There are seven rods erected at the vertices of a regular heptagonal area. The top of each rod is connected to the top of its second neighbor by a straight piece of wire so that, looking from above, one sees each wire crossing exactly two others. Is it possible to set the respective heights of the rods in such a way that no four tops of the rods are coplanar and each wire passes one of the crossings from above and the other one from below?

A corner with arm $n$ is a figure made of $2n-1$ unit squares, such that 2 rectangles $1$ x $(n-1)$ are connected to two adjacent sides of a square $1$ x $1$, so that their unit sides coincide. The squares or a chessboard $100$ x $100$ are colored in 15 colors. We say that a corner with arm 8 is [i]“multicolored”[/i], if it contains each of the colors on the board. What’s the greatest number of corners with arm 8 which could be [i]“mutlticolored”[/i]?

The plane is partitioned into regions by a finite number of lines no three of which are concurrent. Two regions are called "neighbors" if the intersection of their boundaries is a segment, or half-line or a line (a point is not a segment). An integer is to be assigned to each region in such a way that: i) the product of the integers assigned to any two neighbors is less than their sum; ii) for each of the given lines, and each of the half-planes determined by it, the sum of the integers, assigned to all of the regions lying on this half-plane equal to zero. Prove that this is possible if and only if not all of the lines are parallel.

Find the number of even permutations of $ \{1,2,\ldots,n\}$ with no fixed points.

There are $6$ points on the plane such that no three of them are collinear. It's known that between every $4$ points of them, there exists a point that it's power with respect to the circle passing through the other three points is a constant value $k$.(Power of a point in the interior of a circle has a negative value.) Prove that $k=0$ and all $6$ points lie on a circle. [i]Proposed by Morteza Saghafian[/I]

Ten people apply for a job. The manager decides to interview the candidates one by one according to the following conditions: i) the first three candidates will not be employed; ii) from the fourth candidates onwards, if a candidate's comptence surpasses the competence of all those who preceded him, then that candidate is employed; iii) if the first nine candidates are not employed, then the tenth candidate will be employed. We assume that none of the $10$ applicants have the same competence, and these competences can be ranked from the first to tenth. Let $P_k$ represent the probability that the $k$th-ranked applicant in competence is employed. Prove that: i) $P_1>P_2>\ldots>P_8=P_9=P_{10}$; ii) $P_1+P_2+P_3>0.7$ iii) $P_8+P_9+P_{10}\le 0.1$. [i]Su Chun[/i]

Anne has thought of a finite set $A \subseteq \mathbb{R}^2 $ . Bob does not know how many elements $A$ has, but his goal is to completely determine $A$. To achieve this, Bob can chooseany point $b \in \mathbb{R}^2 $ and ask Anne how far it is from$ A$ . Anne replies with the distance, defined as $min \{d(a, b) | a \in A\}$. (Here $d(a, b)$ denotes the distance between points $a, b \in \mathbb{R}$ .) Bob can ask as many questions of this type as he wants, until he can determine A with certainty. a) Can Bob achieve his goal with finitely many questions? b) What if Anne tells Bob in advance that all points of A have both coordinates in the interval$\ [0, 1]\ $? Note: $\mathbb{R}^2$ is the set of points in the plane.

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Players $A$ and $B$ play the following game: $A$ tosses a coin $n$ times, and $B$ does $n+1$ times. The player who obtains more ”heads” wins; or in the case of equal balances, $A$ is assigned victory. Find the values of $n$ for which this game is fair (i.e. both players have equal chances for victory).

Prove that for all natural $n\ge 6 000$ any convex $1994$-gon can be cut into $n$ such quadrilaterals thata circle can be circumscribed around each of them

Let $f:\mathbb{Z}\rightarrow\{ 1,2,\ldots ,n\}$ be a function such that $f(x)\not= f(y)$, for all $x,y\in\mathbb{Z}$ such that $|x-y|\in\{2,3,5\}$. Prove that $n\ge 4$. [i]Ioan Tomescu[/i]

