Let $n> 2$ be an integer. A child has $n^2$ candies, which are distributed in $n$ boxes. An operation consists in choosing two boxes that together contain an even number of candies and redistribute the candy from those boxes so that both contain the same amount of candy. Determine all the values of $n$ for which the child, after some operations, can get each box containing $n$ candies, no matter which the initial distribution of candies is.

The positive integers from 1 to \(n\) are arranged in a sequence, initially in ascending order. In one move, we can swap the positions of two of the numbers, provided they share a common divisor greater than 1. Let \(s_n\) be the number of sequences that can be obtained with a finite number of moves. Prove that \(s_n = a_n!\), where the sequence of positive integers \((a_n)_{n\geq 1}\) is such that for any \(\delta > 0\), there exists an integer \(N\), for which for all \(n\geq N\), the following is true: \[ n - \left(\frac{1}{2}+\delta\right)\frac{n}{\log n} < a_n < n - \left(\frac{1}{2}-\delta\right)\frac{n}{\log n}. \]

Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can not use a needle to peer through the tiling? Determine if there is a way to tile a $5 \times 6$ unit square board by dominos such that one can use a needle to through the tiling? What if it is a $6 \times 6$ board?

A digital calendar displays the date: day, month, and year, with $2$ digits for the day, $2$ digits for the month, and $2$ digits for the year. For example, $01-01-01$ is January $1$, $2001$ and $05-25-23$ is May $25$, $2023$. In front of the calendar is a mirror. The digits of the calendar are as in the figure [img]https://cdn.artofproblemsolving.com/attachments/c/5/a08a4e34071fff4d33b95b23690254f55b33e1.gif[/img] If $0, 1, 2, 5$, and $8$ are reflected, respectively, in $0, 1, 5, 2$, and $8$, and the other digits lose meaning when reflected, determine how many days of the century, when reflected in the mirror, also correspond to a date.

Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\] Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.

Three tokens $A$, $B$, $C$ are, each one in a vertex of an equilateral triangle of side $n$. Its divided on equilateral triangles of side 1, such as it is shown in the figure for the case $n=3$ Initially, all the lines of the figure are painted blue. The tokens are moving along the lines painting them of red, following the next two rules: [b](1) [/b]First $A$ moves, after that $B$ moves, and then $C$, by turns. On each turn, the token moves over exactly one line of one of the little triangles, form one side to the other. [b](2)[/b] Non token moves over a line that is already painted red, but it can rest on one endpoint of a side that is already red, even if there is another token there waiting its turn. Show that for every positive integer $n$ it is possible to paint red all the sides of the little triangles.

The teacher wrote a non-zero natural number on the board. The teacher explained students that they can delete the number written on the board and can write a number instead naturally new, whenever they want, applying one of the following each time rules: 1) Instead of the current number $n$ write $3n + 13$ 2) Instead of the current number $n$ write the number $\sqrt{n}$, if $n$ is a perfect square . a) If the number $256$ was originally written on the board, is it possible that after a finite number of steps to get the number $55$ on the board? b) If the number $55$ was originally written on the board, is it possible that after a number finished the steps to get the number $256$ on the board?

