For $100$ straight lines on a plane, let $T$ be the set of all right-angled triangles bounded by some $3$ lines. Determine, with proof, the maximum value of $|T|$.

Let $n$ be a positive integer. In a village, $n$ boys and $n$ girls are living. For the yearly ball, $n$ dancing couples need to be formed, each of which consists of one boy and one girl. Every girl submits a list, which consists of the name of the boy with whom she wants to dance the most, together with zero or more names of other boys with whom she wants to dance. It turns out that $n$ dancing couples can be formed in such a way that every girl is paired with a boy who is on her list. Show that it is possible to form $n$ dancing couples in such a way that every girl is paired with a boy who is on her list, and at least one girl is paired with the boy with whom she wants to dance the most.

We have $n$ keys, each of them belonging to exactly one of $n$ locked chests. Our goal is to decide which key opens which chest. In one try we may choose a key and a chest, and check whether the chest can be opened with the key. Find the minimal number $p(n)$ with the property that using $p(n)$ tries, we can surely discover which key belongs to which chest.

[hide=C stands for Cantor, G stands for Gauss]they had two problem sets under those two names[/hide] [u] Set 4[/u] [b]C.16 / G.6[/b] Let $a, b$, and $c$ be real numbers. If $a^3 + b^3 + c^3 = 64$ and $a + b = 0$, what is the value of $c$? [b]C.17 / G.8[/b] Bender always turns $60$ degrees clockwise. He walks $3$ meters, turns, walks $2$ meters, turns, walks $1$ meter, turns, walks $4$ meters, turns, walks $1$ meter, and turns. How many meters does Bender have to walk to get back to his original position? [b]C.18 / G.13[/b] Guang has $4$ identical packs of gummies, and each pack has a red, a blue, and a green gummy. He eats all the gummies so that he finishes one pack before going on to the next pack, but he never eats two gummies of the same color in a row. How many different ways can Guang eat the gummies? [b]C.19[/b] Find the sum of all digits $q$ such that there exists a perfect square that ends in $q$. [b]C.20 / G.14[/b] The numbers $5$ and $7$ are written on a whiteboard. Every minute Stev replaces the two numbers on the board with their sum and difference. After $2017$ minutes the product of the numbers on the board is $m$. Find the number of factors of $m$. [u]Set 5[/u] [b]C.21 / G.10[/b] On the planet Alletas, $\frac{32}{33}$ of the people with silver hair have purple eyes and $\frac{8}{11}$ of the people with purple eyes have silver hair. On Alletas, what is the ratio of the number of people with purple eyes to the number of people with silver hair? [b]C.22 / G.15[/b] Let $P$ be a point on $y = -1$. Let the clockwise rotation of $P$ by $60^o$ about $(0, 0)$ be $P'$. Find the minimum possible distance between $P'$ and $(0, -1)$. [b]C.23 / G.18[/b] How many triangles can be made from the vertices and center of a regular hexagon? Two congruent triangles with different orientations are considered distinct. [b]C.24[/b] Jeremy and Kevin are arguing about how cool a sweater is on a scale of $1-5$. Jeremy says “$3$”, and Kevin says “$4$”. Jeremy angrily responds “$3.5$”, to which Kevin replies “$3.75$”. The two keep going at it, responding with the average of the previous two ratings. What rating will they converge to (and settle on as the coolness of the sweater)? [b]C.25 / G.20[/b] An even positive integer $n$ has an [i]odd factorization[/i] if the largest odd divisor of $n$ is also the smallest odd divisor of $n$ greater than $1$. Compute the number of even integers $n$ less than $50$ with an odd factorization. [u]Set 6[/u] [b]C.26 / G.26[/b] When $2018! = 2018 \times 2017 \times ... \times 1$ is multiplied out and written as an integer, find the number of $4$’s. If the correct answer is $A$ and your answer is $E$, you will receive $12 \min\, \, (A/E, E/A)^3$points. [b]C.27 / G.27[/b] A circle of radius $10$ is cut into three pieces of equal area with two parallel cuts. Find the width of the center piece. [img]https://cdn.artofproblemsolving.com/attachments/e/2/e0ab4a2d51052ee364dd14336677b053a40352.png[/img] If the correct answer is $A$ and your answer is $E$, you will receive $\max \, \,(0, 12 - 6|A - E|)$points. [b]C.28 / G.28[/b] An equilateral triangle of side length $1$ is randomly thrown onto an infinite set of lines, spaced $1$ apart. On average, how many times will the boundary of the triangle intersect one of the lines? [img]https://cdn.artofproblemsolving.com/attachments/0/1/773c3d3e0dfc1df54945824e822feaa9c07eb7.png[/img] For example, in the above diagram, the boundary of the triangle intersects the lines in $2$ places. If the correct answer is $A$ and your answer is $E$, you will receive $\max\, \,(0, 12-120|A-E|/A)$ points. [b]C.29 / G.29[/b] Call an ordered triple of integers $(a, b, c)$ nice if there exists an integer $x$ such that $ax^2 + bx + c = 0$. How many nice triples are there such that $-100 \le a, b, c \le 100$? If the correct answer is $A$ and your answer is $E$, you will receive $12 \min\, \,(A/E, E/A)$ points. [b]C.30 / G.30[/b] Let $f(i)$ denote the number of MBMT volunteers to be born in the $i$th state to join the United States. Find the value of $1f(1) + 2f(2) + 3f(3) + ... + 50f(50)$. Note 1: Maryland was the $7$th state to join the US. Note 2: Last year’s MBMT competition had $42$ volunteers. If the correct answer is $A$ and your answer is $E$, you will receive $\max\, \,(0, 12 - 500(|A -E|/A)^2)$ points. PS. You should use hide for answers. C1-15/ G1-10 have been posted [url=https://artofproblemsolving.com/community/c3h2790674p24540132]here [/url] and G16-25 [url=https://artofproblemsolving.com/community/c3h2790679p24540159]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In Greifswald there are three schools called $A,B$ and $C$, each of which is attended by at least one student. Among any three students, one from $A$, one from $B$ and one from $C$, there are two knowing each other and two not knowing each other. Prove that at least one of the following holds: [list] [*]Some student from $A$ knows all students from $B$. [*]Some student from $B$ knows all students from $C$. [*] Some student from $C$ knows all students from $A$.[/list]

