There are $2024$ points on a circle. A purple elephant labels the points $P_1,P_2,\ldots,P_{2024}$ in some order, and walks along the points from $P_1$ to $P_{2024}$ in this order, while laying some eggs. To ensure the elephant does not step on the eggs it laid, the chords $P_1P_2, P_2P_3, \ldots, P_{2023}P_{2024}$ must not intersect each other except possibly at their endpoints. How many labellings are there? (Note: Two labellings are the same if one is a rotation of the other.) [i](Proposed by Ho Janson)[/i]

Let $n\ge 3$ be an integer and $a_1,a_2,\dots ,a_n$ be a finite sequence of positive integers, such that, for $k=2,3,\dots ,n$ $$n(a_k+1)-(n-1)a_{k-1}=1.$$ Prove that $a_n$ is not divisible by $(n-1)^2$.

Place numbers from $1$ to $9$ in the circles of the figure (see Fig. ) so that the sum of four numbers, finding located in the circles at the tops of all squares (there are six of them), was constant , [img]https://cdn.artofproblemsolving.com/attachments/8/8/5fe1e8c5949903dd9500b992c8139277cebe7f.png[/img]

[b]p1.[/b] How many real solutions does the following system of equations have? Justify your answer. $$x + y = 3$$ $$3xy -z^2 = 9$$ [b]p2.[/b] After the first year the bank account of Mr. Money decreased by $25\%$, during the second year it increased by $20\%$, during the third year it decreased by $10\%$, and during the fourth year it increased by $20\%$. Does the account of Mr. Money increase or decrease during these four years and how much? [b]p3.[/b] Two circles are internally tangent. A line passing through the center of the larger circle intersects it at the points $A$ and $D$. The same line intersects the smaller circle at the points $B$ and $C$. Given that $|AB| : |BC| : |CD| = 3 : 7 : 2$, find the ratio of the radiuses of the circles. [b]p4.[/b] Find all integer solutions of the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{19}$ [b]p5.[/b] Is it possible to arrange the numbers $1, 2, . . . , 12$ along the circle so that the absolute value of the difference between any two numbers standing next to each other would be either $3$, or $4$, or $5$? Prove your answer. [b]p6.[/b] Nine rectangles of the area $1$ sq. mile are located inside the large rectangle of the area $5$ sq. miles. Prove that at least two of the rectangles (internal rectangles of area $1$ sq. mile) overlap with an overlapping area greater than or equal to $\frac19$ sq. mile PS. You should use hide for answers.

Path graph $U$ is given, and a blindfolded player is standing on one of its vertices. The vertices of $U$ are labeled with positive integers between 1 and $n$, not necessarily in the natural order. In each step of the game, the game master tells the player whether he is in a vertex with degree 1 or with degree 2. If he is in a vertex with degree 1, he has to move to its only neighbour, if he is in a vertex with degree 2, he can decide whether he wants to move to the adjacent vertex with the lower or with the higher number. All the information the player has during the game after $k$ steps are the $k$ degrees of the vertices he visited and his choice for each step. Is there a strategy for the player with which he can decide in finitely many steps how many vertices the path has?

Determine all positive integers $n$ such that it is possible to fill the $n \times n$ table with numbers $1, 2$ and $-3$ so that the sum of the numbers in each row and each column is $0$.

In the plane $xOy$, a lot of points are considered $$X = \{P (a, b) | (a, b) \in \{1, 2,..., 10\} \times \{1, 2,..., 10 \}\}$$ Determine the number of different lines that can be obtained by joining two of them between the points of the set $X$; so that any two lines are not parallel.

Let $N$ be an integer greater than $1$. A deck has $N^3$ cards, each card has one of $N$ colors, has one of $N$ figures and has one of $N$ numbers (there are no two identical cards). A collection of cards of the deck is "complete" if it has cards of every color, or if it has cards of every figure or of all numbers. How many non-complete collections are there such that, if you add any other card from the deck, the collection becomes complete?