[b]p1.[/b] Find the area of a right triangle with legs of lengths $20$ and $18$. [b]p2.[/b] How many $4$-digit numbers (without leading zeros) contain only $2,0,1,8$ as digits? Digits can be used more than once. [b]p3.[/b] A rectangle has perimeter $24$. Compute the largest possible area of the rectangle. [b]p4.[/b] Find the smallest positive integer with $12$ positive factors, including one and itself. [b]p5.[/b] Sammy can buy $3$ pencils and $6$ shoes for $9$ dollars, and Ben can buy $4$ pencils and $4$ shoes for $10$ dollars at the same store. How much more money does a pencil cost than a shoe? [b]p6.[/b] What is the radius of the circle inscribed in a right triangle with legs of length $3$ and $4$? [b]p7.[/b] Find the angle between the minute and hour hands of a clock at $12 : 30$. [b]p8.[/b] Three distinct numbers are selected at random fromthe set $\{1,2,3, ... ,101\}$. Find the probability that $20$ and $18$ are two of those numbers. [b]p9.[/b] If it takes $6$ builders $4$ days to build $6$ houses, find the number of houses $8$ builders can build in $9$ days. [b]p10.[/b] A six sided die is rolled three times. Find the probability that each consecutive roll is less than the roll before it. [b]p11.[/b] Find the positive integer $n$ so that $\frac{8-6\sqrt{n}}{n}$ is the reciprocal of $\frac{80+6\sqrt{n}}{n}$. [b]p12.[/b] Find the number of all positive integers less than $511$ whose binary representations differ from that of $511$ in exactly two places. [b]p13.[/b] Find the largest number of diagonals that can be drawn within a regular $2018$-gon so that no two intersect. [b]p14.[/b] Let $a$ and $b$ be positive real numbers with $a > b $ such that $ab = a +b = 2018$. Find $\lfloor 1000a \rfloor$. Here $\lfloor x \rfloor$ is equal to the greatest integer less than or equal to $x$. [b]p15.[/b] Let $r_1$ and $r_2$ be the roots of $x^2 +4x +5 = 0$. Find $r^2_1+r^2_2$ . [b]p16.[/b] Let $\vartriangle ABC$ with $AB = 5$, $BC = 4$, $C A = 3$ be inscribed in a circle $\Omega$. Let the tangent to $\Omega$ at $A$ intersect $BC$ at $D$ and let the tangent to $\Omega$ at $B$ intersect $AC$ at $E$. Let $AB$ intersect $DE$ at $F$. Find the length $BF$. [b]p17.[/b] A standard $6$-sided die and a $4$-sided die numbered $1, 2, 3$, and $4$ are rolled and summed. What is the probability that the sum is $5$? [b]p18.[/b] Let $A$ and $B$ be the points $(2,0)$ and $(4,1)$ respectively. The point $P$ is on the line $y = 2x +1$ such that $AP +BP$ is minimized. Find the coordinates of $P$. [b]p19.[/b] Rectangle $ABCD$ has points $E$ and $F$ on sides $AB$ and $BC$, respectively. Given that $\frac{AE}{BE}=\frac{BF}{FC}= \frac12$, $\angle ADE = 30^o$, and $[DEF] = 25$, find the area of rectangle $ABCD$. [b]p20.[/b] Find the sum of the coefficients in the expansion of $(x^2 -x +1)^{2018}$. [b]p21.[/b] If $p,q$ and $r$ are primes with $pqr = 19(p+q+r)$, find $p +q +r$ . [b]p22.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B$ is acute and $AB < AC$. Let $D$ be the foot of altitude from $A$ to $BC$ and $F$ be the foot of altitude from $E$, the midpoint of $BC$, to $AB$. If $AD = 16$, $BD = 12$, $AF = 5$, find the value of $AC^2$. [b]p23.[/b] Let $a,b,c$ be positive real numbers such that (i) $c > a$ (ii) $10c = 7a +4b +2024$ (iii) $2024 = \frac{(a+c)^2}{a}+ \frac{(c+a)^2}{b}$. Find $a +b +c$. [b]p24.[/b] Let $f^1(x) = x^2 -2x +2$, and for $n > 1$ define $f^n(x) = f ( f^{n-1}(x))$. Find the greatest prime factor of $f^{2018}(2019)-1$. [b]p25.[/b] Let $I$ be the incenter of $\vartriangle ABC$ and $D$ be the intersection of line that passes through $I$ that is perpendicular to $AI$ and $BC$. If $AB = 60$, $C A =120$, and $CD = 100$, find the length of $BC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An illusionist and his assistant are about to perform the following magic trick. Let $k$ be a positive integer. A spectator is given $n=k!+k-1$ balls numbered $1,2,…,n$. Unseen by the illusionist, the spectator arranges the balls into a sequence as he sees fit. The assistant studies the sequence, chooses some block of $k$ consecutive balls, and covers them under her scarf. Then the illusionist looks at the newly obscured sequence and guesses the precise order of the $k$ balls he does not see. Devise a strategy for the illusionist and the assistant to follow so that the trick always works. (The strategy needs to be constructed explicitly. For instance, it should be possible to implement the strategy, as described by the solver, in the form of a computer program that takes $k$ and the obscured sequence as input and then runs in time polynomial in $n$. A mere proof that an appropriate strategy exists does not qualify as a complete solution.)

An infinite triangular lattice is given, such that the distance between any two adjacent points is always equal to $1$. Points $A$, $B$, and $C$ are chosen on the lattice such that they are the vertices of an equilateral triangle of side length $L$, and the sides of $ABC$ contain no points from the lattice. Prove that, inside triangle $ABC$, there are exactly $\frac{L^2-1}{2}$ points from the lattice.

Given a simple graph with $ 4038 $ vertices. Assume we arbitrarily choose $ 2019 $ vertices as a group (the other $ 2019 $ is another group, of course), there are always $ k $ edges that connect two groups. Find all possible value of $ k $.

Different points $A_1, A_2,..., A_n$ in the plane ($n> 3$) are such that the triangle $A_iA_jA_k$ is obtuse for all the different $i,j,k \in\{1,2,...,n\}$. Prove that there is a point $A_{n + 1}$ in the plane, such that the triangle $A_iA_jA_{n + 1}$ is obtuse for all different $i,j \in\{1,2,...,n\}$

Given are $50$ distinct sets of positive integers, each of size $30$, such that every $30$ of them have a common element. Prove that all of them have a common element.

