Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

In the figure we see the paths connecting the square of a city (point $P$) with the school (point $S$). In the square there are $k$ pupils starting to go to the school. They have the ability to move only to the right and up. If the pupils are free to choose any allowed path (in order to get to school), determine the minimum value of $k$ so that in any case at least two pupils follow the same path. [img]https://cdn.artofproblemsolving.com/attachments/e/2/b5d6c6db5942cb706428cb63af3ca15590727f.png[/img]

A number from $1$ to $100$ is intended. In what is the smallest number of questions one can surely guess the intended number, if one is allowed to lie once? (Questions are asked like: “Does the intended number belong to such and such a numerical set?” The only possible answers are “Yes” and “No.”)

Mathematicians $M$ and $N$ each have their own favorite collection of manuals on the book, which he often uses in his work. Once they decided to make a statement in which each mathematician proves at each turn any theorem from his handbook which neither has yet been proven. Everything is done in turn, the mathematician starts $M$. The theorems of the handbook can win first all proven; if the theorems of both manuals can proved at once, wins the last theorem proved by a mathematician. Let $m$ be a theorem in the mathematician's handbook $M$. Find all values of $m$ for which the mathematician $M$ has a winning strategy if is It is known that there are $222$ theorems in the mathematician's handbook $N$ and $101$ of them also appears in the mathematician's $M$ handbook.

[b]p1.[/b] The following number is the product of the divisors of $n$. $$2^63^3$$ What is $n$? [b]p2.[/b] Let a right triangle have the sides $AB =\sqrt3$, $BC =\sqrt2$, and $CA = 1$. Let $D$ be a point such that $AD = BD = 1$. Let $E$ be the point on line $BD$ that is equidistant from $D$ and $A$. Find the angle $\angle AEB$. [b]p3.[/b] There are twelve indistinguishable blackboards that are distributed to eight different schools. There must be at least one board for each school. How many ways are there of distributing the boards? [b]p4.[/b] A Nishop is a chess piece that moves like a knight on its first turn, like a bishop on its second turn, and in general like a knight on odd-numbered turns and like a bishop on even-numbered turns. A Nishop starts in the bottom-left square of a $3\times 3$-chessboard. How many ways can it travel to touch each square of the chessboard exactly once? [b]p5.[/b] Let a Fibonacci Spiral be a spiral constructed by the addition of quarter-circles of radius $n$, where each $n$ is a term of the Fibonacci series: $$1, 1, 2, 3, 5, 8,...$$ (Each term in this series is the sum of the two terms that precede it.) What is the arclength of the maximum Fibonacci spiral that can be enclosed in a rectangle of area $714$, whose side lengths are terms in the Fibonacci series? [b]p6.[/b] Suppose that $a_1 = 1$ and $$a_{n+1} = a_n -\frac{2}{n + 2}+\frac{4}{n + 1}-\frac{2}{n}$$ What is $a_{15}$? [b]p7.[/b] Consider $5$ points in the plane, no three of which are collinear. Let $n$ be the number of circles that can be drawn through at least three of the points. What are the possible values of $n$? [b]p8.[/b] Find the number of positive integers $n$ satisfying $\lfloor n /2014 \rfloor =\lfloor n/2016 \rfloor$. [b]p9.[/b] Let $f$ be a function taking real numbers to real numbers such that for all reals $x \ne 0, 1$, we have $$f(x) + f \left( \frac{1}{1 - x}\right)= (2x - 1)^2 + f\left( 1 -\frac{1}{ x}\right)$$ Compute $f(3)$. [b]p10.[/b] Alice and Bob split $5$ beans into piles. They take turns removing a positive number of beans from a pile of their choice. The player to take the last bean loses. Alice plays first. How many ways are there to split the piles such that Alice has a winning strategy? [b]p11.[/b] Triangle $ABC$ is an equilateral triangle of side length $1$. Let point $M$ be the midpoint of side $AC$. Another equilateral triangle $DEF$, also of side length $1$, is drawn such that the circumcenter of $DEF$ is $M$, point $D$ rests on side $AB$. The length of $AD$ is of the form $\frac{a+\sqrt{b}}{c}$ , where $b$ is square free. What is $a + b + c$? [b]p12.[/b] Consider the function $f(x) = \max \{-11x- 37, x - 1, 9x + 3\}$ defined for all real $x$. Let $p(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with x values $t_1$, $t_2$ and $t_3$ Compute the maximum value of $t_1 + t_2 + t_3$ over all possible $p$. [b]p13.[/b] Circle $J_1$ of radius $77$ is centered at point $X$ and circle $J_2$ of radius $39$ is centered at point $Y$. Point $A$ lies on $J1$ and on line $XY$ , such that A and Y are on opposite sides of $X$. $\Omega$ is the unique circle simultaneously tangent to the tangent segments from point $A$ to $J_2$ and internally tangent to $J_1$. If $XY = 157$, what is the radius of $\Omega$ ? [b]p14.[/b] Find the smallest positive integer $n$ so that for any integers $a_1, a_2,..., a_{527}$,the number $$\left( \prod^{527}_{j=1} a_j\right) \cdot\left( \sum^{527}_{j=1} a^n_j\right)$$ is divisible by $527$. [b]p15.[/b] A circle $\Omega$ of unit radius is inscribed in the quadrilateral $ABCD$. Let circle $\omega_A$ be the unique circle of radius $r_A$ externally tangent to $\Omega$, and also tangent to segments $AB$ and $DA$. Similarly define circles $\omega_B$, $\omega_C$, and $\omega_D$ and radii $r_B$, $r_C$, and $r_D$. Compute the smallest positive real $\lambda$ so that $r_C < \lambda$ over all such configurations with $r_A > r_B > r_C > r_D$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The edges of a cube are numbered from $1$ to $12$, then is calculated for each vertex the sum of the numbers of the edges going out from it. a) Prove that these sums cannot all be the same. b) Can eight equal sums result after one of the edge numbers is replaced by the number $13$ ?

