Consider a table consisting of $2\times 7$ squares. Each little square is surrounded by walls (each internal wall belongs to two squares). We would like to remove some internal walls to make it possible to get from any square to any other one without crossing walls. How many ways can we do this while removing the minimal possible number of internal walls? [i]The figure shows a possible configuration, the remaining walls are marked in red, the removed ones are marked in light pink. Two configurations are considered the same if the same walls are removed.[/i] [img]https://cdn.artofproblemsolving.com/attachments/d/c/1a3d9ab0d0971929e6d484a970d4b1f36f0031.png[/img]

A [i]site[/i] is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones. [i]Proposed by Gurgen Asatryan, Armenia[/i]

Let $(X, <)$ be an arbitrary ordered set. Show that the elements of $X$ can be coloured by two colours in such a way that between any two points of the same colour there is a point of the opposite colour. (translated by L. Erdős)

$n$ diameters divide a disk into $2n$ equal sectors. $n$ of the sectors are coloured blue , and the other $n$ are coloured red (in arbitrary order) . Blue sectors are numbered from $1$ to $n$ in the anticlockwise direction, starting from an arbitrary blue sector, and red sectors are numbered from $1$ to $n$ in the clockwise direction, starting from an arbitrary red sector. Prove that there is a semi-disk containing sectors with all numbers from $1$ to $n$. (V Proizvolov)

Every point on a circle is coloured in blue or yellow such there is at least a point of each color. Prove that for every colouring of the circle there is always an isosceles triangle inscribed inside the circle 1) with all vertexes of the same colour. 2) with vertexes of both colours. For every colouring of the circle is there an equilateral triangle inscribed inside the circle 3) with all vertexes of the same colour? 4) with vertexes of both colours?

How many squares must be painted at least on a $5 \times 5$ board such that in each row, in each column and in each $2 \times 2$ square is there at least one square painted?

For each permutation $\sigma$ of the set $\{1, 2, \ldots , N\}$ we define its [i]correctness[/i] as the number of triples $1 \leqslant i < j < k \leqslant N$ such that the number $\sigma(j)$ lies between the numbers $\sigma(i)$ and $\sigma(k)$. Find the difference between the number of permutations with even correctness and the number of permutations with odd correctness if a) $N = 2018$ and b) $N = 2019$.

For a pair of integers $a$ and $b$, with $0<a<b<1000$, a set $S\subset \begin{Bmatrix}1,2,3,...,2024\end{Bmatrix}$ $escapes$ the pair $(a,b)$ if for any elements $s_1,s_2\in S$ we have $\left|s_1-s_2\right| \notin \begin{Bmatrix}a,b\end{Bmatrix}$. Let $f(a,b)$ be the greatest possible number of elements of a set that escapes the pair $(a,b)$. Find the maximum and minimum values of $f$.

A graph has $17$ points and each point has $4$ edges. Show that there are two points which are not joined and which are not both joined to the same point.

[b]p1A[/b] Positive reals $x$, $y$, and $z$ are such that $x/y +y/x = 7$ and $y/z +z/y = 7$. There are two possible values for $z/x + x/z;$ find the greater value. [b]p1B[/b] Real values $x$ and $y$ are such that $x+y = 2$ and $x^3+y^3 = 3$. Find $x^2+y^2$. [b]p2[/b] Set $A = \{5, 6, 8, 13, 20, 22, 33, 42\}$. Let $\sum S$ denote the sum of the members of $S$; then $\sum A = 149$. Find the number of (not necessarily proper) subsets $B$ of $A$ for which $\sum B \ge 75$. [b]p3[/b] $99$ dots are evenly spaced around a circle. Call two of these dots ”close” if they have $0$, $1$, or $2$ dots between them on the circle. We wish to color all $99$ dots so that any two dots which are close are colored differently. How many such colorings are possible using no more than $4$ different colors? [b]p4[/b] Given a $9 \times 9$ grid of points, count the number of nondegenerate squares that can be drawn whose vertices are in the grid and whose center is the middle point of the grid. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The Mictlán is an $n\times n$ board and each border of each $1\times 1$ cell is painted either purple or orange. Initially, a catrina lies inside a $1\times 1$ cell and may move in four directions (up, down, left, right) into another cell with the condition that she may move from one cell to another only if the border that joins them is painted orange. We know that no matter which cell the catrina chooses as a starting point, she may reach all other cells through a sequence of valid movements and she may not abandon the Mictlán (she may not leave the board). What is the maximum number of borders that could have been colored purple? [i]Proposed by CDMX[/i]

For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac1n$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most most $99+\frac12$, prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$.

