[b]8.1 / 7.4[/b] What number needs to be put in place * so that the next the problem had a unique solution: “There are n straight lines on the plane, intersecting at * points. Find n.” ? [b]8.2 / 7.3[/b] Prove that for any natural number $n$ the number $ n(2n+1)(3n+1)...(1966n + 1) $ is divisible by every prime number less than $1966$. [b]8.3 / 7.6[/b] There are $n$ points on the plane so that any triangle with vertices at these points has an area less than $1$. Prove that all these points can be enclosed in a triangle of area $4$. [b]8.4[/b] Prove that the sum of all divisors of the number $n^2$ is odd. [b]8.5[/b] A quadrilateral has three obtuse angles. Prove that the larger of its two diagonals emerges from the vertex of an acute angle. [b]8.6[/b] Numbers $x_1, x_2, . . . $ are constructed according to the following rule: $$x_1 = 2, x_2 = (x^5_1 + 1)/5x_1, x_3 = (x^5_2 + 1)/5x_2, ...$$ Prove that no matter how much we continued this construction, all the resulting numbers will be no less $1/5$ and no more than $2$. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].

Given positive integer $n \geq 2$. Now Cindy fills each cell of the $n*n$ grid with a positive integer not greater than $n$ such that the numbers in each row are in a non-decreasing order (from left to right) and numbers in each column is also in a non-decreasing order (from top to bottom). We call two adjacant cells form a $domino$ , if they are filled with the same number. Now Cindy wants the number of $domino$s as small as possible. Find the smallest number of $dominos$ Cindy can reach. (Here, two cells are called adjacant if they share one common side)

Certain king of a certain state wants to build $n$ cities and $n-1$ roads, connecting them to provide a possibility to move from every city to every city. (Each road connects two cities, the roads do not intersect, and don't come through another city.) He wants also, to make the shortests distances between the cities, along the roads, to be $1,2,3,...,n(n-1)/2$ kilometres. Is it possible for a) $n=6$ b) $n=1986$ ?

In the kindergarten there is a big box with balls of three colors: red, blue and green, 100 balls in total. Once Pasha took out of the box 30 red, 10 blue, and 20 green balls and played with them. Then he lost five balls and returned the others back into the box. The next day, Sasha took out of the box 8 red, 18 blue, and 48 green balls. Is it possible to determine the color of at least one lost ball?

Determine all positive integers $n$, $n\ge2$, such that the following statement is true: If $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.

A polyomino $P$ occupies $n$ cells of an infinite grid of unit squares. In each move, we lift $P$ off the grid and then we place it back into a new position, possibly rotated and reflected, so that the preceding and the new position have $n-1$ cells in common. We say that $P$ is a caterpillar of area $n$ if, by means of a series of moves, we can free up all cells initially occupied by $P$. How many caterpillars of area $n=10^{6}+1$ are there? Proposed by Nikolai Beluhov, Bulgaria

The set $A$ has exactly $n>4$ elements. Ann chooses $n+1$ distinct subsets of $A$, such that every subset has exactly $3$ elements. Prove that there exist two subsets chosen by Ann which have exactly one common element.