[b]p1.[/b] Consider the system of simultaneous equations $$(x+y)(x+z)=a^2$$ $$(x+y)(y+z)=b^2$$ $$(x+z)(y+z)=c^2$$ , where $abc \ne 0$. Find all solutions $(x,y,z)$ in terms of $a$,$b$, and $c$. [b]p2.[/b] Shown in the figure is a triangle $PQR$ upon whose sides squares of areas $13$, $25$, and $36$ sq. units have been constructed. Find the area of the hexagon $ABCDEF$ . [img]https://cdn.artofproblemsolving.com/attachments/b/6/ab80f528a2691b07430d407ff19b60082c51a1.png[/img] [b]p3.[/b] Suppose $p,q$, and $r$ are positive integers no two of which have a common factor larger than $1$. Suppose $P,Q$, and $R$ are positive integers such that $\frac{P}{p}+\frac{Q}{q}+\frac{R}{r}$ is an integer. Prove that each of $P/p$, $Q/q$, and $R/r$ is an integer. [b]p4.[/b] An isosceles tetrahedron is a tetrahedron in which opposite edges are congruent. Prove that all face angles of an isosceles tetrahedron are acute angles. [img]https://cdn.artofproblemsolving.com/attachments/7/7/62c6544b7c3651bba8a9d210cd0535e82a65bd.png[/img] [b]p5.[/b] Suppose that $p_1$, $p_2$, $p_3$ and $p_4$ are the centers of four non-overlapping circles of radius $1$ in a plane and that, $p$ is any point in that plane. Prove that $$\overline{p_1p}^2+\overline{p_2p}^2+\overline{p_3p}^2+\overline{p_4p}^2 \ge 6.$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $N$ cities in a country. Any two of them are connected either by a road or by an airway. A tourist wants to visit every city exactly once and return to the city at which he started the trip. Prove that he can choose a starting city and make a path, changing means of transportation at most once.

Every two of the $n$ cities of Ruritania are connected by a direct flight of one from two airlines. Promonopoly Committee wants at least $k$ flights performed by one company. To do this, he can at least every day to choose any three cities and change the ownership of the three flights connecting these cities each other (that is, to take each of these flights from a company that performs it, and pass the other). What is the largest $k$ committee knowingly will be able to achieve its goal in no time, no matter how the flights are distributed hour?