Bertalan thought about a $4$-digit positive number. Then he draw a simple graph on $4$ vertices and wrote the digits of the number to the vertices of the graph in such a way that every vertex received exactly the degree of the vertex. In how many ways could he think about? In a simple graph every edge connects two different vertices, and between two vertices at most one edge can go.

Determine the number of sets $A = \{a_1,a_2,...,a_{1000}\}$ of positive integers satisfying $a_1 < a_2 <...< a_{1000} \le 2014$, for which we have that the set $S = \{a_i + a_j | 1 \le i, j \le 1000$ with $i + j \in A\}$ is a subset of $A$.

$N$ teams participated in a national basketball championship in which every two teams played exactly one game. Of the $N$ teams, $251$ are from California. It turned out that a Californian team Alcatraz is the unique Californian champion (Alcatraz has won more games against Californian teams than any other team from California). However, Alcatraz ended up being the unique loser of the tournament because it lost more games than any other team in the nation! What is the smallest possible value for $N$?

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [u]Set 3[/u] [b]P3.11[/b] Find all possible values of $c$ in the following system of equations: $$a^2 + ab + c^2 = 31$$ $$b^2 + ab - c^2 = 18$$ $$a^2 - b^2 = 7$$ [b]P3.12 / R5.25[/b] In square $ABCD$ with side length $13$, point $E$ lies on segment $CD$. Segment $AE$ divides $ABCD$ into triangle $ADE$ and quadrilateral $ABCE$. If the ratio of the area of $ADE$ to the area of $ABCE$ is $4 : 11$, what is the ratio of the perimeter of $ADE$ to the perimeter of$ ABCE$? [b]P3.13[/b] Thomas has two distinct chocolate bars. One of them is $1$ by $5$ and the other one is $1$ by $3$. If he can only eat a single $1$ by $1$ piece off of either the leftmost side or the rightmost side of either bar at a time, how many different ways can he eat the two bars? [b]P3.14[/b] In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. The entire triangle is revolved about side $BC$. What is the volume of the swept out region? [b]P3.15[/b] Find the number of ordered pairs of positive integers $(a, b)$ that satisfy the equation $a(a -1) + 2ab + b(b - 1) = 600$. [u]Set 4[/u] [b]P4.16[/b] Compute the sum of the digits of $(10^{2017} - 1)^2$ . [b]P4.17[/b] A right triangle with area $210$ is inscribed within a semicircle, with its hypotenuse coinciding with the diameter of the semicircle. $2$ semicircles are constructed (facing outwards) with the legs of the triangle as their diameters. What is the area inside the $2$ semicircles but outside the first semicircle? [b]P4.18[/b] Find the smallest positive integer $n$ such that exactly $\frac{1}{10}$ of its positive divisors are perfect squares. [b]P4.19[/b] One day, Sambuddha and Jamie decide to have a tower building competition using oranges of radius $1$ inch. Each player begins with $14$ oranges. Jamie builds his tower by making a $3$ by $3$ base, placing a $2$ by $2$ square on top, and placing the last orange at the very top. However, Sambuddha is very hungry and eats $4$ of his oranges. With his remaining $10$ oranges, he builds a similar tower, forming an equilateral triangle with $3$ oranges on each side, placing another equilateral triangle with $2$ oranges on each side on top, and placing the last orange at the very top. What is the positive difference between the heights of these two towers? [b]P4.20[/b] Let $r, s$, and $t$ be the roots of the polynomial $x^3 - 9x + 42$. Compute the value of $(rs)^3 + (st)^3 + (tr)^3$. [u]Set 5[/u] [b]P5.21[/b] For all integers $k > 1$, $\sum_{n=0}^{\infty}k^{-n} =\frac{k}{k -1}$. There exists a sequence of integers $j_0, j_1, ...$ such that $\sum_{n=0}^{\infty}j_n k^{-n} =\left(\frac{k}{k -1}\right)^3$ for all integers $k > 1$. Find $j_{10}$. [b]P5.22[/b] Nimi is a triangle with vertices located at $(-1, 6)$, $(6, 3)$, and $(7, 9)$. His center of mass is tied to his owner, who is asleep at $(0, 0)$, using a rod. Nimi is capable of spinning around his center of mass and revolving about his owner. What is the maximum area that Nimi can sweep through? [b]P5.23[/b] The polynomial $x^{19} - x - 2$ has $19$ distinct roots. Let these roots be $a_1, a_2, ..., a_{19}$. Find $a^{37}_1 + a^{37}_2+...+a^{37}_{19}$. [b]P5.24[/b] I start with a positive integer $n$. Every turn, if $n$ is even, I replace $n$ with $\frac{n}{2}$, otherwise I replace $n$ with $n-1$. Let $k$ be the most turns required for a number $n < 500$ to be reduced to $1$. How many values of $n < 500$ require k turns to be reduced to $1$? [b]P5.25[/b] In triangle $ABC$, $AB = 13$, $BC = 14$, and $AC = 15$. Let $I$ and $O$ be the incircle and circumcircle of $ABC$, respectively. The altitude from $A$ intersects $I$ at points $P$ and $Q$, and $O$ at point $R$, such that $Q$ lies between $P$ and $R$. Find $PR$. PS. You should use hide for answers. R1-15 /P1-5 have been posted [url=https://artofproblemsolving.com/community/c3h2786721p24495629]here[/url], and R16-30 /P6-10/ P26-30 [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Points $A$, $B$, $C$, $D$ and $E$ are placed on the circle. In how many ways can the resulting five arcs be designated by the letters $a$, $b$, $c$, $d$ and $e$, if it is forbidden to designate an arc with the same letter as one of its ends? (For example, an arc with ends $A$ and $B$ cannot be designated by the letter $a$ or $b$.)