[b]p1.[/b] Will is distributing his money to three friends. Since he likes some friends more than others, the amount of money he gives each is in the ratio of $5 : 3 : 2$. If the person who received neither the least nor greatest amount of money was given $42$ dollars, how many dollars did Will distribute in all? [b]p2.[/b] Fan, Zhu, and Ming are driving around a circular track. Fan drives $24$ times as fast as Ming and Zhu drives $9$ times as fast as Ming. All three drivers start at the same point on the track and keep driving until Fan and Zhu pass Ming at the same time. During this interval, how many laps have Fan and Zhu driven together? [b]p3.[/b] Mr. DoBa is playing a game with Gunga the Gorilla. They both agree to think of a positive integer from $1$ to $120$, inclusive. Let the sum of their numbers be $n$. Let the remainder of the operation $\frac{n^2}{4}$ be $r$. If $r$ is $0$ or $1$, Mr. DoBa wins. Otherwise, Gunga wins. Let the probability that Mr. DoBa wins a given round of this game be $p$. What is $120p$? [b]p4.[/b] Let S be the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. How many subsets of $S$ are there such that if $a$ is the number of even numbers in the subset and $b$ is the number of odd numbers in the subset, then $a$ and $b$ are either both odd or both even? By definition, subsets of $S$ are unordered and only contain distinct elements that belong to $S$. [b]p5.[/b] Phillips Academy has five clusters, $WQN$, $WQS$, $PKN$, $FLG$ and $ABB$. The Blue Key heads are going to visit all five clusters in some order, except $WQS$ must be visited before $WQN$. How many total ways can they visit the five clusters? [b]p6.[/b] An astronaut is in a spaceship which is a cube of side length $6$. He can go outside but has to be within a distance of $3$ from the spaceship, as that is the length of the rope that tethers him to the ship. Out of all the possible points he can reach, the surface area of the outer surface can be expressed as $m+n\pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? [b]p7.[/b] Let $ABCD$ be a square and $E$ be a point in its interior such that $CDE$ is an equilateral triangle. The circumcircle of $CDE$ intersects sides $AD$ and $BC$ at $D$, $F$ and $C$, $G$, respectively. If $AB = 30$, the area of $AFGB$ can be expressed as $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and c is not divisible by the square of any prime. Find $a + b + c$. [b]p8.[/b] Suppose that $x, y, z$ satisfy the equations $$x + y + z = 3$$ $$x^2 + y^2 + z^2 = 3$$ $$x^3 + y^3 + z^3 = 3$$ Let the sum of all possible values of $x$ be $N$. What is $12000N$? [b]p9.[/b] In circle $O$ inscribe triangle $\vartriangle ABC$ so that $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be the midpoint of arc $BC$, and let $AD$ intersect $BC$ at $E$. Determine the value of $DE \cdot DA$. [b]p10.[/b] How many ways are there to color the vertices of a regular octagon in $3$ colors such that no two adjacent vertices have the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n\ge 3$ be an integer. Two players, Ana and Beto, play the following game. Ana tags the vertices of a regular $n$- gon with the numbers from $1$ to $n$, in any order she wants. Every vertex must be tagged with a different number. Then, we place a turkey in each of the $n$ vertices. These turkeys are trained for the following. If Beto whistles, each turkey moves to the adjacent vertex with greater tag. If Beto claps, each turkey moves to the adjacent vertex with lower tag. Beto wins if, after some number of whistles and claps, he gets to move all the turkeys to the same vertex. Ana wins if she can tag the vertices so that Beto can't do this. For each $n\ge 3$, determine which player has a winning strategy. [i]Proposed by Victor and Isaías de la Fuente[/i]

Let $ R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $ R_1 \equal{} (1),$ and if $ R_{n - 1} \equal{} (x_1, \ldots, x_s),$ then \[ R_n \equal{} (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).\] For example, $ R_2 \equal{} (1, 2),$ $ R_3 \equal{} (1, 1, 2, 3),$ $ R_4 \equal{} (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $ n > 1,$ then the $ k$th term from the left in $ R_n$ is equal to 1 if and only if the $ k$th term from the right in $ R_n$ is different from 1.

Can a paper circle be cut into pieces and then rearranged into a square of the same area, if only a finite number of cuts is allowed and they must be along segments of straight lines or circular arcs? (A Belov)

In some cells of the table $2025 \times 2025$ crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least $k$ crosses. Find the minimal $k$ for which it is always possible to color crosses in two colors such that any balanced set has crosses of both colors. [i]M. Karpuk[/i]

On a certain board, fractions are always written in their lowest form. Pionaj starts with 2 random positive fractions. After every minute,he replaces one of the previous 2 fractions (at random) with a new fraction that is equal to the sum of their numerators divided by the sum of their denominators. Given that he continues this indefinitely, show that eventually all the resulting fractions would be in their lowest forms even before writing them on the board(recall that he has to reduce each fration to their lowest form beore writing it on the board for the next operation). (for example starting with $\frac{15}{7}$ and $\frac{10}{3}$ he may replace it with $\frac{5}{2}$

Evin picks distinct points $A, B, C, D, E$, and $F$ on a circle. What is the probability that there are exactly two intersections among the line segments $AB$, $CD$, and $EF$? [i]Proposed by Evin Liang[/i]

Given a convex polyhedron with an even number of edges. Prove that we can attach an arrow to each edge, such that for every vertex of the polyhedron, the number of the arrows ending in this vertex is even.

