It is possible to paint with the colors red and blue the squares of a grid board $1999\times 1999$, so that in each of the $1999$ rows, in each of the $1999$ columns and each of the the $2$ diagonals are exactly $1000$ squares painted red?

[hide=E stands for Euclid , L stands for Lobachevsky]they had two problem sets under those two names[/hide] [b]E1.[/b] How many positive divisors does $72$ have? [b]E2 / L2.[/b] Raymond wants to travel in a car with $3$ other (distinguishable) people. The car has $5$ seats: a driver’s seat, a passenger seat, and a row of $3$ seats behind them. If Raymond’s cello must be in a seat next to him, and he can’t drive, but every other person can, how many ways can everyone sit in the car? [b]E3 / L3.[/b] Peter wants to make fruit punch. He has orange juice ($100\%$ orange juice), tropical mix ($25\%$ orange juice, $75\%$ pineapple juice), and cherry juice ($100\%$ cherry juice). If he wants his final mix to have $50\%$ orange juice, $10\%$ cherry juice, and $40\%$ pineapple juice, in what ratios should he mix the $3$ juices? Please write your answer in the form (orange):(tropical):(cherry), where the three integers are relatively prime. [b]E4 / L4.[/b] Points $A, B, C$, and $D$ are chosen on a circle such that $m \angle ACD = 85^o$, $m\angle ADC = 40^o$,and $m\angle BCD = 60^o$. What is $m\angle CBD$? [b]E5.[/b] $a, b$, and $c$ are positive real numbers. If $abc = 6$ and $a + b = 2$, what is the minimum possible value of $a + b + c$? [b]E6 / L5.[/b] Circles $A$ and $B$ are drawn on a plane such that they intersect at two points. The centers of the two circles and the two intersection points lie on another circle, circle $C$. If the distance between the centers of circles $A$ and $B$ is $20$ and the radius of circle $A$ is $16$, what is the radius of circle $B$? [b]E7.[/b] Point $P$ is inside rectangle $ABCD$. If $AP = 5$, $BP = 6$, and $CP = 7$, what is the length of $DP$? [b]E8 / L6.[/b] For how many integers $n$ is $n^2 + 4$ divisible by $n + 2$? [b]E9. [/b] How many of the perfect squares between $1$ and $10000$, inclusive, can be written as the sum of two triangular numbers? We define the $n$th triangular number to be $1 + 2 + 3 + ... + n$, where $n$ is a positive integer. [b]E10 / L7.[/b] A small sphere of radius $1$ is sitting on the ground externally tangent to a larger sphere, also sitting on the ground. If the line connecting the spheres’ centers makes a $60^o$ angle with the ground, what is the radius of the larger sphere? [b]E11 / L8.[/b] A classroom has $12$ chairs in a row and $5$ distinguishable students. The teacher wants to position the students in the seats in such a way that there is at least one empty chair between any two students. In how many ways can the teacher do this? [b]E12 / L9.[/b] Let there be real numbers $a$ and $b$ such that $a/b^2 + b/a^2 = 72$ and $ab = 3$. Find the value of $a^2 + b^2$. [b]E13 / L10.[/b] Find the number of ordered pairs of positive integers $(x, y)$ such that $gcd \, (x, y)+lcm \, (x, y) =x + y + 8$. [b]E14 / L11.[/b] Evaluate $\sum_{i=1}^{\infty}\frac{i}{4^i}=\frac{1}{4} +\frac{2}{16} +\frac{3}{64} +...$ [b]E15 / L12.[/b] Xavier and Olivia are playing tic-tac-toe. Xavier goes first. How many ways can the game play out such that Olivia wins on her third move? The order of the moves matters. [b]L1.[/b] What is the sum of the positive divisors of $100$? [b]L13.[/b] Let $ABCD$ be a convex quadrilateral with $AC = 20$. Furthermore, let $M, N, P$, and $Q$ be the midpoints of $DA, AB, BC$, and $CD$, respectively. Let $X$ be the intersection of the diagonals of quadrilateral $MNPQ$. Given that $NX = 12$ and $XP = 10$, compute the area of $ABCD$. [b]L14.[/b] Evaluate $(\sqrt3 + \sqrt5)^6$ to the nearest integer. [b]L15.[/b] In Hatland, each citizen wears either a green hat or a blue hat. Furthermore, each citizen belongs to exactly one neighborhood. On average, a green-hatted citizen has $65\%$ of his neighbors wearing green hats, and a blue-hatted citizen has $80\%$ of his neighbors wearing blue hats. Each neighborhood has a different number of total citizens. What is the ratio of green-hatted to blue-hatted citizens in Hatland? (A citizen is his own neighbor.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a tennis ball factory there are $4$ machines $m_1 , m_2 , m_3 , m_4$, which produce, respectively, $10\%$, $20\%$, $30\%$ and $40\%$ of the balls that come out of the factory. The machine $m_1$ introduces defects in $1\%$ of the balls it manufactures, the machine $m_2$ in $2\%$, $m_3$ in $4\%$ and $m_4$ in $15\%$. Of the balls manufactured In one day, one is chosen at random and it turns out to be defective. What is the probability that Has this ball been made by the machine $ m_3$ ?