$9004$ lemmings, including an equal number of both sexes, cross in rank and file the new bridge from Eyjafjallajokull to Katla. The entire column therefore moves equidistantly and at a constant speed about the bridge, whereby this is able to hold exactly half of the lemmings for the present distance. The bridge should fulfill as many lemming dreams as possible and at the same time preserve the species be opened briefly at some point in order to halve the total population. However, the law to prevent gender discrimination requires that exact half is female. Show that these sufficient claims are also can be done. [hide=original wording]9004 Lemminge, davon gleich viele von beiden Geschlechtern, uberqueren in Reih und Glied die neue Brucke vom Eyjafjallajokull zum Katla. Die gesammte Kolonne bewegt sich also aquidistant und mit konstanter Geschwindigkeit uber die Brucke, wobei diese fur den vorliegenden Abstand genau die Halfte der Lemminge zu fassen vermag. Zur Erfullung moglichst vieler Lemmingtraume und gleichzeitiger Arterhaltung soll die Brucke irgendwann einmalig kurzzeitig aufgeklappt werden, um die Gesamtpopulation zu halbieren. Das Gesetz zur Verhinderung geschlechtsspezifischer Diskriminierung erfordert jedoch, dass davon ex akt die Halfte weiblich ist. Man zeige, dass diesen Anspruchen auch Genuge getan werden kann.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/f/8/3bf1ef0f90d3eb3761ca3db04ed48480c8aab5.png[/img]

Show that there exists a proper non-empty subset $S$ of the set of real numbers such that, for every real number $x$, the set $\{nx + S : n \in N\}$ is finite, where $nx + S =\{nx + s : s \in S\}$

With $76$ tiles, of which some are white, other blue and the remaining red, they form a rectangle of $4 \times 19$. Show that there is a rectangle, inside the largest, that has its vertices of the same color.