[b]p1.[/b] If $x$ is $20\%$ of $23$ and $y$ is $23\%$ of $20$, compute $xy$ . [b]p2.[/b] Pablo wants to eat a banana, a mango, and a tangerine, one at a time. How many ways can he choose the order to eat the three fruits? [b]p3.[/b] Let $a$, $b$, and $c$ be $3$ positive integers. If $a + \frac{b}{c} = \frac{11}{6}$ , what is the minimum value of $a + b + c$? [b]p4.[/b] A rectangle has an area of $12$. If all of its sidelengths are increased by $2$, its area becomes $32$. What is the perimeter of the original rectangle? [b]p5.[/b] Rohit is trying to build a $3$-dimensional model by using several cubes of the same size. The model’s front view and top view are shown below. Suppose that every cube on the upper layer is directly above a cube on the lower layer and the rotations are considered distinct. Compute the total number of different ways to form this model. [img]https://cdn.artofproblemsolving.com/attachments/b/b/40615b956f3d18313717259b12fcd6efb74cf8.png[/img] [b]p6.[/b] Priscilla has three octagonal prisms and two cubes, none of which are touching each other. If she chooses a face from these five objects in an independent and uniformly random manner, what is the probability the chosen face belongs to a cube? (One octagonal prism and cube are shown below.) [img]https://cdn.artofproblemsolving.com/attachments/0/0/b4f56a381c400cae715e70acde2cdb315ee0e0.png[/img] [b]p7.[/b] Let triangle $\vartriangle ABC$ and triangle $\vartriangle DEF$ be two congruent isosceles right triangles where line segments $\overline{AC}$ and $\overline{DF}$ are their respective hypotenuses. Connecting a line segment $\overline{CF}$ gives us a square $ACFD$ but with missing line segments $\overline{AC}$, $\overline{AD}$, and $\overline{DF}$. Instead, $A$ and $D$ are connected by an arc defined by the semicircle with endpoints $A$ and $D$. If $CF = 1$, what is the perimeter of the whole shape $ABCFED$ ? [img]https://cdn.artofproblemsolving.com/attachments/2/5/098d4f58fee1b3041df23ba16557ed93ee9f5b.png[/img] [b]p8.[/b] There are two moles that live underground, and there are five circular holes that the moles can hop out of. The five holes are positioned as shown in the diagram below, where $A$, $B$, $C$, $D$, and $E$ are the centers of the circles, $AE = 30$ cm, and congruent triangles $\vartriangle ABC$, $\vartriangle CBD$, and $\vartriangle CDE$ are equilateral. The two moles randomly choose exactly two of the five holes, hop out of the two chosen holes, and hop back in. What is the probability that the holes that the two moles hop out of have centers that are exactly $15$ cm apart? [img]https://cdn.artofproblemsolving.com/attachments/c/e/b46ba87b954a1904043020d7a211477caf321d.png[/img] [b]p9.[/b] Carson is planning a trip for $n$ people. Let $x$ be the number of cars that will be used and $y$ be the number of people per car. What is the smallest value of $n$ such that there are exactly $3$ possibilities for $x$ and $y$ so that $y$ is an integer, $x < y$, and exactly one person is left without a car? [b]p10.[/b] Iris is eating an ice cream cone, which consists of a hemisphere of ice cream with radius $r > 0$ on top of a cone with height $12$ and also radius $r$. Iris is a slow eater, so after eating one-third of the ice cream, she notices that the rest of the ice cream has melted and completely filled the cone. Assuming the ice cream did not change volume after it melted, what is the value of $r$? [b]p11.[/b] As Natasha begins eating brunch between $11:30$ AM and $12$ PM, she notes that the smaller angle between the minute and hour hand of the clock is $27$ degrees. What is the number of degrees in the smaller angle between the minute and hour hand when Natasha finishes eating brunch $20$ minutes later? [b]p12.[/b] On a regular hexagon $ABCDEF$, Luke the frog starts at point $A$, there is food on points $C$ and $E$ and there are crocodiles on points $B$ and $D$. When Luke is on a point, he hops to any of the five other vertices with equal probability. What is the probability that Luke will visit both of the points with food before visiting any of the crocodiles? [b]p13.[/b] $2023$ regular unit hexagons are arranged in a tessellating lattice, as follows. The first hexagon $ABCDEF$ (with vertices in clockwise order) has leftmost vertex $A$ at the origin, and hexagons $H_2$ and $H_3$ share edges $\overline{CD}$ and $\overline{DE}$ with hexagon $H_1$, respectively. Hexagon $H_4$ shares edges with both hexagons $H_2$ and $H_3$, and hexagons $H_5$ and $H_6$ are constructed similarly to hexagons H_2 and $H_3$. Hexagons $H_7$ to $H_{2022}$ are constructed following the pattern of hexagons $H_4$, $H_5$, $H_6$. Finally, hexagon H_{2023} is constructed, sharing an edge with both hexagons H2021 and H2022. Compute the perimeter of the resulting figure. [img]https://cdn.artofproblemsolving.com/attachments/1/d/eaf0d04676bac3e3c197b4686dcddd08fce9ac.png[/img] [b]p14.[/b] Aditya’s favorite number is a positive two-digit integer. Aditya sums the integers from $5$ to his favorite number, inclusive. Then, he sums the next $12$ consecutive integers starting after his favorite number. If the two sums are consecutive integers and the second sum is greater than the first sum, what is Aditya’s favorite number? [b]p15.[/b] The $100^{th}$ anniversary of BMT will fall in the year $2112$, which is a palindromic year. Compute the sum of all years from $0000$ to $9999$, inclusive, that are palindromic when written out as four-digit numbers (including leading zeros). Examples include $2002$, $1991$, and $0110$. [b]p16.[/b] Points $A$, $B$, $C$, $D$, and $E$ lie on line $r$, in that order, such that $DE = 2DC$ and $AB = 2BC$. Let $M$ be the midpoint of segment $\overline{AC}$. Finally, let point $P$ lie on $r$ such that $PE = x$. If $AB = 8x$, $ME = 9x$, and $AP = 112$, compute the sum of the two possible values of $CD$. [b]p17.[/b] A parabola $y = x^2$ in the xy-plane is rotated $180^o$ about a point $(a, b)$. The resulting parabola has roots at $x = 40$ and $x = 48$. Compute $a + b$. [b]p18.[/b] Susan has a standard die with values $1$ to $6$. She plays a game where every time she rolls the die, she permanently increases the value on the top face by $1$. What is the probability that, after she rolls her die 3 times, there is a face on it with a value of at least $7$? [b]p19.[/b] Let $N$ be a $6$-digit number satisfying the property that the average value of the digits of $N^4$ is $5$. Compute the sum of the digits of $N^4$. [b]p20.[/b] Let $O_1$, $O_2$, $...$, $O_8$ be circles of radius $1$ such that $O_1$ is externally tangent to $O_8$ and $O_2$ but no other circles, $O_2$ is externally tangent to $O_1$ and $O_3$ but no other circles, and so on. Let $C$ be a circle that is externally tangent to each of $O_1$, $O_2$, $...$, $O_8$. Compute the radius of $C$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A positive integer $n$ is called [i]nice[/i] if it has at least $3$ proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is [i]nice[/i] because its largest three proper divisors are $3,2,1$ and $6=3+2+1$. Find the number of [i]nice[/i] integers not greater than $3000$.