Some of the $n + 1$ cities in a country (including the capital city) are connected by one-way or two-way airlines. No two cities are connected by both a one-way airline and a two-way airline, but there may be more than one two-way airline between two cities. If $d_A$ denotes the number of airlines from a city $A$, then $d_A \le n$ for any city $A$ other than the capital city and $d_A + d_B \le n$ for any two cities $A$ and $B$ other than the capital city which are not connected by a two-way airline. Every airline has a return, possibly consisting of several connected flights. Find the largest possible number of two-way airlines and all configurations of airlines for which this largest number is attained.

Ana and Bruno have an $8 \times 8$ checkered board. Ana paints each of the $64$ squares with some color. Then Bruno chooses two rows and two columns on the board and looks at the $4$ squares where they intersect. Bruno's goal is for these $4$ squares to be the same color. How many colors, at least, must Ana use so that Bruno can't fulfill his objective? Show how you can paint the board with this amount of colors and explain because if you use less colors then Bruno can always fulfill his goal.

Let $n>1$ be an integer. Suppose we are given $2n$ points in the plane such that no three of them are collinear. The points are to be labelled $A_1, A_2, \dots , A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2$. We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.

Given are $N$ points on the plane such that no three of them are collinear,which are coloured red,green and black.We consider all the segments between these points and give to each segment a [i]"value"[/i] according to the following conditions: [b]i.[/b]If at least one of the endpoints of a segment is black then the segment's [i]"value"[/i] is $0$. [b]ii.[/b]If the endpoints of the segment have the same colour,re or green,then the segment's [i]"value"[/i] is $1$. [b]iii.[/b]If the endpoints of the segment have different colours but none of them is black,then the segment's [i]"value"[/i] is $-1$. Determine the minimum possible sum of the [i]"values"[/i] of the segments.

Let $n$ be positive integer and fix $2n$ distinct points on a circle. Determine the number of ways to connect the points with $n$ arrows (oriented line segments) such that all of the following conditions hold: [list] [*]each of the $2n$ points is a startpoint or endpoint of an arrow; [*]no two arrows intersect; and [*]there are no two arrows $\overrightarrow{AB}$ and $\overrightarrow{CD}$ such that $A$, $B$, $C$ and $D$ appear in clockwise order around the circle (not necessarily consecutively). [/list]

[b]p1.[/b] Consider the following function $f(x) = \left(\frac12 \right)^x - \left(\frac12 \right)^{x+1}$. Evaluate the infinite sum $f(1) + f(2) + f(3) + f(4) +...$ [b]p2.[/b] Find the area of the shape bounded by the following relations $$y \le |x| -2$$ $$y \ge |x| - 4$$ $$y \le 0$$ where |x| denotes the absolute value of $x$. [b]p3.[/b] An equilateral triangle with side length $6$ is inscribed inside a circle. What is the diameter of the largest circle that can fit in the circle but outside of the triangle? [b]p4.[/b] Given $\sin x - \tan x = \sin x \tan x$, solve for $x$ in the interval $(0, 2\pi)$, exclusive. [b]p5.[/b] How many rectangles are there in a $6$ by $6$ square grid? [b]p6.[/b] Find the lateral surface area of a cone with radius $3$ and height $4$. [b]p7.[/b] From $9$ positive integer scores on a $10$-point quiz, the mean is $ 8$, the median is $ 8$, and the mode is $7$. Determine the maximum number of perfect scores possible on this test. [b]p8.[/b] If $i =\sqrt{-1}$, evaluate the following continued fraction: $$2i +\frac{1}{2i +\frac{1}{2i+ \frac{1}{2i+...}}}$$ [b]p9.[/b] The cubic polynomial $x^3-px^2+px-6$ has roots $p, q$, and $r$. What is $(1-p)(1-q)(1-r)$? [b]p10.[/b] (Variant on a Classic.) Gilnor is a merchant from Cutlass, a town where $10\%$ of the merchants are thieves. The police utilize a lie detector that is $90\%$ accurate to see if Gilnor is one of the thieves. According to the device, Gilnor is a thief. What is the probability that he really is one? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The International Mathematical Olympiad is being organized in Japan, where a folklore belief is that the number $4$ brings bad luck. The opening ceremony takes place at the Grand Theatre where each row has the capacity of $55$ seats. What is the maximum number of contestants that can be seated in a single row with the restriction that no two of them are $4$ seats apart (so that bad luck during the competition is avoided)?

In Wonderland there are at least $5$ towns. Some towns are connected directly by roads or railways. Every town is connected to at least one other town and for any four towns there exists some direct connection between at least three pairs of towns among those four. When entering the public transportation network of this land, the traveller must insert one gold coin into a machine, which lets him use a direct connection to go to the next town. But if the traveller continues travelling from some town with the same method of transportation that took him there, and he has paid a gold coin to get to this town, then going to the next town does not cost anything, but instead the traveller gains the coin he last used back. In other cases he must pay just like when starting travelling. Prove that it is possible to get from any town to any other town by using at most $2$ gold coins.

