There are three sticks, each of which has an integer length which is at least $n$; the sum of their lengths is $n(n + 1)/2$. Prove that it is possible to break the sticks (possibly several times) so that the resulting sticks have length $1, 2,\dots, n$. [i]Note: a stick of length $a + b$ can be broken into sticks of lengths $a$ and $b$.[/i]

[b]7.1[/b] Construct a trapezoid given four sides. [b]7.2[/b] Prove that $$(1 + x + x^2 + ...+ x^{100})(1 + x^{102}) - 102x^{101} \ge 0 .$$ [b]7.3 [/b] In a quadrilateral $ABCD$, $M$ is the midpoint of AB, $N$ is the midpoint of $CD$. Lines $AD$ and BC intersect $MN$ at points $P$ and $Q$, respectively. Prove that if $\angle BQM = \angle APM$ , then $BC=AD$. [img]https://cdn.artofproblemsolving.com/attachments/a/2/1c3cbc62ee570a823b5f3f8d046da9fbb4b0f2.png[/img] [b]7.4 / 6.4[/b] Each of the eight given different natural numbers less than $16$. Prove that among their pairwise differences there is at least at least three are the same. [b]7.5 / 8.4[/b] An entire arc of circle is drawn through the vertices $A$ and $C$ of the rectangle $ABCD$ lying inside the rectangle. Draw a line parallel to $AB$ intersecting $BC$ at point $P$, $AD$ at point $Q$, and the arc $AC$ at point $R$ so that the sum of the areas of the figures $AQR$ and $CPR$ is the smallest. [img]https://cdn.artofproblemsolving.com/attachments/1/4/9b5a594f82a96d7eff750e15ca6801a5fc0bf1.png[/img] [b]7.6 / 6.5 [/b]The distance AB is 100 km. From A and B , cyclists simultaneously ride towards each other at speeds of 20 km/h and 30 km/hour accordingly. Together with the first A, a fly flies out with speed 50 km/h, she flies until she meets the cyclist from B, after which she turns around and flies back until she meets the cyclist from A, after which turns around, etc. How many kilometers will the fly fly in the direction from A to B until the cyclists meet? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988083_1967_leningrad_math_olympiad]here[/url].

Each of the positive integers $1,2,\dots,n$ is colored in one of the colors red, blue or yellow regarding the following rules: (1) A Number $x$ and the smallest number larger than $x$ colored in the same color as $x$ always have different parities. (2) If all colors are used in a coloring, then there is exactly one color, such that the smallest number in that color is even. Find the number of possible colorings.

Zeus’s lightning is made of a copper rod of length $60$ by bending it $4$ times in alternating directions so that the angle between two adjacent parts is always $60^o$. What is the minimum value of the square of the distance between the two endpoints of the lightning? All five segments of the lightning lie in the same plane. [img]https://cdn.artofproblemsolving.com/attachments/5/1/a18206df4fde561421022c0f2b4332f5ac44a2.png[/img]

In a plane a finite number of equal circles are given. These circles are mutually nonintersecting (they may be externally tangent). Prove that one can use at most four colors for coloring these circles so that two circles tangent to each other are of different colors. What is the smallest number of circles that requires four colors?

Let $f(k)$ be the number of integers $n$ satisfying the following conditions: (i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed; (ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$. Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$. [i]Proposed by Dirk Laurie, South Africa[/i]

On a plane is given a finite set of points. Prove that the points can be covered by open squares $Q_1,Q_2,...,Q_n$ such that $1 \le\frac{N_j}{S_j} \le 4$ for $j = 1,...,n,$ where $N_j$ is the number of points from the set inside square $Q_j$ and $S_j$ is the area of $Q_j$.