[b]p1.[/b] An airplane travels between two cities. The first half of the distance between the cities is traveled at a constant speed of $600$ mi/hour, and the second half of the distance is traveled at a a constant speed of $900$ mi/hour. Find the average speed of the plane. [b]p2.[/b] The figure below shows two egg cartons, $A$ and $B$. Carton $A$ has $6$ spaces (cell) and has $3$ eggs. Carton $B$ has $12$ cells and $3$ eggs. Tow cells from the total of $18$ cells are selected at random and the contents of the selected cells are interchanged. (Not that one or both of the selected cells may be empty.) [img]https://cdn.artofproblemsolving.com/attachments/6/7/2f7f9089aed4d636dab31a0885bfd7952f4a06.png[/img] (a) Find the number of selections/interchanges that produce a decrease in the number of eggs in cartoon $A$- leaving carton $A$ with $2$ eggs. (b) Assume that the total number of eggs in cartons $A$ and $B$ is $6$. How many eggs must initially be in carton $A$ and in carton $B$ so that the number of selections/interchanges that lead to an increase in the number of eggs in $A$ equals the number of selections/interchanges that lead to an increase in the number of eggs in $B$. $\bullet$ In other words, find the initial distribution of $6$ eggs between $A$ and $B$ so that the likelihood of an increase in A equals the likelihood of an increase in $B$ as the result of a selection/interchange. Prove your answer. [b]p3.[/b] Divide the following figure into four equal parts (parts should be of the same shape and of the same size, they may be rotated by different angles however they may not be disjoint and reconnected). [img]https://cdn.artofproblemsolving.com/attachments/f/b/faf0adbf6b09b5aaec04c4cfd7ab1d6397ad5d.png[/img] [b]p4.[/b] Find the exact numerical value of $\sqrt[3]{5\sqrt2 + 7}- \sqrt[3]{5\sqrt2 - 7}$ (do not use a calculator and do not use approximations). [b]p5.[/b] The medians of a triangle have length $9$, $12$ and $15$ cm respectively. Find the area of the triangle. [b]p6. [/b]The numbers $1, 2, 3, . . . , 82$ are written in an arbitrary order. Prove that it is possible to cross out $72$ numbers in such a sway the remaining number will be either in increasing order or in decreasing order. PS. You should use hide for answers.

Find the greatest integer $k\leq 2023$ for which the following holds: whenever Alice colours exactly $k$ numbers of the set $\{1,2,\dots, 2023\}$ in red, Bob can colour some of the remaining uncoloured numbers in blue, such that the sum of the red numbers is the same as the sum of the blue numbers. Romania

Let $m$ and $n$ be positive integers. Player $A$ has a field of $m \times n$, and player $B$ has a $1 \times n$ field (the first is the number of rows). On the first move, each player places on each square of his field white or black chip as he pleases. At each next on the move, each player can change the color of randomly chosen pieces on your field to the opposite, provided that in no row for this move will not change more than one chip (it is allowed not to change not a single chip). The moves are made in turn, player $A$ starts. Player $A$ wins if there is such a position that in the only row player $B$'s squares, from left to right, are the same as in some row of player's field $A$. Prove that player $A$ has the ability to win for any game of player $B$ if and only if $n <2m$.

Find the number of functions $f$ that map the set $\{1,2, 3,4\}$ into itself such that the range of the function $f(x)$ is the same as the range of the function $f(f(x))$.

Let $A$ and $B$ be nonempty subsets of $\mathbb{N}$. The sum of $2$ distinct elements in $A$ is always an element of $B$. Furthermore, the result of the division of $2$ distinct elements in $B$ (where the larger number is divided by the smaller number) is always a member of $A$. Determine the maximum number of elements in $A \cup B$.

Let $a_1,a_2,\ldots,a_7, b_1,b_2,\ldots,b_7\geq 0$ be real numbers satisfying $a_i+b_i\le 2$ for all $i=\overline{1,7}$. Prove that there exist $k\ne m$ such that $|a_k-a_m|+|b_k-b_m|\le 1$. Thanks for show me the mistake typing

A group of people is divided over two busses in such a way that there are as many seats in total as people. The chance that two friends are seated on the same bus is $\frac{1}{2}$. a) Show that the number of people in the group is a square. b) Show that the number of seats on each bus is a triangular number.

A school has two classes $A$ and $B$ which have $m$ and $n$ students each. The students of the two classes sit in a circle. Each student is then given a number of candies equal to the number of consecutive students sitting to the left of him that are from his same class. After distributing the candies, the teacher decides to group the students such that in each group, all the students receive the same amount of candies, and any two students from two different groups should receive a different amount of candies. a) What is the maximum number of students that a group can have? b) Excluding the group where every student receives no candies, what is the maximum number of students that a group can have?