[b]p1.[/b] How many lines of symmetry does a square have? [b]p2.[/b] Compute$ 1/2 + 1/6 + 1/12 + 1/4$. [b]p3.[/b] What is the maximum possible area of a rectangle with integer side lengths and perimeter $8$? [b]p4.[/b] Given that $1$ printer weighs $400000$ pennies, and $80$ pennies weighs $2$ books, what is the weight of a printer expressed in books? [b]p5.[/b] Given that two sides of a triangle are $28$ and $3$ and all three sides are integers, what is the sum of the possible lengths of the remaining side? [b]p6.[/b] What is half the sum of all positive integers between $1$ and $15$, inclusive, that have an even number of positive divisors? [b]p7.[/b] Austin the Snowman has a very big brain. His head has radius $3$, and the volume of his torso is one third of his head, and the volume of his legs combined is one third of his torso. If Austin's total volume is $a\pi$ where $a$ is an integer, what is $a$? [b]p8.[/b] Neethine the Kiwi says that she is the eye of the tiger, a fighter, and that everyone is gonna hear her roar. She is standing at point $(3, 3)$. Neeton the Cat is standing at $(11,18)$, the farthest he can stand from Neethine such that he can still hear her roar. Let the total area of the region that Neeton can stand in where he can hear Neethine's roar be $a\pi$ where $a$ is an integer. What is $a$? [b]p9.[/b] Consider $2018$ identical kiwis. These are to be divided between $5$ people, such that the first person gets $a_1$ kiwis, the second gets $a_2$ kiwis, and so forth, with $a_1 \le a_2 \le a_3 \le a_4 \le a_5$. How many tuples $(a_1, a_2, a_3, a_4, a_5)$ can be chosen such that they form an arithmetic sequence? [b]p10.[/b] On the standard $12$ hour clock, each number from $1$ to $12$ is replaced by the sum of its divisors. On this new clock, what is the number of degrees in the measure of the non-reflex angle between the hands of the clock at the time when the hour hand is between $7$ and $6$ while the minute hand is pointing at $15$? [b]p11.[/b] In equiangular hexagon $ABCDEF$, $AB = 7$, $BC = 3$, $CD = 8$, and $DE = 5$. The area of the hexagon is in the form $\frac{a\sqrt{b}}{c}$ with $b$ square free and $a$ and $c$ relatively prime. Find $a+b+c$ where $a, b,$ and $c$ are integers. [b]p12.[/b] Let $\frac{p}{q} = \frac15 + \frac{2}{5^2} + \frac{3}{5^3} + ...$ . Find $p + q$, where $p$ and $q$ are relatively prime positive integers. [b]p13.[/b] Two circles $F$ and $G$ with radius $10$ and $4$ respectively are externally tangent. A square $ABMC$ is inscribed in circle $F$ and equilateral triangle $MOP$ is inscribed in circle $G$ (they share vertex $M$). If the area of pentagon $ABOPC$ is equal to $a + b\sqrt{c}$, where $a$, $b$, $c$ are integers $c$ is square free, then find $a + b + c$. [b]p14.[/b] Consider the polynomial $P(x) = x^3 + 3x^2 + ax + 8$. Find the sum of all integer $a$ such that the sum of the squares of the roots of $P(x)$ divides the sum of the coecients of $P(x)$. [b]p15.[/b] Nithin and Antonio play a number game. At the beginning of the game, Nithin picks a prime $p$ that is less than $100$. Antonio then tries to find an integer $n$ such that $n^6 + 2n^5 + 2n^4 + n^3 + (n^2 + n + 1)^2$ is a multiple of $p$. If Antonio can find such a number n, then he wins, otherwise, he loses. Nithin doesn't know what he is doing, and he always picks his prime randomly while Antonio always plays optimally. The probability of Antonio winning is $a/b$ where $a$ and $b$ are relatively prime positive integers. Find$a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Evaluate $1 \oplus 2 \oplus \dots \oplus 987654321$ where $\oplus$ is bitwise exclusive OR. ($A\oplus B$ in binary has an $n$-th digit equal to $1$ if the $n$-th binary digits of $A$ and $B$ differ and $0$ otherwise. For example, $5 \oplus 9 = 0101_{2} \oplus 1001_{2} = 1100_2= 12$ and $6 \oplus 7 = 110_2 + 111_2 = 001_2 = 1$.) [i]Proposed by Jacob Weiner[/i]

