[b]p1.[/b] What is the smallest possible value for the product of two real numbers that differ by ten? [b]p2.[/b] Determine the number of positive integers $n$ with $1 \le n \le 400$ that satisfy the following: $\bullet$ $n$ is a square number. $\bullet$ $n$ is one more than a multiple of $5$. $\bullet$ $n$ is even. [b]p3.[/b] How many positive integers less than $2019$ are either a perfect cube or a perfect square but not both? [b]p4.[/b] Felicia draws the heart-shaped figure $GOAT$ that is made of two semicircles of equal area and an equilateral triangle, as shown below. If $GO = 2$, what is the area of the figure? [img]https://cdn.artofproblemsolving.com/attachments/3/c/388daa657351100f408ab3f1185f9ab32fcca5.png[/img] [b]p5.[/b] For distinct digits $A, B$, and $ C$: $$\begin{tabular}{cccc} & A & A \\ & B & B \\ + & C & C \\ \hline A & B & C \\ \end{tabular}$$ Compute $A \cdot B \cdot C$. [b]p6 [/b] What is the difference between the largest and smallest value for $lcm(a,b,c)$, where $a,b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive? [b]p7.[/b] Let $A$ and $B$ be points on the circumference of a circle with center $O$ such that $\angle AOB = 100^o$. If $X$ is the midpoint of minor arc $AB$ and $Y$ is on the circumference of the circle such that $XY\perp AO$, find the measure of $\angle OBY$ . [b]p8. [/b]When Ben works at twice his normal rate and Sammy works at his normal rate, they can finish a project together in $6$ hours. When Ben works at his normal rate and Sammy works as three times his normal rate, they can finish the same project together in $4$ hours. How many hours does it take Ben and Sammy to finish that project if they each work together at their normal rates? [b][b]p9.[/b][/b] How many positive integer divisors $n$ of $20000$ are there such that when $20000$ is divided by $n$, the quotient is divisible by a square number greater than $ 1$? [b]p10.[/b] What’s the maximum number of Friday the $13$th’s that can occur in a year? [b]p11.[/b] Let circle $\omega$ pass through points $B$ and $C$ of triangle $ABC$. Suppose $\omega$ intersects segment $AB$ at a point $D \ne B$ and intersects segment $AC$ at a point $E \ne C$. If $AD = DC = 12$, $DB = 3$, and $EC = 8$, determine the length of $EB$. [b]p12.[/b] Let $a,b$ be integers that satisfy the equation $2a^2 - b^2 + ab = 18$. Find the ordered pair $(a,b)$. [b]p13.[/b] Let $a,b,c$ be nonzero complex numbers such that $a -\frac{1}{b}= 8, b -\frac{1}{c}= 10, c -\frac{1}{a}= 12.$ Find $abc -\frac{1}{abc}$ . [b]p14.[/b] Let $\vartriangle ABC$ be an equilateral triangle of side length $1$. Let $\omega_0$ be the incircle of $\vartriangle ABC$, and for $n > 0$, define the infinite progression of circles $\omega_n$ as follows: $\bullet$ $\omega_n$ is tangent to $AB$ and $AC$ and externally tangent to $\omega_{n-1}$. $\bullet$ The area of $\omega_n$ is strictly less than the area of $\omega_{n-1}$. Determine the total area enclosed by all $\omega_i$ for $i \ge 0$. [b]p15.[/b] Determine the remainder when $13^{2020} +11^{2020}$ is divided by $144$. [b]p16.[/b] Let $x$ be a solution to $x +\frac{1}{x}= 1$. Compute $x^{2019} +\frac{1}{x^{2019}}$ . [b]p17. [/b]The positive integers are colored black and white such that if $n$ is one color, then $2n$ is the other color. If all of the odd numbers are colored black, then how many numbers between $100$ and $200$ inclusive are colored white? [b]p18.[/b] What is the expected number of rolls it will take to get all six values of a six-sided die face-up at least once? [b]p19.[/b] Let $\vartriangle ABC$ have side lengths $AB = 19$, $BC = 2019$, and $AC = 2020$. Let $D,E$ be the feet of the angle bisectors drawn from $A$ and $B$, and let $X,Y$ to be the feet of the altitudes from $C$ to $AD$ and $C$ to $BE$, respectively. Determine the length of $XY$ . [b]p20.[/b] Suppose I have $5$ unit cubes of cheese that I want to divide evenly amongst $3$ hungry mice. I can cut the cheese into smaller blocks, but cannot combine blocks into a bigger block. Over all possible choices of cuts in the cheese, what’s the largest possible volume of the smallest block of cheese? PS. You had better use hide for answers.