[b]p1.[/b] Suppose $\frac{x}{y} = 0.\overline{ab}$ where $x$ and $y$ are relatively prime positive integers and $ab + a + b + 1$ is a multiple of $12$. Find the sum of all possible values of $y$. [b]p2.[/b] Let $A$ be the set of points $\{(0, 0), (2, 0), (0, 2),(2, 2),(3, 1),(1, 3)\}$. How many distinct circles pass through at least three points in $A$? [b]p3.[/b] Jack and Jill need to bring pails of water home. The river is the $x$-axis, Jack is initially at the point $(-5, 3)$, Jill is initially at the point $(6, 1)$, and their home is at the point $(0, h)$ where $h > 0$. If they take the shortest paths home given that each of them must make a stop at the river, they walk exactly the same total distance. What is $h$? [b]p4.[/b] What is the largest perfect square which is not a multiple of $10$ and which remains a perfect square if the ones and tens digits are replaced with zeroes? [b]p5.[/b] In convex polygon $P$, each internal angle measure (in degrees) is a distinct integer. What is the maximum possible number of sides $P$ could have? [b]p6.[/b] How many polynomials $p(x)$ of degree exactly $3$ with real coefficients satisfy $$p(0), p(1), p(2), p(3) \in \{0, 1, 2\}?$$ [b]p7.[/b] Six spheres, each with radius $4$, are resting on the ground. Their centers form a regular hexagon, and adjacent spheres are tangent. A seventh sphere, with radius $13$, rests on top of and is tangent to all six of these spheres. How high above the ground is the center of the seventh sphere? [b]p8.[/b] You have a paper square. You may fold it along any line of symmetry. (That is, the layers of paper must line up perfectly.) You then repeat this process using the folded piece of paper. If the direction of the folds does not matter, how many ways can you make exactly eight folds while following these rules? [b]p9.[/b] Quadrilateral $ABCD$ has $\overline{AB} = 40$, $\overline{CD} = 10$, $\overline{AD} = \overline{BC}$, $m\angle BAD = 20^o$, and $m \angle ABC = 70^o$. What is the area of quadrilateral $ABCD$? [b]p10.[/b] We say that a permutation $\sigma$ of the set $\{1, 2,..., n\}$ preserves divisibilty if $\sigma (a)$ divides $\sigma (b)$ whenever $a$ divides $b$. How many permutations of $\{1, 2,..., 40\}$ preserve divisibility? (A permutation of $\{1, 2,..., n\}$ is a function $\sigma$ from $\{1, 2,..., n\}$ to itself such that for any $b \in \{1, 2,..., n\}$, there exists some $a \in \{1, 2,..., n\}$ satisfying $\sigma (a) = b$.) [b]p11.[/b] In the diagram shown at right, how many ways are there to remove at least one edge so that some circle with an “A” and some circle with a “B” remain connected? [img]https://cdn.artofproblemsolving.com/attachments/8/7/fde209c63cc23f6d3482009cc6016c7cefc868.png[/img] [b]p12.[/b] Let $S$ be the set of the $125$ points in three-dimension space of the form $(x, y, z)$ where $x$, $y$, and $z$ are integers between $1$ and $5$, inclusive. A family of snakes lives at the point $(1, 1, 1)$, and one day they decide to move to the point $(5, 5, 5)$. Snakes may slither only in increments of $(1,0,0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Given that at least one snake has slithered through each point of $S$ by the time the entire family has reached $(5, 5, 5)$, what is the smallest number of snakes that could be in the family? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $N$ be the number of sequences of natural numbers $d_1,d_2,\dots,d_{10}$ such that the following conditions hold: $d_1|d_2$, $\dots$, $d_9|d_{10}$ and $d_{10}|6^{2018}$. Evaluate the remainder when $N$ is divided by $2017$.

Two players take turns alternatively and remove a number from $1,2,\dots,1000$. Players can not remove a number that differ with a number already removed by $1$ also they can not remove a number such that it sums up with another removed number to $1001$. The player who can not move loses. Determine the winner.

The siblings Robb, Arya and Sansa have received seven sealed bags from an unknown donor with varying number of beads. Six of the bags have labels indicating the number beads: $7, 9, 11, 13, 15, 18$, but the seventh bag lacks etiquette. The sensor has set certain requirements: Robb must have three bags and his sisters two bags each. In addition, Arya will have the bag that contains $7$ beads. The bags should be distributed so that each of the siblings get the same number of pearls (this is possible according to the donor). How many pearls are there in the bag without a label, how many pearls are there in total and how should the bags be distributed?

Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that: $(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$; $(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$; $(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$ $(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$. Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.