A robot initially at position $0$ along a number line has a [i]movement function[/i] $f(u, v)$. It rolls a fair $26$-sided die repeatedly, with the $k$-th roll having value $r_k$. For $k \ge 2$, if $r_k > r_{k-1}$, it moves $f(r_k, r_{k-1})$ units in the positive direction. If $r_k < r_{k-1}$, it moves $f(r_k, r_{k-1})$ units in the negative direction. If $r_k = r_{k-1}$, all movement and die-rolling stops and the robot remains at its final position $x$. If $f(u, v) = (u^2 - v^2)^2 + (u - 1)(v + 1)$, compute the expected value of $x$.

There are $5$ boxes arranged in a circle. At first, there is one a box containing one ball, while the other boxes are empty. At each step, we can do one of the following two operations: i. select one box that is not empty, remove one ball from the box and add one ball into both boxes next to the box, ii. select an empty box next to a non-empty box, from the box the non-empty one moves one ball to the empty box. Is it possible, that after a few steps, obtained conditions where each box contains exactly $17^{5^{2016}}$ balls?

Given $10$ points in the space such that each $4$ points are not lie on a plane. Connect some points with some segments such that there are no triangles or quadrangles. Find the maximum number of the segments.

[b]p1.[/b] An $18$oz glass of apple juice is $6\%$ sugar and a $6$oz glass of orange juice is $12\%$ sugar. The two glasses are poured together to create a cocktail. What percent of the cocktail is sugar? [b]p2.[/b] Find the number of positive numbers that can be expressed as the difference of two integers between $-2$ and $2012$ inclusive. [b]p3.[/b] An annulus is defined as the region between two concentric circles. Suppose that the inner circle of an annulus has radius $2$ and the outer circle has radius $5$. Find the probability that a randomly chosen point in the annulus is at most $3$ units from the center. [b]p4.[/b] Ben and Jerry are walking together inside a train tunnel when they hear a train approaching. They decide to run in opposite directions, with Ben heading towards the train and Jerry heading away from the train. As soon as Ben finishes his $1200$ meter dash to the outside, the front of the train enters the tunnel. Coincidentally, Jerry also barely survives, with the front of the train exiting the tunnel as soon as he does. Given that Ben and Jerry both run at $1/9$ of the train’s speed, how long is the tunnel in meters? [b]p5.[/b] Let $ABC$ be an isosceles triangle with $AB = AC = 9$ and $\angle B = \angle C = 75^o$. Let $DEF$ be another triangle congruent to $ABC$. The two triangles are placed together (without overlapping) to form a quadrilateral, which is cut along one of its diagonals into two triangles. Given that the two resulting triangles are incongruent, find the area of the larger one. [b]p6.[/b] There is an infinitely long row of boxes, with a Ditto in one of them. Every minute, each existing Ditto clones itself, and the clone moves to the box to the right of the original box, while the original Ditto does not move. Eventually, one of the boxes contains over $100$ Dittos. How many Dittos are in that box when this first happens? [b]p7.[/b] Evaluate $$26 + 36 + 998 + 26 \cdot 36 + 26 \cdot 998 + 36 \cdot 998 + 26 \cdot 36 \cdot 998.$$ [b]p8. [/b]There are $15$ students in a school. Every two students are either friends or not friends. Among every group of three students, either all three are friends with each other, or exactly one pair of them are friends. Determine the minimum possible number of friendships at the school. [b]p9.[/b] Let $f(x) = \sqrt{2x + 1 + 2\sqrt{x^2 + x}}$. Determine the value of $$\frac{1}{f(1)}+\frac{1}{f(1)}+\frac{1}{f(3)}+...+\frac{1}{f(24)}.$$ [b]p10.[/b] In square $ABCD$, points $E$ and $F$ lie on segments $AD$ and $CD$, respectively. Given that $\angle EBF = 45^o$, $DE = 12$, and $DF = 35$, compute $AB$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On a horizontal line, $2005$ points are marked, each of which is either white or black. For every point, one finds the sum of the number of white points on the right of it and the number of black points on the left of it. Among the $2005$ sums, exactly one number occurs an odd number of times. Find all possible values of this number.