Let $n \geq 5$ be an integer. A shop sells balls in $n$ different colors. Each of $n + 1 $ children bought three balls with different colors, but no two children bought exactly the same color combination. Show that there are at least two children who bought exactly one ball of the same color.

A football tournament is played between 5 teams, each two of which playing exactly one match. 5 points are awarded for a victory and 0 – for a loss. In case of a draw 1 point is awarded to both teams, if no goals are scored, and 2 – if they have scored any. In the final ranking the five teams had points that were 5 consecutive numbers. Determine the least number of goals that could be scored in the tournament.

There are $n$ holes in a circle. The holes are numbered $1,2,3$ and so on to $n$. In the beginning, there is a peg in every hole except for hole $1$. A peg can jump in either direction over one adjacent peg to an empty hole immediately on the other side. After a peg moves, the peg it jumped over is removed. The puzzle will be solved if all pegs disappear except for one. For example, if $n=4$ the puzzle can be solved in two jumps: peg $3$ jumps peg $4$ to hole $1$, then peg $2$ jumps the peg in $1$ to hole $4$. (See illustration below, in which black circles indicate pegs and white circles are holes.) [center][img]http://i.imgur.com/4ggOa8m.png[/img][/center] [list=a] [*]Can the puzzle be solved for $n=5$? [*]Can the puzzle be solved for $n=2014$? [/list] In each part (a) and (b) either describe a sequence of moves to solve the puzzle or explain why it is impossible to solve the puzzle.

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

10.4, 11.3 Given $N\geq 3$ points enumerated with 1, 2, ..., $N$. Each two numbers are connected by mean of arrow from a lesser number to a greater one. A coloring of all arrows into red and blue is called [i]monochromatic[/i] iff for any numbers $A$ and $B$ there are [color=red]no[/color] two monochromatic paths from $A$ to $B$ of different colors. Find the number of monochromatic colorings. ([i]I. Bogdanov, G. Chelnokov[/i])

Let $n$ be an even positive integer. Show that there is a permutation $\left(x_{1},x_{2},\ldots,x_{n}\right)$ of $\left(1,\,2,\,\ldots,n\right)$ such that for every $i\in\left\{1,\ 2,\ ...,\ n\right\}$, the number $x_{i+1}$ is one of the numbers $2x_{i}$, $2x_{i}-1$, $2x_{i}-n$, $2x_{i}-n-1$. Hereby, we use the cyclic subscript convention, so that $x_{n+1}$ means $x_{1}$.

Each field of the $1987\times 1987$ board is filled with numbers, which absolute value is not greater than one. The sum of all the numbers in every $2\times 2$ square equals $0$. Prove that the sum of all the numbers is not greater than $1987$.

Albrecht fills in each cell of an $8 \times 8$ table with a $0$ or a $1$. Then at the end of each row and column he writes down the sum of the $8$ digits in that row or column, and then he erases the original digits in the table. Afterwards, he claims to Berthold that given only the sums, it is possible to restore the $64$ digits in the table uniquely. Show that the $8 \times 8$ table contained either a row full of $0$’s or a column full of $1$’s