Consider an $m\times n$ board where $m, n \ge 3$ are positive integers, divided into unit squares. Initially all the squares are white. What is the minimum number of squares that need to be painted red such that each $3\times 3$ square contains at least two red squares? Andrei Eckstein and Alexandru Mihalcu

The World Cup, featuring $17$ teams from Europe and South America, as well as $15$ other teams that honestly don’t have a chance, is a soccer tournament that is held once every four years. As we speak, Croatia andMorocco are locked in a battle that has no significance whatsoever on the winner, but if you would like live score updates nonetheless, feel free to ask your proctor, who has no obligation whatsoever to provide them. [b]p1.[/b] During the group stage of theWorld Cup, groups of $4$ teams are formed. Every pair of teams in a group play each other once. Each team earns $3$ points for each win and $1$ point for each tie. Find the greatest possible sum of the points of each team in a group. [b]p2.[/b] In the semi-finals of theWorld Cup, the ref is bad and lets $11^2 = 121$ players per team go on the field at once. For a given team, one player is a goalie, and every other player is either a defender, midfielder, or forward. There is at least one player in each position. The product of the number of defenders, midfielders, and forwards is a mulitple of $121$. Find the number of ordered triples (number of defenders, number of midfielders, number of forwards) that satisfy these conditions. [b]p3.[/b] Messi is playing in a game during the Round of $16$. On rectangular soccer field $ABCD$ with $AB = 11$, $BC = 8$, points $E$ and $F$ are on segment $BC$ such that $BE = 3$, $EF = 2$, and $FC = 3$. If the distance betweenMessi and segment $EF$ is less than $6$, he can score a goal. The area of the region on the field whereMessi can score a goal is $a\pi +\sqrt{b} +c$, where $a$, $b$, and $c$ are integers. Find $10000a +100b +c$. [b]p4.[/b] The workers are building theWorld Cup stadium for the $2022$ World Cup in Qatar. It would take 1 worker working alone $4212$ days to build the stadium. Before construction started, there were 256 workers. However, each day after construction, $7$ workers disappear. Find the number of days it will take to finish building the stadium. [b]p5.[/b] In the penalty kick shootout, $2$ teams each get $5$ attempts to score. The teams alternate shots and the team that scores a greater number of times wins. At any point, if it’s impossible for one team to win, even before both teams have taken all $5$ shots, the shootout ends and nomore shots are taken. If each team does take all $5$ shots and afterwards the score is tied, the shootout enters sudden death, where teams alternate taking shots until one team has a higher score while both teams have taken the same number of shots. If each shot has a $\frac12$ chance of scoring, the expected number of times that any team scores can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An ant crawls along the grid lines of an infinite quadrille notebook. One grid point is marked red, this is its starting point. Every time the ant reaches a grid point, it continues forward with probability $\frac13$ , left with probability $\frac13$ , and right with probability $\frac13$. What is the chance that it is after its third turn, but not after its fourth turn that it returns to the red point? If the answer is $\frac{p}{q}$ , where $p$ and $q$ are coprime positive integers, then your answer should be $p + q$. [i]The steps of the ant are independent.[/i]

Consider a convex polyhedron whose faces are triangles. Prove that it is possible to color its edges by either red or blue, in a way that the following property is satisfied: one can travel from any vertex to any other vertex while passing only along red edges, and can also do this while passing only along blue edges.

2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.

Call a positive integer [i]challenging[/i] if it can be expressed as $2^a(2^b+1)$, where $a,b$ are positive integers. Prove that if $X$ is a set of challenging numbers smaller than $2^n (n$ is a given positive integer) and $|X|\ge \frac{4}{3}(n-1)$, there exist two disjoint subsets $A,B\subset X$ such that $|A|=|B|$ and $\sum_{a\in A}a=\sum_{b \in B}b$.