Petya and Vasya are playing on an $100\times 100$ board. Initially, all the cells of the board are white. With each of his moves, Petya paints one or more white cells standing on the same diagonal in black. With each of his moves, Vasya paints one or more white cells standing on the same column in black. Petya makes the first move. The one who can't make a move loses. Who has a winning strategy? [i]Proposed by M. Didin[/i]

A square of side length $1$ centimeter is cut into three convex polygons. Is it possible that the diameter of each of them does not exceed [list][b]a)[/b] $1$ centimeter; [b]b)[/b] $1.01$ centimeters; [b]c)[/b] $1.001$ centimeters?[/list]

A segment $d$ is said to divide a segment $s$ if there is a natural number $n$ such that $s = nd = d+d+ ...+d$ ($n$ times). (a) Prove that if a segment $d$ divides segments $s$ and $s'$ with $s < s'$, then it also divides their difference $s'-s$. (b) Prove that no segment divides the side $s$ and the diagonal $s'$ of a regular pentagon (consider the pentagon formed by the diagonals of the given pentagon without explicitly computing the ratios).

Given a circle $S$ and its $121$ chords $P_i (i=1,2,\ldots,121)$, each with a point $A_i(i=1,2,\ldots,121)$ on it. Prove that there exists a point $X$ on the circumference of $S$ such that: there exist $29$ distinct indices $1\le k_1\le k_2\le\ldots\le k_{29}\le 121$, such that the angle formed by ${A_{k_j}}X$ and ${P_{k_j}}$ is smaller than $21$ degrees for every $j=1,2,\ldots,29$.

Let $n \ge 2$ be a positive integer. An $n\times n$ grid of squares has been colored as a chessboard. Let a [i]move[/i] consist of picking a square from the board and then changing the colors to the opposite for all squares that lie in the same row as the chosen square, as well as for all squares that lie in the same column (the chosen square itself is also changed to the opposite color). Find all values of $n$ for which it is possible to make all squares of the grid be the same color in a finite sequence of moves.

There is a piece on each square of the solitaire board shown except for the central square. A move can be made when there are three adjacent squares in a horizontal or vertical line with two adjacent squares occupied and the third square vacant. The move is to remove the two pieces from the occupied squares and to place a piece on the third square. (One can regard one of the pieces as hopping over the other and taking it.) Is it possible to end up with a single piece on the board, on the square marked $X$?

Let $n>1$ be a positive integer. Ana and Bob play a game with other $n$ people. The group of $n$ people form a circle, and Bob will put either a black hat or a white one on each person's head. Each person can see all the hats except for his own one. They will guess the color of his own hat individually. Before Bob distribute their hats, Ana gives $n$ people a strategy which is the same for everyone. For example, it could be "guessing the color just on your left" or "if you see an odd number of black hats, then guess black; otherwise, guess white". Ana wants to maximize the number of people who guesses the right color, and Bob is on the contrary. Now, suppose Ana and Bob are clever enough, and everyone forms a strategy strictly. How many right guesses can Ana guarantee? [i]Proposed by China.[/i]

On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on. If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?

Half the cells of a $2m \times n$ board are colored black and the other half are colored white. The cells at the opposite ends of the main diagonal are different colors. The center of each black cell is connected to the center of every other black cell by a straight line segment, and similarly for the white cells. Show that we can place an arrow on each segment so that it becomes a vector and the vectors sum to zero.

Six different small books and three different large books sit on a shelf. Three children may each take either two small books or one large book. Find the number of ways the three children can select their books.