How many ways can you roll three 20-sided dice such that the sum of the three rolls is exactly $42$? Here the order of the rolls matter. [i](Note that a 20-sided die is is very much like a regular 6-sided die other than the fact that it has $20$ faces instead of $6$)[/i]

Let $k \leq 2022$ be a positive integer. Alice and Bob play a game on a $2022 \times 2022$ board. Initially, all cells are white. Alice starts and the players alternate. In her turn, Alice can either color one white cell in red or pass her turn. In his turn, Bob can either color a $k \times k$ square of white cells in blue or pass his turn. Once both players pass, the game ends and the person who colored more cells wins (a draw can occur). For each $1 \leq k \leq 2022$, determine which player (if any) has a winning strategy.

Find the smallest integer $n$ such that an $n\times n$ square can be partitioned into $40\times 40$ and $49\times 49$ squares, with both types of squares present in the partition, if a) $40|n$; b) $49|n$; c) $n\in \mathbb N$.

In each square of a $101\times 101$ board, except the central one, is placed either a sign " turn" or a sign " straight". The chess piece " car" can enter any square on the boundary of the board from outside (perpendicularly to the boundary). If the car enters a square with the sign " straight" then it moves to the next square in the same direction, otherwise (in case it enters a square with the sign " turn") it turns either to the right or to the left ( its choice). Can one place the signs in such a way that the car never enter the central square?

Yuri is looking at the great Mayan table. The table has $200$ columns and $2^{200}$ rows. Yuri knows that each cell of the table depicts the sun or the moon, and any two rows are different (i.e. differ in at least one column). Each cell of the table is covered with a sheet. The wind has blown aways exactly two sheets from each row. Could it happen that now Yuri can find out for at least $10000$ rows what is depicted in each of them (in each of the columns)? [i]Proposed by I. Bogdanov, K. Knop[/i]

A gang of robbers took away a bag of coins from a merchant . Each coin is worth an integer number of pennies. It is known that if any single coin is removed from the bag, then the remaining coins can be divided fairly among the robbers (that is, they all get coins with the same total value in pennies) . Prove that after one coin is removed, the number of the remaining coins is divisible by the number of robbers. (Folklore, modified by A Shapovalov)