For 31 years, n (>6) tennis players have records of wins. It turns out that for every two players, there is a third player who has won over them before. Prove that for every integer $k,l$ such that $2^{2^k+1}-1>n, 1<l<2k+1$, there exist $l$ players ($A_1, A_2, ... , A_l$) such that every player $A_{i+1}$ won over $A_i$. ($A_{l+1}$ is same as $A_1$)

Let $\Lambda$ be the set of points in the plane whose coordinates are integers and let $F$ be the collection of all functions from $\Lambda$ to $\{1,-1\}.$ We call a function $f$ in $F$ [i]perfect[/i] if every function $g$ in $F$ that differs from $f$ at finitely many points satisfies the condition \[ \sum_{0<d(P,Q)<2010} \frac{f(P)f(Q)-g(P)g(Q)}{d(P,Q)} \geq 0 \] where $d(P,Q)$ denotes the distance between $P$ and $Q.$ Show that there exist infinitely many [i]perfect[/i] functions that are not translates of each other.

Find the number of $(a_1,a_2, ... ,a_{2014})$ permutations of the $(1,2, . . . ,2014)$ such that, for all $1\leq i<j\leq2014$, $i+a_i \leq j+a_j$.

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$

Each box in a $2x2$ table can be colored black or white. How many different colorings of the table are there? (A): $4$, (B): $8$, (C): $16$, (D): $32$, (E) None of the above.

Along the direct highway Tmutarakan - Uryupinsk at points $ A_1 $, $ A_2 $, $ \dots $, $ A_ {100} $ are the towers of the DPS mobile operator, and in points $ B_1 $, $ B_2 $, $ \dots $, $ B_ {100} $ are the towers of the "Horn" company. (Tower numbering may not coincide with the order of their location along the highway.) Each tower operates at a distance of $10$ km in both directions along the highway. It is known that $ A_iA_k \geq B_iB_k $ for any $ i $, $ k \leq 100 $. Prove that the total length of all sections of the highway covered by the DPS network is not less than the length of the sections covered by the Horn network .

Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine the minimum number of integers in a complete sequence of $n$ numbers.

Several irrational numbers are written on a blackboard. It is known that for every two numbers $ a$ and $ b$ on the blackboard, at least one of the numbers $ a\over b\plus{}1$ and $ b\over a\plus{}1$ is rational. What maximum number of irrational numbers can be on the blackboard? [i]Author: Alexander Golovanov[/i]

Denote by $ \mathcal{C}$ the family of all configurations $ C$ of $ N > 1$ distinct points on the sphere $ S^2,$ and by $ \mathcal{H}$ the family of all closed hemispheres $ H$ of $ S^2.$ Compute: $ \displaystyle\max_{H\in\mathcal{H}}\displaystyle\min_{C\in\mathcal{C}}|H\cap C|$, $ \displaystyle\min_{H\in\mathcal{H}}\displaystyle\max_{C\in\mathcal{C}}|H\cap C|$ $ \displaystyle\max_{C\in\mathcal{C}}\displaystyle\min_{H\in\mathcal{H}}|H\cap C|$ and $ \displaystyle\min_{C\in\mathcal{C}}\displaystyle\max_{H\in\mathcal{H}}|H\cap C|.$

Let $n \geq 2$ be a positive integer. Let $\mathcal{R}$ be a connected set of unit squares on a grid. A [i]bar[/i] is a rectangle of length or width $1$ which is fully contained in $\mathcal{R}$. A bar is [i]special[/i] if it is not fully contained within any larger bar. Given that $\mathcal{R}$ contains special bars of sizes $1 \times 2,1 \times 3,\ldots,1 \times n$, find the smallest possible number of unit squares in $\mathcal{R}$. [i]Srinivas Arun[/i]

Katie has a chocolate bar that is a $5$-by-$5$ grid of square pieces, but she only wants to eat the center piece. To get to it, she performs the following operations: i. Take a gridline on the chocolate bar, and split the bar along the line. ii. Remove the piece that doesn’t contain the center. iii. With the remaining bar, repeat steps $1$ and $2$. Determine the number of ways that Katie can perform this sequence of operations so that eventually she ends up with just the center piece.

In a country, there are $100$ towns. Some pairs of towns are joined by roads. The roads do not intersect one another except meeting at towns. It is possible to go from any town to any other town by road. Prove that it is possible to pave some of the roads so that the number of paved roads at each town is odd.

Suppose you have a $6$ sided dice with $3$ faces colored red, $2$ faces colored blue, and $1$ face colored green. You roll this dice $20$ times and record the color that shows up on top. What is the expected value of the product of the number of red faces, blue faces, and green faces? [i]Proposed by Daniel Li[/i]