Snow White has, in her huge basket, $2021$ apples, and she knows that exactly one of them has a deadly poison, capable of killing a human being hours after ingesting just a measly piece. Contrary to what the fairy tales say, Snow White is more malevolent than the Evil Queen, and doesn't care about the lives of the seven dwarfs. Therefore, she decided to use them to find out which apple is poisonous. To this end, at the beginning of each day, Snow White prepares some apple salads (each salad is a mixture of pieces of some apples chosen by her), and forces some of the dwarfs (possibly all) to eat a salad each. At the end of the day, she notes who died and who survived, and the next day she again prepares some apple salads, forcing some of the surviving dwarves (possibly all) to eat a salad each. And she continues to do this, day after day, until she discovers the poisoned apple or until all the dwarves die. (a) Prove that there is a strategy in which Snow White manages to discover the poisoned apple after a few days. (b) What is the minimum number of days Snow White needs to find the poisoned apple, no matter how lucky she is?

Consider the following one-person game: A player starts with score $0$ and writes the number $20$ on an empty whiteboard. At each step, she may erase any one integer (call it a) and writes two positive integers (call them $b$ and $c$) such that $b + c = a$. The player then adds $b\times c$ to her score. She repeats the step several times until she ends up with all $1$'s on the whiteboard. Then the game is over, and the final score is calculated. Let $M, m$ be the maximum and minimum final score that can be possibly obtained respectively. Find $M-m$.

Let $X_1, X_2,...,X_n$ be points in the plane. For every $i$, let $A_i$ be the list of $n-1$ distances from $X_i$ to the remaining points. Find all arrangements of the $n$ points such all of these lists are the same, except for the order.

Given a $9\times 9$ grid, we assign either $+1$ or $-1$ to each square on the grid. We define an [i]adjustment[/i] as follow: for each square on the grid, we make a product of all numbers of those squares which share a common side with the square (excluding itself).Then we have $81$ products. Next we replace all the squares’ values with their corresponding products. Determine if we can make all values in the grid equal to $1$ through finite [i]adjustments[/i].

Let $(m,r,s,t)$ be positive integers such that $m\geq s+1$ and $r\geq t$. Consider $m$ sets $A_1, A_2, \dots, A_m$ with $r$ elements each one. Suppose that, for each $1\leq i\leq m$, there exist at least $t$ elements of $A_i$, such that each one(element) belongs to (at least) $s$ sets $A_j$ where $j\neq i$. Determine the greatest quantity of elements in the following set $A_1 \cup A_2 \cup A_3 \dots \cup A_m$.

Given a $n \times n$ square table. Exactly one beetle sits in each cell of the table. At $12.00$ all beetles creeps to some neighbouring cell (two cells are neighbouring if they have the common side). Find the greatest number of cells which can become empty (i.e. without beetles) if a) $n=8$ b) $n=9$ Problem Committee of BMO 2011

In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers $ m$ and $ n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $ m$ and $ n$, lie along edges of the squares. Let $ S_1$ be the total area of the black part of the triangle and $ S_2$ be the total area of the white part. Let $ f(m,n) \equal{} | S_1 \minus{} S_2 |$. a) Calculate $ f(m,n)$ for all positive integers $ m$ and $ n$ which are either both even or both odd. b) Prove that $ f(m,n) \leq \frac 12 \max \{m,n \}$ for all $ m$ and $ n$. c) Show that there is no constant $ C\in\mathbb{R}$ such that $ f(m,n) < C$ for all $ m$ and $ n$.

In a football season, even number $n$ of teams plays a simple series, i.e. each team plays once against each other team. Show that ona can group the series into $n-1$ rounds such that in every round every team plays exactly one match.

[u]Polyhedron Hopping[/u] [b]p1.[/b] Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he's currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started? [b]p2.[/b] In terms of $k$, for $k > 0$ how likely is he to be back where he started after $2k$ minutes? [b]p3.[/b] While Travis is having fun on cubes, Sherry is hopping in the same manner on an octahedron. An octahedron has six vertices and eight regular triangular faces. After ve minutes, how likely is Sherry to be one edge away from where she started? [b]p4.[/b] In terms of $k$, for $k > 0$, how likely is it that after $k$ minutes Sherry is at the vertex opposite the vertex where she started?

$3n-1$ points are given in the plane, no three are collinear. Prove that one can select $2n$ of them whose convex hull is not a triangle.

A token is placed in one square of a $m\times n$ board, and is moved according to the following rules: [list] [*]In each turn, the token can be moved to a square sharing a side with the one currently occupied. [*]The token cannot be placed in a square that has already been occupied. [*]Any two consecutive moves cannot have the same direction.[/list] The game ends when the token cannot be moved. Determine the values of $m$ and $n$ for which, by placing the token in some square, all the squares of the board will have been occupied in the end of the game.

Of the $9$ people who reached the final stage of the competition, only $4$ should receive a prize. The candidates were renumbered and lined up in a circle. Then a certain number $m$ (possibly greater than $9$) and the direction of reference were determined. People began to be counted, starting from the first. Each one became a winner and was eliminated from the drawing, and counting, starting from the next, continued until four winners were identified. The first three prizes were awarded to three people who had numbers $2$, $7$ and $5$ in the original lineup (they were eliminated in that order). What number did the fourth winner of the competition have in the initial lineup?