For an integer $n \geq 5,$ two players play the following game on a regular $n$-gon. Initially, three consecutive vertices are chosen, and one counter is placed on each. A move consists of one player sliding one counter along any number of edges to another vertex of the $n$-gon without jumping over another counter. A move is legal if the area of the triangle formed by the counters is strictly greater after the move than before. The players take turns to make legal moves, and if a player cannot make a legal move, that player loses. For which values of $n$ does the player making the first move have a winning strategy?

In a group of $20$ people, each person sends a letter to $10$ of the others. Prove that there are two persons who send a letter to each other.

In the sixth-year exams of a Center, they pass Physics at least$70\%$ of the students, Mathematics at least $75\%$; Philosophy at least, the $90\%$ and the Language at least, $85\%$. How many students, at least, pass these four subjects?

In a tennis tournament there are participants from $n$ different countries. Each team consists of a coach and a player whom should settle in a hotel. The rooms considered for the settlement of coaches are different from players' ones. Each player wants to be in a room whose roommates are [b][u]all[/u][/b] from countries which have a defense agreement with the player's country. Conversely, each coach wants to be in a room whose roommates are [b][u]all[/u][/b] from countries which don't have a defense agreement with the coach's country. Find the minimum number of the rooms such that we can [u][b]always[/b][/u] grant everyone's desire. [i]proposed by Seyed Reza Hosseini and Mohammad Amin Ghiasi[/i]

Let $P$ be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers $a_1,a_2,a_3$ represent the number of unit squares that have exactly $1,2\text{ or } 3$ edges on the boundary of $P$ respectively. Find the largest real number $k$ such that the inequality $a_1+a_2>ka_3$ holds for each polygon constructed with these conditions.

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

Let $n, b$ and $c$ be positive integers. A group of $n$ pirates wants to fairly split their treasure. The treasure consists of $c \cdot n$ identical coins distributed over $b \cdot n$ bags, of which at least $n-1$ bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most $n-1$ moves and then split the bags among the pirates such that each pirate gets $b$ bags and $c$ coins.

Consider $2018$ pairwise crossing circles no three of which are concurrent. These circles subdivide the plane into regions bounded by circular $edges$ that meet at $vertices$. Notice that there are an even number of vertices on each circle. Given the circle, alternately colour the vertices on that circle red and blue. In doing so for each circle, every vertex is coloured twice- once for each of the two circle that cross at that point. If the two colours agree at a vertex, then it is assigned that colour; otherwise, it becomes yellow. Show that, if some circle contains at least $2061$ yellow points, then the vertices of some region are all yellow. Proposed by [i]India[/i]

We have thrown $k$ white dice and $m$ black dice. Find the probability that the remainder modulo $7$ of the sum of the numbers on the white dice is equal to the remainder modulo $7$ of the sum of the numbers on the black dice.

A knight stands on an infinite chess board. Find all places it can reach in exactly $2n$ moves.

A $10$-digit number is called a $\textit{cute}$ number if its digits belong to the set $\{1,2,3\}$ and the difference of every pair of consecutive digits is $1$. a) Find the total number of cute numbers. b) Prove that the sum of all cute numbers is divisibel by $1408$.

Let $n \ge 3$ cells be arranged into a circle. Each cell can be occupied by $0$ or $1$. The following operation is admissible: Choose any cell $C$ occupied by a $1$, change it into a $0$ and simultaneously reverse the entries in the two cells adjacent to $C$ (so that $x,y$ become $1 - x$, $1 - y$). Initially, there is a $1$ in one cell and zeros elsewhere. For which values of $n$ is it possible to obtain zeros in all cells in a finite number of admissible steps?