Let $N$ be an odd number, $N\geq 3$. $N$ tennis players take part in a championship. Before starting the championship, a commission puts the players in a row depending on how good they think the players are. During the championship, every player plays with every other player exactly once, and each match has a winner. A match is called [i]suprising[/i] if the winner was rated lower by the commission. At the end of the tournament, players are arranged in a line based on the number of victories they have achieved. In the event of a tie, the commission's initial order is used to decide which player will be higher. It turns out that the final order is exactly the same as the commission's initial order. What is the maximal number of suprising matches that could have happened.

[b]p1.[/b] Each box represents $1$ square unit. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/0/0/f8f8ad6d771f3bbbc59b374a309017cecdce5a.png[/img] [b]p2.[/b] Evaluate $(3^3)\sqrt{5^2-2^4} -5 \cdot 9$. [b]p3.[/b] Find the last two digits of $21^3$. [b]p4.[/b] Let $L$, $M$, and $T$ be distinct prime numbers. Find the least possible odd value of$ L+M +T$ . [b]p5.[/b]Two circles have areas that sum to $20\pi$ and diameters that sum to $12$. Find the radius of the smaller circle. [b]p6.[/b] Zach and Evin each independently choose a date in the year $2022$, uniformly and randomly. The probability that at least one of the chosen dates is December $17$, $2022$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $A$. [b]p7.[/b] Let $L$ be a list of $2023$ real numbers with medianm. When any two numbers are removed from $L$, its median is still $m$. Find the greatest possible number of distinct values in $L$. [b]p8.[/b] Some children and adults are eating a delicious pile of sand. Children comprise $20\%$ of the group and combined, they consume $80\%$ of the sand. Given that on average, each child consumes $N$ pounds of sand and on average, each adult consumes $M$ pounds of sand, find $\frac{N}{M}$. [b]p9.[/b] An integer $N$ is chosen uniformly and randomly from the set of positive integers less than $100$. The expectedm number of digits in the base-$10$-representation of $N$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p10.[/b] Dunan is taking a calculus course in which the final exam counts for $15\%$ of the total grade. Dunan wishes to have an $A$ in the course, which is defined as a grade of $93\%$ or above. When counting everything but the final exam, he currently has a $92\%$ in the course. What is the minimum integer grade Dunan must get on the final exam in order to get an $A$ in the course? [b]p11.[/b] Norbert, Eorbert, Sorbert, andWorbert start at the origin of the Cartesian Plane and walk in the positive $y$, positive $x$, negative $y$, and negative $x$ directions respectively at speeds of $1$, $2$, $3$, and $4$ units per second respectively. After how many seconds will the quadrilateral with a vertex at each person’s location have area $300$? [b]p12.[/b] Find the sum of the unique prime factors of $1020201$. [b]p13.[/b] HacoobaMatata rewrites the base-$10$ integers from $0$ to $30$ inclusive in base $3$. How many times does he write the digit $1$? [b]p14.[/b] The fractional part of $x$ is $\frac17$. The greatest possible fractional part of $x^2$ can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p15.[/b] For howmany integers $x$ is $-2x^2 +8 \ge x^2 -3x +2$? [b]p16.[/b] In the figure below, circle $\omega$ is inscribed in square $EFGH$, which is inscribed in unit square $ABCD$ such that $\overline{EB} = 2\overline{AE}$. If the minimum distance from a point on $\omega$ to $ABCD$ can be written as $\frac{P-\sqrt{Q}}{R}$ with $Q$ square-free, find $10000P +100Q +R$. [img]https://cdn.artofproblemsolving.com/attachments/a/1/c6e5400bc508ab14f34987c9f5f4039daaa4d6.png[/img] [b]p17.[/b] There are two base number systems in use in the LHS Math Team. One member writes “$13$ people usemy base, while $23$ people use the other, base $12$.” Another member writes “out of the $34$ people in the club, $10$ use both bases while $9$ use neither.” Find the sum of all possible numbers ofMath Team members, as a regular decimal number. [b]p18.[/b] Sam is taking a test with $100$ problems. On this test the questions gradually get harder in such a way that for question $i$ , Sam has a $\frac{(101-i)^2}{ 100} \%$ chance to get the question correct. Suppose the expected number of questions Sam gets correct can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p19.[/b] In an ordered $25$-tuple, each component is an integer chosen uniformly and randomly from $\{1,2,3,4,5\}$. Ephram and Zach both copy this tuple into a $5\times 5$ grid, both starting from the top-left corner. Ephram writes five components from left to right to fill one row before continuing down to the next row. Zach writes five components from top to bottom to fill one column before continuing right to the next column. Find the expected number of spaces on their grids where Zach and Ephram have the same integer written. [b]p20.[/b] In $\vartriangle ABC$ with circumcenter $O$ and circumradius $8$, $BC = 10$. Let $r$ be the radius of the circle that passes through $O$ and is tangent to $BC$ at $C$. The value of $r^2$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $1000m+n$. [b]p21.[/b] Find the number of integer values of $n$ between $1$ and $100$ inclusive such that the sum of the positive divisors of $2n$ is at least $220\%$ of the sum of the divisors of $n$. [b]p22.[/b] Twenty urns containing one ball each are arranged in a circle. Ernie then moves each ball either $1$, $2$ or $3$ urns clockwise, chosen independently, uniformly, and randomly. The expected number of empty urns after this process is complete can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p23.[/b] Hannah the cat begins at $0$ on a number line. Every second, Hannah jumps $1$ unit in the positive or negative direction, chosen uniformly at random. After $7$ seconds,Hannah‘s expected distance from $0$, in units, can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p24.[/b] Find the product of all primes $p < 30$ for which there exists an integer $n$ such that $p$ divides $n +(n +1)^{-1}\,\, (mod \,\,p)$. [b]p25.[/b] In quadrilateral $ABCD$, $\angle ABD = \angle CBD = \angle C AD$, $AB = 9$, $BC = 6$, and $AC = 10$. The area of $ABCD$ can be expressed as $\frac{P\sqrt{Q}}{R}$ with $Q$ squarefree and $P$ and $R$ relatively prime. Find $10000P +100Q +R$. [img]https://cdn.artofproblemsolving.com/attachments/4/8/28569605b262c8f26e685e27f5f261c70a396c.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A special kind of parallelogram tile is made up by attaching the legs of two right isosceles triangles of side length $1$. We want to put a number of these tiles on the floor of an $n\times n$ room such that the distance from each vertex of each tile to the sides of the room is an integer and also no two tiles overlap. Prove that at least an area $n$ of the room will not be covered by the tiles. [i]Proposed by Ali Khezeli[/i]