[b]p1.[/b] What is $20\times 20 - 19\times 19$? [b]p2.[/b] Andover has a total of $1440$ students and teachers as well as a $1 : 5$ teacher-to-student ratio (for every teacher, there are exactly $5$ students). In addition, every student is either a boarding student or a day student, and $70\%$ of the students are boarding students. How many day students does Andover have? [b]p3.[/b] The time is $2:20$. If the acute angle between the hour hand and the minute hand of the clock measures $x$ degrees, find $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/a/a18b089ae016b15580ec464c3e813d5cb57569.png[/img] [b]p4.[/b] Point $P$ is located on segment $AC$ of square $ABCD$ with side length $10$ such that $AP >CP$. If the area of quadrilateral $ABPD$ is $70$, what is the area of $\vartriangle PBD$? [b]p5.[/b] Andrew always sweetens his tea with sugar, and he likes a $1 : 7$ sugar-to-unsweetened tea ratio. One day, he makes a $100$ ml cup of unsweetened tea but realizes that he has run out of sugar. Andrew decides to borrow his sister's jug of pre-made SUPERSWEET tea, which has a $1 : 2$ sugar-to-unsweetened tea ratio. How much SUPERSWEET tea, in ml,does Andrew need to add to his unsweetened tea so that the resulting tea is his desired sweetness? [b]p6.[/b] Jeremy the architect has built a railroad track across the equator of his spherical home planet which has a radius of exactly $2020$ meters. He wants to raise the entire track $6$ meters off the ground, everywhere around the planet. In order to do this, he must buymore track, which comes from his supplier in bundles of $2$ meters. What is the minimum number of bundles he must purchase? Assume the railroad track was originally built on the ground. [b]p7.[/b] Mr. DoBa writes the numbers $1, 2, 3,..., 20$ on the board. Will then walks up to the board, chooses two of the numbers, and erases them from the board. Mr. DoBa remarks that the average of the remaining $18$ numbers is exactly $11$. What is the maximum possible value of the larger of the two numbers that Will erased? [b]p8.[/b] Nathan is thinking of a number. His number happens to be the smallest positive integer such that if Nathan doubles his number, the result is a perfect square, and if Nathan triples his number, the result is a perfect cube. What is Nathan's number? [b]p9.[/b] Let $S$ be the set of positive integers whose digits are in strictly increasing order when read from left to right. For example, $1$, $24$, and $369$ are all elements of $S$, while $20$ and $667$ are not. If the elements of $S$ are written in increasing order, what is the $100$th number written? [b]p10.[/b] Find the largest prime factor of the expression $2^{20} + 2^{16} + 2^{12} + 2^{8} + 2^{4} + 1$. [b]p11.[/b] Christina writes down all the numbers from $1$ to $2020$, inclusive, on a whiteboard. What is the sum of all the digits that she wrote down? [b]p12.[/b] Triangle $ABC$ has side lengths $AB = AC = 10$ and $BC = 16$. Let $M$ and $N$ be the midpoints of segments $BC$ and $CA$, respectively. There exists a point $P \ne A$ on segment $AM$ such that $2PN = PC$. What is the area of $\vartriangle PBC$? [b]p13.[/b] Consider the polynomial $$P(x) = x^4 + 3x^3 + 5x^2 + 7x + 9.$$ Let its four roots be $a, b, c, d$. Evaluate the expression $$(a + b + c)(a + b + d)(a + c + d)(b + c + d).$$ [b]p14.[/b] Consider the system of equations $$|y - 1| = 4 -|x - 1|$$ $$|y| =\sqrt{|k - x|}.$$ Find the largest $k$ for which this system has a solution for real values $x$ and $y$. [b]p16.[/b] Let $T_n = 1 + 2 + ... + n$ denote the $n$th triangular number. Find the number of positive integers $n$ less than $100$ such that $n$ and $T_n$ have the same number of positive integer factors. [b]p17.[/b] Let $ABCD$ be a square, and let $P$ be a point inside it such that $PA = 4$, $PB = 2$, and $PC = 2\sqrt2$. What is the area of $ABCD$? [b]p18.[/b] The Fibonacci sequence $\{F_n\}$ is defined as $F_0 = 0$, $F_1 = 1$, and $F_{n+2}= F_{n+1} + F_n$ for all integers $n \ge 0$. Let $$ S =\dfrac{1}{F_6 + \frac{1}{F_6}}+\dfrac{1}{F_8 + \frac{1}{F_8}}+\dfrac{1}{F_{10} +\frac{1}{F_{10}}}+\dfrac{1}{F_{12} + \frac{1}{F_{12}}}+ ... $$ Compute $420S$. [b]p19.[/b] Let $ABCD$ be a square with side length $5$. Point $P$ is located inside the square such that the distances from $P$ to $AB$ and $AD$ are $1$ and $2$ respectively. A point $T$ is selected uniformly at random inside $ABCD$. Let $p$ be the probability that quadrilaterals $APCT$ and $BPDT$ are both not self-intersecting and have areas that add to no more than $10$. If $p$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. Note: A quadrilateral is self-intersecting if any two of its edges cross. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all finite sets $S$ of points in the plane with the following property: for any three distinct points $A,B,$ and $C$ in $S,$ there is a fourth point $D$ in $S$ such that $A,B,C,$ and $D$ are the vertices of a parallelogram (in some order).

Let $T$ be a tree. Prove that there is a constant $c>0$ (independent of $n$) such that every graph with $n$ vertices that does not contain a subgraph isomorphic to $T$ has at most $cn$ edges. [i]David Yang.[/i]

The following game is played on a rectangular chessboard $ 5 \times 9$ (with five rows and nine columns). Initially, a number of discs are randomly placed on some of the squares of the chessboard, with at most one disc on each square. A complete move consists of the moving every disc subject to the following rules: $ (1)$ Each disc may be moved one square up, down, left or right; $ (2)$ If a particular disc is moved up or down as part of a complete move, then it must be moved left or right in the next complete move; $ (3)$ If a particular disc is moved left or right as part of a complete move, then it must be moved up or down in the next complete move; $ (4)$ At the end of a complete move, no two discs can be on the same square. The game stops if it becomes impossible to perform a complete move. Prove that if initially $ 33$ discs are placed on the board then the game must eventually stop. Prove also that it is possible to place $ 32$ discs on the boards in such a way that the game could go on forever.