Determine the largest possible number of groups one can compose from the integers $1,2,3,..., 19,20$, so that the product of the numbers in each group is a perfect square. (The group may contain exactly one number, in that case the product equals this number, each number must be in exactly one group.) (E. Barabanov, I. Voronovich)

[b]p1.[/b] In a $3$-person race, how many different results are possible if ties are allowed? [b]p2.[/b] An isosceles trapezoid has lengths $5$, $5$, $5$, and $8$. What is the sum of the lengths of its diagonals? [b]p3.[/b] Let $f(x) = (1 + x + x^2)(1 + x^3 + x^6)(1 + x^9 + x^{18})...$. Compute $f(4/5)$. [b]p4.[/b] Compute the largest prime factor of $3^{12} - 1$. [b]p5.[/b] Taren wants to throw a frisbee to David, who starts running perpendicular to the initial line between them at rate $1$ m/s. Taren throws the frisbee at rate $2$ m/s at the same instant David begins to run. At what angle should Taren throw the frisbee? [b]p6.[/b] The polynomial $p(x)$ leaves remainder $6$ when divided by $x-5$, and $5$ when divided by $x-6$. What is the remainder when $p(x)$ is divided by $(x - 5)(x - 6)$? [b]p7.[/b] Find the sum of the cubes of the roots of $x^{10} + x^9 + ... + x + 1 = 0$. [b]p8.[/b] A circle of radius $1$ is inscribed in a the parabola $y = x^2$. What is the $x$-coordinate of the intersection in the first quadrant? [b]p9.[/b] You are stuck in a cave with $3$ tunnels. The first tunnel leads you back to your starting point in $5$ hours, and the second tunnel leads you back there in $7$ hours. The third tunnel leads you out of the cave in $4$ hours. What is the expected number of hours for you to exit the cave, assuming you choose a tunnel randomly each time you come across your point of origin? [b]p10.[/b] What is the minimum distance between the line $y = 4x/7 + 1/5$ and any lattice point in the plane? (lattice points are points with integer coordinates) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Several zeros and ones are written down in a row. Consider all pairs of digits (not necessarily adjacent) such that the left digit is $1$ while the right digit is $0$. Let $M$ be the number of the pairs in which $1$ and $0$ are separated by an even number of digits (possibly zero), and let $N$ be the number of the pairs in which $1$ and $0$ are separated by an odd number of digits. Prove that $M \ge N$.

Let $n \geq 3$ be an integer. Two players play a game on an empty graph with $n + 1$ vertices, consisting of the vertices of a regular n-gon and its center. They alternately select a vertex of the n-gon and draw an edge (that has not been drawn) to an adjacent vertex on the n-gon or to the center of the n-gon. The player who first makes the graph connected wins. Between the player who goes first and the player who goes second, who has a winning strategy? [i]Note: an empty graph is a graph with no edges.[/i]

A polygon is $\textit{monochromatic}$ if all its vertices are coloured by the same colour. Suppose now every point of the plane is coloured red or blue. Show that there exists either a monochromatic equilateral triangle of side length $2$, or a monochromatic equilateral triangle of side length $\sqrt{3}$, or a monochromatic rhombus of side length $1$.

Let $m, n$ be positive integers. Consider a sequence of positive integers $a_1, a_2, ... , a_n$ that satisfies $m = a_1 \ge a_2\ge ... \ge a_n \ge 1$. Then define, for $1\le  i\le  m$, $b_i =$ # $\{ j \in \{1, 2, ... , n\}: a_j \ge i\}$, so $b_i$ is the number of terms $a_j $ of the given sequence for which $a_j  \ge i$. Similarly, we define, for $1\le   j \le  n$, $c_j=$ # $\{ i \in \{1, 2, ... , m\}: b_i \ge j\}$ , thus $c_j$ is the number of terms bi in the given sequence for which $b_i \ge j$. E.g.: If $a$ is the sequence $5, 3, 3, 2, 1, 1$ then $b$ is the sequence $6, 4, 3, 1, 1$. (a) Prove that $a_j = c_j $ for $1  \le j  \le n$. (b) Prove that for $1\le  k \le m$: $\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j$.

All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of$ k$ for which the same $k$-digit number can be found in at least two different places in this string?

For which numbers $n$ is it possible to put marks on a stick such that all distances $1$ cm, $2$ cm, . . . , $n$ cm each appear exactly once as the distance between two of the marks, and no other distance appears as such a distance?