Let $n$ be a positive integer, and let $A$ be a subset of $\{ 1,\cdots ,n\}$. An $A$-partition of $n$ into $k$ parts is a representation of n as a sum $n = a_1 + \cdots + a_k$, where the parts $a_1 , \cdots , a_k $ belong to $A$ and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set $\{ a_1 , a_2 , \cdots , a_k \} $. We say that an $A$-partition of $n$ into $k$ parts is optimal if there is no $A$-partition of $n$ into $r$ parts with $r<k$. Prove that any optimal $A$-partition of $n$ contains at most $\sqrt[3]{6n}$ different parts.

In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.

A board that has some of its squares painted black is called [i]acceptable [/i] if there are no four black squares that form a $2 \times 2$ subboard. Find the largest real number $\lambda$ such that for every positive integer $n$ the following proposition holds: mercy: if an $n \times n$ board is acceptable and has fewer than $\lambda n^2$ dark squares, then an additional square black can be painted so that the board is still acceptable.

Several zeros, ones and twos are written on the blackboard. An anonymous clean in turn pairs of different numbers, writing, instead of cleaned, the number not equal to each. ($0$ instead of pair $\{1,2\}, 1$ instead of $\{0,2\}, 2$ instead of $\{0,1\}$). Prove that if there remains one number only, it does not depend on the processing order.

A substitute teacher lead a groop of students to go for a trip. The teacher who in charge of the groop of the students told the substitude teacher that there are two students who always lie, and the others always tell the truth. But the substitude teacher don't know who are the two students always lie. They get lost in a forest. Finally the are in a crossroad which has four roads. The substitute teacher knows that their camp is on one road, and the distence is $20$ minutes' walk. The students have to go to the camp before it gets dark. $(1)$ If there are $8$ students, and $60$ minutes before it gets dark, give a plan that all students can get back to the camp. $(2)$ If there are $4$ students, and $100$ minutes before it gets dark, is there a plan that all students can get back to the camp?

There are $20$ people in a certain community. $10$ of them speak English, $10$ of them speak German, and $10$ of them speak French. We call a [i]committee[/i] to a $3$-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset. At most how many commitees are there in this community? $ \textbf{(A)}\ 120 \qquad\textbf{(B)}\ 380 \qquad\textbf{(C)}\ 570 \qquad\textbf{(D)}\ 1020 \qquad\textbf{(E)}\ 1140 $

The captain and his three sailors get $2009$ golden coins with the same value . The four people decided to divide these coins by the following rules : sailor $1$,sailor $2$,sailor $3$ everyone write down an integer $b_1,b_2,b_3$ , satisfies $b_1\ge b_2\ge b_3$ , and ${b_1+b_2+b_3=2009}$; the captain dosen't know what the numbers the sailors have written . He divides $2009$ coins into $3$ piles , with number of coins: $a_1,a_2,a_3$ , and $a_1\ge a_2\ge a_3$ . For sailor $k$ ($k=1,2,3$) , if $b_k<a_k$ , then he can take $b_k$ coins from the $k$th pile ; if $b_k\ge a_k$ , then he can't take any coins away . At last , the captain own the rest of the coins .If no matter what the numbers the sailors write , the captain can make sure that he always gets $n$ coins . Find the largest possible value of $n$ and prove your conclusion .

How many ways are there to fill in the $ 8$ spots in the picture with letters $A, B, C$ and $D$, using two copies of each letter, such that the spots with identical letters can be connected with a continuous line that stays within the box, without these four lines crossing each other or going through other spots? The lines do not have to be straight. [img]https://cdn.artofproblemsolving.com/attachments/f/f/66c30eaf6fa3b42c5197d0e3a3d59e9160bb8e.png[/img]