_2 + \dots + k_n < 3n. \]

Let $C_n$ denote the $n$-dimensional unit cube, consisting of the $2^n$ points $$\{(x_1, x_2, \ldots, x_n) \mid x_i \in \{0, 1\} \text{ for all } 1 \le i \le n\}.$$ A tetrahedron is [i]equilateral[/i] if all six side lengths are equal. Find the smallest positive integer $n$ for which there are four distinct points in $C_n$ that form a non-equilateral tetrahedron with integer side lengths.

For some positive integer $n$, consider the board $n\times n$. On this board you can put any rectangles with sides along the sides of the grid. What is the smallest number of such rectangles that must be placed so that all the cells of the board are covered by distinct numbers of rectangles (possibly $0$)? The rectangles are allowed to have the same sizes. [i]Proposed by Anton Trygub[/i]

A class consists of $7$ boys and $13$ girls. During the first three months of the school year, each boy has communicated with each girl at least once. Prove that there exist two boys and two girls such that both boys communicated with both girls first time in the same month.

$ 8$ students took part in exam that contains $ 8$ questions. If it is known that each question was solved by at least $ 5$ students, prove that we can always find $ 2$ students such that each of questions was solved by at least one of them.

A finite number of chords is drawn in a circle $1$ cm in diameter so that any diameter of the circle intersects at most $N$ of these chords. Prove that the sum of the lengths of all chords is less than $3.15 \cdot N$ cm.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] The number $123456789$ is written on the blackboard. At each step it is allowed to choose its digits $a$ and $b$ of the same parity and to replace each of them by $\frac{a+b}{2}.$ Is it possible to obtain a number larger then a)$800000000$; b)$880000000$ by such replacements?

Let $n\geqslant 2$ be an integer and $A{}$ a set of $n$ points in the plane. Find all integers $1\leqslant k\leqslant n-1$ with the following property: any two circles $C_1$ and $C_2$ in the plane such that $A\cap\text{Int}(C_1)\neq A\cap\text{Int}(C_2)$ and $|A\cap\text{Int}(C_1)|=|A\cap\text{Int}(C_2)|=k$ have at least one common point. [i]Cristi Săvescu[/i]

Consider a set of $2016$ distinct points in the plane, no four of which are collinear. Prove that there is a subset of $63$ points among them such that no three of these $63$ points are collinear.

There are 94 safes and 94 keys. Each key can open only one safe, and each safe can be opened by only one key. We place randomly one key into each safe. 92 safes are then randomly chosen, and then locked. What is the probability that we can open all the safes with the two keys in the two remaining safes? (Once a safe is opened, the key inside the safe can be used to open another safe.)

[b]p1.[/b] Show that $9x + 5y$ is a multiple of$ 17$ whenever $2x + 3y$ is a multiple of $17$. [b]p2.[/b] Express the five distinct fifth roots of $1$ in terms of radicals. [b]p3.[/b] Prove that the three perpendiculars dropped to the three sides of an equilateral triangle from any point inside the triangle have a constant sum. [b]p4.[/b] Find the volume of a sphere which circumscribes a regular tetrahedron of edge $a$. [b]p5.[/b] For any integer $n$ greater than $1$, show that $n^2-2n + 1$ is a factor at $n^{n-1}-1$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given positive integer $ m \geq 17$, $ 2m$ contestants participate in a circular competition. In each round, we devide the $ 2m$ contestants into $ m$ groups, and the two contestants in one group play against each other. The groups are re-divided in the next round. The contestants compete for $ 2m\minus{}1$ rounds so that each contestant has played a game with all the $ 2m\minus{}1$ players. Find the least possible positive integer $ n$, so that there exists a valid competition and after $ n$ rounds, for any $ 4$ contestants, non of them has played with the others or there have been at least $ 2$ games played within those $ 4$.

Carson the farmer has a plot of land full of crops in the shape of a $6 \times 6$ grid of squares. Each day, he uniformly at random chooses a row or a column of the plot that he hasn’t chosen before and harvests all of the remaining crops in the row or column. Compute the expected number of connected components that the remaining crops form after $6$ days. If all crops have been harvested, we say there are $0$ connected components.