A finite number of stones are [i]good[/i] when the weight of each of these stones is less than the total weight of the rest. It is known that arbitrary $n-1$ of the given $n$ stones is [i]good[/i]. Prove that it is possible to choose a [i]good[/i] triple from these stones.

A set $X$ consisting of $n$ positive integers is called $\textit{good}$ if the following condition holds: For any two different subsets of $X$, say $A$ and $B$, the number $s(A) - s(B)$ is not divisible by $2^n$. (Here, for a set $A$, $s(A)$ denotes the sum of the elements of $A$) Given $n$, find the number of good sets of size $n$, all of whose elements is strictly less than $2^n$.

Prove that each finite set of integers can be arranged without intersection.

[b]p1.[/b] Find all positive integers $n$ such that $n^3 - 14n^2 + 64n - 93$ is prime. [b]p2.[/b] Let $a, b, c$ be real numbers such that $0 \le a, b, c \le 1$. Find the maximum value of $$\frac{a}{1 + bc}+\frac{b}{1 + ac}+\frac{c}{1 + ab}$$ [b]p3.[/b] Find the maximum value of the function $f(x, y, z) = 4x + 3y + 2z$ on the ellipsoid $16x^2 + 9y^2 + 4z^2 = 1$ [b]p4.[/b] Let $x_1,..., x_n$ be numbers such that $x_1+...+x_n = 2009$. Find the minimum value of $x^2_1+...+x^2_n$ (in term of $n$). [b]p5.[/b] Find the number of odd integers between $1000$ and $9999$ that have at least 3 distinct digits. [b]p6.[/b] Let $A_1,A_2,...,A_{2^n-1}$ be all the possible nonempty subsets of $\{1, 2, 3,..., n\}$. Find the maximum value of $a_1 + a_2 + ... + a_{2^n-1}$ where $a_i \in A_i$ for each $i = 1, 2,..., 2^n - 1$. [b]p7.[/b] Find the rightmost digit when $41^{2009}$ is written in base $7$. [b]p8.[/b] How many integral ordered triples $(x, y, z)$ satisfy the equation $x+y+z = 2009$, where $10 \le x < 31$, $100 < z < 310$ and $y \ge 0$. [b]p9.[/b] Scooby has a fair six-sided die, labeled $1$ to $6$, and Shaggy has a fair twenty-sided die, labeled $1$ to $20$. During each turn, they both roll their own dice at the same time. They keep rolling the die until one of them rolls a 5. Find the probability that Scooby rolls a $5$ before Shaggy does. [b]p10.[/b] Let $N = 1A323492110877$ where $A$ is a digit in the decimal expansion of $N$. Suppose $N$ is divisible by $7$. Find $A$. [b]p11.[/b] Find all solutions $(x, y)$ of the equation $\tan^4(x+y)+\cot^4(x+y) = 1-2x-x^2$, where $-\frac{\pi}{2} \le x; y \le \frac{\pi}{2}$ [b]p12.[/b] Find the remainder when $\sum^{50}_{k=1}k!(k^2 + k - 1)$ is divided by $1008$. [b]p13.[/b] The devil set of a positive integer $n$, denoted $D(n)$, is defined as follows: (1) For every positive integer $n$, $n \in D(n)$. (2) If $n$ is divisible by $m$ and $m < n$, then for every element $a \in D(m)$, $a^3$ must be in $D(n)$. Furthermore, call a set $S$ scary if for any $a, b \in S$, $a < b$ implies that $b$ is divisible by $a$. What is the least positive integer $n$ such that $D(n)$ is scary and has at least $2009$ elements? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the numerical line, arrange a system of closed segments of length $1$ without common points (endpoints included) so that any infinite arithmetic progression with any non zero difference and any first term has a common point with a segment of the system.