During a game of chess, at a certain point, in each row and column of the board there is an odd number of pieces. Prove that the number of pieces that are on black squares is even. (Note: a chessboard has $8$ rows and $8$ columns)

A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? $ \textbf{(A)} \ 8! \qquad \textbf{(B)} \ 2^8 \cdot 8! \qquad \textbf{(C)} \ (8!)^2 \qquad \textbf{(D)} \ \frac {16!}{2^8} \qquad \textbf{(E)} \ 16!$

Let $f(n)$ be defined on the positive integers and satisfy: $f(prime) = 1$, $f(ab) = a f(b) + f(a) b$. Show that $f$ is unique and find all $n$ such that $n = f(n)$.

A convex $n$-gon $A_0A_1\dots A_{n-1}$ has been partitioned into $n-2$ triangles by certain diagonals not intersecting inside the $n$-gon. Prove that these triangles can be labeled $\triangle_1,\triangle_2,\dots,\triangle_{n-2}$ in such a way that $A_i$ is a vertex of $\triangle_i$, for $i=1,2,\dots,n-2$. Find the number of all such labellings.

Consider trapezoid $[ABCD]$ which has $AB\parallel CD$ with $AB = 5$ and $CD = 9$. Moreover, $\angle C = 15^\circ$ and $\angle D = 75^\circ$. Let $M_1$ be the midpoint of $AB$ and $M_2$ be the midpoint of $CD$. What is the distance $M_1M_2$? [i]Proposed by Daniel Li[/i]

A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$? $ \textbf{(A) }5 \qquad \textbf{(B) }6 \qquad \textbf{(C) }7 \qquad \textbf{(D) }8 \qquad \textbf{(E) }9 \qquad $

Let $n$ be a positive integer. A square $ABCD$ is partitioned into $n^2$ unit squares. Each of them is divided into two triangles by the diagonal parallel to $BD$. Some of the vertices of the unit squares are colored red in such a way that each of these $2n^2$ triangles contains at least one red vertex. Find the least number of red vertices. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 4)[/i]

In a square field of side length $12$ there is a source that contains a system of straight irrigation ditches. This is laid out in such a way that for every point of the field the distance to the next ditch is at most $1$. Here, the source is as a point and are the ditches to be regarded as stretches. It must be verified that the total length of the irrigation ditches is greater than $70$ m. The sketch shows an example of a trench system of the type indicated. [img]https://cdn.artofproblemsolving.com/attachments/6/5/5b51511da468cf14b5823c6acda3c4d2fe8280.png[/img]

In an isosceles triangle $ABC$ ($AB = BC$), the midline parallel to side $BC$ intersects the incircle at a point $F$ that does not lie on the base $AC$. Prove that the tangent to the circle at point $F$ intersects the bisector of angle $C$ on side $AB$.

An [i]$n$-folding process[/i] on a rectangular piece of paper with sides aligned vertically and horizontally consists of repeating the following process $n$ times: [list] [*]Take the piece of paper and fold it in half vertically (choosing to either fold the right side over the left, or the left side over the right). [*]Rotate the paper $90^\circ$ clockwise. [/list] A $10$-folding process is performed on a piece of paper, resulting in a $1$-by-$1$ square base consisting of many stacked layers of paper. Let $d(i,j)$ be the Euclidean distance between the center of the $i$th square from the top and the center of the $j$th square from the top when the paper is unfolded. Determine the maximum possible value of $\sum_{i=1}^{1023} d(i, i+1).$

In each of the squares of an $n \times n$ board, with $n \ge 3$, a positive integer is written in such a way that the absolute value of the difference of the numbers written in any two neighboring cells is less than or equal to $2$ (two neighboring cells are those that have a common side). a) Show a $5 \times 5$ board on which $15$ integers have been written different following the indicated rule. b) Find, as a function of $n$, the maximum number of different numbers that can have the board of $n \times n$ squares.

For any non-empty subset $X$ of $M=\{1,2,3,...,2021\}$, let $a_X$ be the sum of the greatest and smallest elements of $X$. Determine the arithmetic mean of all the values of $a_X$, as $X$ covers all the non-empty subsets of $M$.

Given a finite sequence of integers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ for $n\geq 2.$ Show that there exists a subsequence $a_{k_{1}},$ $a_{k_{2}},$ $...,$ $a_{k_{m}},$ where $1\leq k_{1}\leq k_{2}\leq...\leq k_{m}\leq n,$ such that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}$ is divisible by $n.$ [b]Note by Darij:[/b] Of course, the $1\leq k_{1}\leq k_{2}\leq ...\leq k_{m}\leq n$ should be understood as $1\leq k_{1}<k_{2}<...<k_{m}\leq n;$ else, we could take $m=n$ and $k_{1}=k_{2}=...=k_{m},$ so that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}=n^{2}a_{k_{1}}^{2}$ will surely be divisible by $n.$

Using a compass and a ruler, construct a point $K$ inside an acute-angled triangle $ABC$ so that $\angle KBA = 2\angle KAB$ and $ \angle KBC = 2\angle KCB$.