In each box of a $ 1 \times 2009$ grid, we place either a $ 0$ or a $ 1$, such that the sum of any $ 90$ consecutive boxes is $ 65$. Determine all possible values of the sum of the $ 2009$ boxes in the grid.

We wish to color the cells of a $n \times n$ chessboard with $k$ different colors such that for every $i\in \{1,2,...,n\}$, the $2n-1$ cells on $i$. row and $i$. column have all different colors. a) Prove that for $n=2001$ and $k=4001$, such coloring is not possible. b) Show that for $n=2^{m}-1$ and $k=2^{m+1}-1$, such coloring is possible.

Natural $n \geq 2$ is given. Group of people calls $n-compact$, if for any men from group, we can found $n$ people (without he), each two of there are familiar. Find maximum $N$ such that for any $n-compact$ group, consisting $N$ people contains subgroup from $n+1$ people, each of two of there are familiar.

Every square on a $n\times n$ chessboard is colored with red, blue, or green. Each red square has at least one green square adjacent to it, each green square has at least one blue square adjacent to it, and each blue square has at least one red square adjacent to it. Let $R$ be the number of red squares. Prove that $\displaystyle \frac{n^2}{11} \le R \le \frac{2n^2}{3}$.

Adamand Topher are playing a game in which each of them starts with $2$ pickles. Each turn, they flip a fair coin: if it lands heads, Topher takes $1$ pickle from Adam; if it lands tails, Adam takes $2$ pickles from Topher. (If Topher has only $1$ pickle left, Adam will just take it.) What’s the probability that Topher will have all $4$ pickles before Adam does?

Let $M$ be a set of six distinct positive integers whose sum is $60$. These numbers are written on the faces of a cube, one number to each face. A [i]move[/i] consists of choosing three faces of the cube that share a common vertex and adding $1$ to the numbers on those faces. Determine the number of sets $M$ for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.

Consider $37$ distinct points in space, all with integer coordinates. Prove that we may find among them three distinct points such that their barycentre has integers coordinates.

Let $n \geq 3$ be an integer. A sequence $P_1, P_2, \ldots, P_n$ of distinct points in the plane is called [i]good[/i] if no three of them are collinear, the polyline $P_1P_2 \ldots P_n$ is non-self-intersecting and the triangle $P_iP_{i + 1}P_{i + 2}$ is oriented counterclockwise for every $i = 1, 2, \ldots, n - 2$. For every integer $n \geq 3$ determine the greatest possible integer $k$ with the following property: there exist $n$ distinct points $A_1, A_2, \ldots, A_n$ in the plane for which there are $k$ distinct permutations $\sigma : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\}$ such that $A_{\sigma(1)}, A_{\sigma(2)}, \ldots, A_{\sigma(n)}$ is good. (A polyline $P_1P_2 \ldots P_n$ consists of the segments $P_1P_2, P_2P_3, \ldots, P_{n - 1}P_n$.)

Let $n$ be a positive integer. A frog starts on the number line at $0$. Suppose it makes a finite sequence of hops, subject to two conditions: [list] [*]The frog visits only points in $\{1, 2, \dots, 2^n-1\}$, each at most once. [*]The length of each hop is in $\{2^0, 2^1, 2^2, \dots\}$. (The hops may be either direction, left or right.) [/list] Let $S$ be the sum of the (positive) lengths of all hops in the sequence. What is the maximum possible value of $S$? [i]Ashwin Sah[/i]

A bean packing plant has a machine that puts a certain amount of beans into bags and then puts a certain amount of bags into boxes, which are then shipped to customers. One day, the machine broke down and the first n bags came out empty, the next $n$ bags came out with $1$ bean, the next $n$ bags came out with $2$ beans,..., and the last $n$ bags came out with $2006$ beans. To provide each customer with the agreed quantity of bags of beans, the person responsible for the unit intends to distribute the bags among the $2007$ boxes that day so that all boxes contain the same number of bags and all boxes contain the same number. number of beans. For what values of $n$ is this possible?

Three players Alexander, Beverley and Catherine participate in a tournament (all of them play the same number of games with each other). Is it possible that Alexander gets more points than the others, Catherine gets less points than the others, but Alexander has a smaller number of wins than the others and Catherine has a greater number of wins than the others? (A win scores $1$ point, a draw scores $\frac12$.) (A Rubin,)

Anja and Bernd take turns in removing stones from a heap, initially consisting of $n$ stones ($n \ge 2$). Anja begins, removing at least one but not all the stones. Afterwards, in each turn the player has to remove at least one stone and at most as many stones as removed in the preceding move. The player removing the last stone wins. Depending on the value of $n$, which player can ensure a win?