In a village $X_0$ there are $80$ tourists who are about to visit $5$ nearby villages $X_1,X_2,X_3,X_4,X_5$.Each of them has chosen to visit only one of them.However,there are cases when the visit in a village forces the visitor to visit other villages among $X_1,X_2,X_3,X_4,X_5$.Each tourist visits only the village he has chosen and the villages he is forced to.If $X_1,X_2,X_3,X_4,X_5$ are totally visited by $40,60,65,70,75$ tourists respectively,then find how many tourists had chosen each one of them and determine all the ordered pairs $(X_i,X_j):i,j\in \{1,2,3,4,5\}$ which are such that,the visit in $X_i$ forces the visitor to visit $X_j$ as well.

Let $A = \{ 1,2, 3 , \ldots, 100 \}$ and $B$ be a subset of $A$ having $53$ elements. Show that $B$ has 2 distinct elements $x$ and $y$ whose sum is divisible by $11$.

Eight guests arrive to a hotel with four rooms. Each guest dislikes at most three other guests and doesn’t want to share a room with any of them (this feeling is mutual). Show that the guests can reside in the four rooms, with two persons in each room

Determine all pairs $\{a, b\}$ of positive integers with the property that, in whatever manner you color the positive integers with two colors $A$ and $B$, there always exist two positive integers of color $A$ having their difference equal to $a$ [b]or[/b] of color $B$ having their difference equal to $b$.

Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$

Let $ABC$ be a given triangle. Consider any painting of points of the plane in red and green. Show that there exist either two red points on the distance $1$, or three green points forming a triangle congruent to triangle $ABC$.

A knight is modified so that it moves $p$ fields horizontally or vertically and $q$ fields in the perpendicular direction. It is placed on an infinite chessboard. If the knight returns to the initial field after $n$ moves, show that $n$ must be even.

Is it possible to divide a $7\times7$ table into a few $\text{connected}$ parts of cells with the same perimeter? ( A group of cells is called $\text{connected}$ if any cell in the group, can reach other cells by passing through the sides of cells.)

Let \( n \geq 2 \) be an integer. Two players, Alice and Bob, play the following game on the complete graph \( K_n \): They take turns to perform operations, where each operation consists of coloring one or two edges that have not been colored yet. The game terminates if at any point there exists a triangle whose three edges are all colored. Prove that there exists a positive number \(\varepsilon\), Alice has a strategy such that, no matter how Bob colors the edges, the game terminates with the number of colored edges not exceeding \[ \left( \frac{1}{4} - \varepsilon \right) n^2 + n. \]

There are arbitrary 7 points in the plane. Circles are drawn through every 4 possible concyclic points. Find the maximum number of circles that can be drawn.

[b]p13.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $1$. Let the unit circles centered at $A$, $B$, and $C$ be $\Omega_A$, $\Omega_B$, and $\Omega_C$, respectively. Then, let $\Omega_A$ and $\Omega_C$ intersect again at point $D$, and $\Omega_B$ and $\Omega_C$ intersect again at point $E$. Line $BD$ intersects $\Omega_B$ at point $F$ where $F$ lies between $B$ and $D$, and line $AE$ intersects $\Omega_A$ at $G$ where $G$ lies between $A$ and $E$. $BD$ and $AE$ intersect at $H$. Finally, let $CH$ and $FG$ intersect at $I$. Compute $IH$. [b]p14.[/b] Suppose Bob randomly fills in a $45 \times 45$ grid with the numbers from $1$ to $2025$, using each number exactly once. For each of the $45$ rows, he writes down the largest number in the row. Of these $45$ numbers, he writes down the second largest number. The probability that this final number is equal to $2023$ can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Compute the value of $p$. [b]p15.[/b] $f$ is a bijective function from the set $\{0, 1, 2, ..., 11\}$ to $\{0, 1, 2, ... , 11\}$, with the property that whenever $a$ divides $b$, $f(a)$ divides $f(b)$. How many such $f$ are there? [i]A bijective function maps each element in its domain to a distinct element in its range. [/i] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$66$ points are given on a plane; collinearity is allowed. There are [b]exactly[/b] $2021$ lines passing by at least two of the given points. Determine the greatest number of points in a same line. Give an example.