Let \(S\) be a set with 10 distinct elements. A set \(T\) of subsets of \(S\) (possibly containing the empty set) is called [i]union-closed[/i] if, for all \(A, B \in T\), it is true that \(A \cup B \in T\). Show that the number of union-closed sets \(T\) is less than \(2^{1023}\). [i]Proposed by Tony Wang[/i]

Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received 36 votes, then how many votes were cast all together? [asy] draw((-1,0)--(0,0)--(0,1)); draw((0,0)--(0.309, -0.951)); filldraw(arc((0,0), (0,1), (-1,0))--(0,0)--cycle, lightgray); filldraw(arc((0,0), (0.309, -0.951), (0,1))--(0,0)--cycle, gray); draw(arc((0,0), (-1,0), (0.309, -0.951))); label("Colby", (-0.5, 0.5)); label("25\%", (-0.5, 0.3)); label("Alicia", (0.7, 0.2)); label("45\%", (0.7, 0)); label("Brenda", (-0.5, -0.4)); label("30\%", (-0.5, -0.6));[/asy] $\textbf{(A) }70\qquad\textbf{(B) }84\qquad\textbf{(C) }100\qquad\textbf{(D) }106\qquad\textbf{(E) }120$

Given six real numbers $a$, $b$, $c$, $x$, $y$, $z$ such that $0 < b-c < a < b+c$ and $ax + by + cz = 0$. What is the sign of the sum $ayz + bzx + cxy$ ?

Ket $f(x) = x^{2} +ax + b$. If for all nonzero real $x$ $$f\left(x + \dfrac{1}{x}\right) = f\left(x\right) + f\left(\dfrac{1}{x}\right)$$ and the roots of $f(x) = 0$ are integers, what is the value of $a^{2}+b^{2}$?

Let $ S$ be the set of points in the Cartesian plane that satisfy \[ \Big|\big|{|x| \minus{} 2}\big| \minus{} 1\Big| \plus{} \Big|\big|{|y| \minus{} 2}\big| \minus{} 1\Big| \equal{} 1. \] If a model of $ S$ were built from wire of negligible thickness, then the total length of wire required would be $ a\sqrt {b},$ where $ a$ and $ b$ are positive integers and $ b$ is not divisible by the square of any prime number. Find $ a \plus{} b.$

Find all pairs of real numbers $(x,y)$ satisfying $$(2x+1)^2+y^2+(y-2x)^2=\frac13.$$

Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.

Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = \{ -1, -2, -1/2 \}$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$, and for all the other permutations of $(1, 2, 3)$, the quadratic equations formed don't have any rational roots. Determine the maximum number of elements in $S(a, b, c)$.

Does there exist any integer $a,b,c$ such that $a^2bc+2,ab^2c+2,abc^2+2$ are perfect squares?

Five airlines operate in a country consisting of $36$ cities. Between any pair of cities exactly one airline operates two way flights. If some airlines operates between cities $A,B$ and $B,C$ we say that the ordered triple $A,B,C$ is properly-connected. Determine the largest possible value of $k$ such that no matter how these flights are arranged there are at least $k$ properly-connected triples.

[asy] draw(ellipse((0,-5),10,3)); fill((-10,-5)--(10,-5)--(10,5)--(-10,5)--cycle,white); draw(ellipse((0,0),10,3)); draw((10,0)--(10,-5)); draw((-10,0)--(-10,-5)); draw((0,0)--(7,-3*sqrt(51)/10)); label("10",(7/2,-3*sqrt(51)/20),NE); label("5",(-10,-3),E); [/asy] Which cylinder has twice the volume of the cylinder shown above? [asy] unitsize(4); draw(ellipse((0,-5),20,6)); fill((-20,-5)--(20,-5)--(20,5)--(-20,5)--cycle,white); draw(ellipse((0,0),20,6)); draw((20,0)--(20,-5)); draw((-20,0)--(-20,-5)); draw((0,0)--(14,-3*sqrt(51)/5)); label("20",(7,-3*sqrt(51)/10),NE); label("5",(-20,-4),E); label("(A)",(0,6),N); draw(ellipse((31,-7),10,3)); fill((21,-7)--(41,-7)--(41,7)--(21,7)--cycle,white); draw(ellipse((31,3),10,3)); draw((41,3)--(41,-7)); draw((21,3)--(21,-7)); draw((31,3)--(38,3-3*sqrt(51)/10)); label("10",(34.5,3-3*sqrt(51)/20),NE); label("10",(21,-4),E); label("(B)",(31,6),N); draw(ellipse((47,-15.5),5,3/2)); fill((42,-15.5)--(42,-15.5)--(42,15.5)--(42,15.5)--cycle,white); draw(ellipse((47,4.5),5,3/2)); draw((42,4.5)--(42,-15.5)); draw((52,4.5)--(52,-15.5)); draw((47,4.5)--(50.5,4.5-3*sqrt(51)/20)); label("5",(48.75,4.5-3*sqrt(51)/40),NE); label("10",(42,-6),E); label("(C)",(47,6),N); draw(ellipse((73,-10),20,6)); fill((53,-10)--(93,-10)--(93,5)--(53,5)--cycle,white); draw(ellipse((73,0),20,6)); draw((53,0)--(53,-10)); draw((93,0)--(93,-10)); draw((73,0)--(87,-3*sqrt(51)/5)); label("20",(80,-3*sqrt(51)/10),NE); label("10",(53,-6),E); label("(D)",(73,6),N); [/asy] $\text{(E)}\ \text{None of the above}$