[b]p1.[/b] Three camps are located in the vertices of an equilateral triangle. The roads connecting camps are along the sides of the triangle. Captain America is inside the triangle and he needs to know the distances between camps. Being able to see the roads he has found that the sum of the shortest distances from his location to the roads is 50 miles. Can you help Captain America to evaluate the distances between the camps? [b]p2.[/b] $N$ regions are located in the plane, every pair of them have a non-empty overlap. Each region is a connected set, that means every two points inside the region can be connected by a curve all points of which belong to the region. Iron Man has one charge remaining to make a laser shot. Is it possible for him to make the shot that goes through all $N$ regions? [b]p3.[/b] Money in Wonderland comes in $\$5$ and $\$7$ bills. (a) What is the smallest amount of money you need to buy a slice of pizza that costs $\$1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do since the pizza man can only give you $\$5$ back. (b) Vending machines in Wonderland accept only exact payment (do not give back change). List all positive integer numbers which CANNOT be used as prices in such vending machines. (That is, find the sums of money that cannot be paid by exact change.) [b]p4.[/b] (a) Put $5$ points on the plane so that each $3$ of them are vertices of an isosceles triangle (i.e., a triangle with two equal sides), and no three points lie on the same line. (b) Do the same with $6$ points. [b]p5.[/b] Numbers $1,2,3,…,100$ are randomly divided in two groups $50$ numbers in each. In the first group the numbers are written in increasing order and denoted $a_1,a_2, ..., a_{50}$. In the second group the numberss are written in decreasing order and denoted $b_1,b_2, ..., b_{50}$. Thus $a_1<a_2<...<a_{50}$ and $ b_1>b_2>...>b_{50}$. Evaluate $|a_1-b_1|+|a_2-b_2|+...+|a_{50}-b_{50}|$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$30$ segments of lengths$$1,\quad \sqrt{3},\quad \sqrt{5},\quad \sqrt{7},\quad \sqrt{9},\quad \ldots ,\quad \sqrt{59} $$ have been drawn on a blackboard. In each step, two of the segments are deleted and a new segment of length equal to the hypotenuse of the right triangle with legs equal to the two deleted segments is drawn. After $29$ steps only one segment remains. Find the possible values of its length.

Albert and Kevin are playing a game. Kevin has a $10\%$ chance of winning any given round in the match. If Kevin wins the first game, he wins the match. If not, he requests that the match be extended to a best of $3$. If he wins the best of $3$, he wins the match. If not, then he requests the match be extended to a best of $5$, and so forth. What is the probability that Kevin eventually wins the match? (A best of $2n+ 1$ match consists of a series of rounds. The first person to reach $n + 1$ winning games wins the match)

You can tile a $2 \times5$ grid of squares using any combination of three types of tiles: single unit squares, two side by side unit squares, and three unit squares in the shape of an L. The diagram below shows the grid, the available tile shapes, and one way to tile the grid. In how many ways can the grid be tiled? [asy] import graph; size(15cm); pen dps = linewidth(1) + fontsize(10); defaultpen(dps); draw((-3,3)--(-3,1)); draw((-3,3)--(2,3)); draw((2,3)--(2,1)); draw((-3,1)--(2,1)); draw((-3,2)--(2,2)); draw((-2,3)--(-2,1)); draw((-1,3)--(-1,1)); draw((0,3)--(0,1)); draw((1,3)--(1,1)); draw((4,3)--(4,2)); draw((4,3)--(5,3)); draw((5,3)--(5,2)); draw((4,2)--(5,2)); draw((5.5,3)--(5.5,1)); draw((5.5,3)--(6.5,3)); draw((6.5,3)--(6.5,1)); draw((5.5,1)--(6.5,1)); draw((7,3)--(7,1)); draw((7,1)--(9,1)); draw((7,3)--(8,3)); draw((8,3)--(8,2)); draw((8,2)--(9,2)); draw((9,2)--(9,1)); draw((11,3)--(11,1)); draw((11,3)--(16,3)); draw((16,3)--(16,1)); draw((11,1)--(16,1)); draw((12,3)--(12,2)); draw((11,2)--(12,2)); draw((12,2)--(13,2)); draw((13,2)--(13,1)); draw((14,3)--(14,1)); draw((14,2)--(15,2)); draw((15,3)--(15,1));[/asy]

Betty has a $4$-by-$4$ square box of chocolates. Every time Betty eats a chocolate, she picks one from a row with the greatest number of remaining chocolates. In how many ways can Betty eat $5$ chocolates from her box, where order matters?