A board of size $3 \times N$ initially has all of its cells painted white. Let $a(N)$ be the maximum number of cells that can be painted black in such a way that no three consecutive cells (either horizontally, vertically, or diagonally) are painted black. Prove that \[ \lim_{N \to \infty} \frac{a(N)}{N} \] exists and determine its value.

[b]p1[/b]. Your Halloween took a bad turn, and you are trapped on a small rock above a sea of lava. You are on rock $1$, and rocks $2$ through $12$ are arranged in a straight line in front of you. You want to get to rock $12$. You must jump from rock to rock, and you can either (1) jump from rock $n$ to $n + 1$ or (2) jump from rock $n$ to $n + 2$. Unfortunately, you are weak from eating too much candy, and you cannot do (2) twice in a row. How many different sequences of jumps will take you to your destination? [b]p2.[/b] Find the number of ordered triples $(p; q; r)$ such that $p, q, r$ are prime, $pq + pr$ is a perfect square and $p + q + r \le 100$. [b]p3.[/b] Let $x, y, z$ be nonzero complex numbers such that $\frac{1}{x}+\frac{1}{y} + \frac{1}{z} \ne 0$ and $$x^2(y + z) + y^2(z + x) + z^2(x + y) = 4(xy + yz + zx) = -3xyz.$$ Find $\frac{x^3 + y^3 + z^3}{x^2 + y^2 + z^2}$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a country with $ n $ ($ n \geq 3 $) airports, the government only licenses air travel to those airlines whose airline system meets the following conditions: a) Each airline must connect any two airports with one and only one one-way airline; b) For each airline there is an airport from which the passenger could fly off and fly back, using the services of only this airline. What is the maximum number of airlines with different airline systems?