p1. Known points $A (-1.2)$, $B (0,2)$, $C (3,0)$, and $D (3, -1)$ as seen in the following picture. Determine the measure of the angle $AOD$ . [img]https://cdn.artofproblemsolving.com/attachments/f/2/ca857aaf54c803db34d8d52505ef9a80e7130f.png[/img] p2. Determine all prime numbers $p> 2$ until $p$ divides $71^2 - 37^2 - 51$. p3. A ball if dropped perpendicular to the ground from a height then it will bounce back perpendicular along the high third again, down back upright and bouncing back a third of its height, and next. If the distance traveled by the ball when it touches the ground the fourth time is equal to $106$ meters. From what height is the ball was dropped? p4. The beam $ABCD.EFGH$ is obtained by pasting two unit cubes $ABCD.PQRS$ and $PQRS.EFGH$. The point K is the midpoint of the edge $AB$, while the point $L$ is the midpoint of the edge $SH$. What is the length of the line segment $KL$? p5. How many integer numbers are no greater than $2004$, with remainder $1$ when divided by $2$, with remainder $2$ when divided by $3$, with remainder $3$ when divided by $4$, and with remainder $4$ when divided by $5$?

We have considered an arbitrary segment from each line in a plane. Show that the set of points of these segments have a subset such that the points of this subset form a triangle in the plane.

Consider an infinite strip, divided into unit squares. A finite number of nuts is placed in some of these squares. In a step, we choose a square $A$ which has more than one nut and take one of them and put it on the square on the right, take another nut (from $A$) and put it on the square on the left. The procedure ends when all squares has at most one nut. Prove that, given the initial configuration, any procedure one takes will end after the same number of steps and with the same final configuration.

During a day $2016$ customers visited the store. Every customer has been only once at the store(a customer enters the store,spends some time, and leaves the store). Find the greatest integer $k$ that makes the following statement always true. We can find $k$ customers such that either all of them have been at the store at the same time, or any two of them have not been at the same store at the same time.

[b]p1.[/b] The clock is $2$ hours $20$ minutes ahead of the correct time each week. The clock is set to the correct time at midnight Sunday to Monday. What time does this clock show at 6pm correct time on Thursday? [b]p2.[/b] Five cities $A,B,C,D$, and $E$ are located along the straight road in the alphabetical order. The sum of distances from $B$ to $A,C,D$ and $E$ is $20$ miles. The sum of distances from $C$ to the other four cities is $18$ miles. Find the distance between $B$ and $C$. [b]p3.[/b] Does there exist distinct digits $a, b, c$, and $d$ such that $\overline{abc}+\overline{c} = \overline{bda}$? Here $\overline{abc}$ means the three digit number with digits $a, b$, and $c$. [b]p4.[/b] Kuzya, Fyokla, Dunya, and Senya participated in a mathematical competition. Kuzya solved $8$ problems, more than anybody else. Senya solved $5$ problem, less than anybody else. Each problem was solved by exactly $3$ participants. How many problems were there? [b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the $5\times 5$ grid $Z^2_5 = \{(a, b) : 0 \le a, b \le 4\}$. Say that two points $(a, b)$,$(x, y)$ are adjacent if $a - x \equiv -1, 0, 1$ (mod $5$) and $b - y \equiv -1, 0, 1$ (mod $5$) . For example, in the diagram, all of the squares marked with $\cdot$ are adjacent to the square marked with $\times$. [img]https://cdn.artofproblemsolving.com/attachments/2/6/c49dd26ab48ddff5e1beecfbd167d5bb9fe16d.png[/img] What is the largest number of $\times$ that can be placed on the grid such that no two are adjacent?

Let $k\ge 1$ be a positive integer. We consider $4k$ chips, $2k$ of which are red and $2k$ of which are blue. A sequence of those $4k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue chips. For example, we can move from $r\underline{bb}br\underline{rr}b$ to $r\underline{rr}br\underline{bb}b$ where $r$ denotes a red chip and $b$ denotes a blue chip. Determine the smallest number $n$ (as a function of $k$) such that starting from any initial sequence of the $4k$ chips, we need at most $n$ moves to reach the state in which the first $2k$ chips are red.

Find the number of all $ 6$-digit natural numbers such that the sum of their digits is $ 10$ and each of the digits $ 0,1,2,3$ occurs at least once in them. [14 points out of 100 for the 6 problems]

Find all natural numbers \( n \) for which it is possible to construct an \( n \times n \) square using only tetrominoes like the one below:

Find the number of ways to arrange the letters in $LE X I NGTON$ such that the string $LE X$ does not appear.

On a blackboard, there are $10$ numbers written: $1, 2, \dots, 10$. Nahyun repeatedly performs the following operations. [b](Operation)[/b] Nahyun chooses two numbers from the 10 numbers on the blackboard that are not in a divisor-multiple relationship, erases them, and writes their GCD and LCM on the blackboard. If every two numbers on the blackboard form a divisor-multiple relationship, Nahyun stops the process. What is the maximum number of operations Nahyun can perform? (Note: $a, b$ are in a divisor-multiple relationship iff $a \mid b$ or $b \mid a$.)

$22$ light bulbs are given. Each light bulb is connected to exactly one switch, but a switch can be connected to one or more light bulbs. Find the least number of switches we should have such that we can turn on whatever number of light bulbs.

Consider a company of $n\ge 4$ people, where everyone knows at least one other person, but everyone knows at most $n-2$ of the others. Prove that we can sit four of these people at a round table such that all four of them know exactly one of their two neighbors. (Knowledge is mutual.)

Given a set $A=\{1;2;...;4044\}$. They color $2022$ numbers of them by white and the rest of them by black. With each $i\in A$, called the [b][i]important number[/i][/b] of $i$ be the number of all white numbers smaller than $i$ and black numbers larger than $i$. With every natural number $m$, find all positive integers $k$ that exist a way to color the numbers that can get $k$ important numbers equal to $m$.

For a special event, the five Vietnamese famous dishes including Phở, (Vietnamese noodle), Nem (spring roll), Bún Chả (grilled pork noodle), Bánh cuốn (stuffed pancake), and Xôi gà (chicken sticky rice) are the options for the main courses for the dinner of Monday, Tuesday, and Wednesday. Every dish must be used exactly one time. How many choices do we have?

All possible vertical lines $x = k$ and horizontal lines $y = m$ are drawn on the coordinate plane, where $k$ and $m$ are integers. Let's imagine that all these straight lines are black. A red straight line is also drawn, the equation of which is $19x+96y= c$. Let us denote by $M$ the number of segments of different lengths formed on the red line when intersecting with the black ones.(The ends of each segment are the intersection points of the red and black lines. There are no such intersection points inside the segment.) What values can $M$ take when $c$ changes?

[b]p1.[/b] What is the smallest positive integer $x$ such that $\frac{1}{x} <\sqrt{12011} - \sqrt{12006}$? [b]p2. [/b] Two soccer players run a drill on a $100$ foot by $300$ foot rectangular soccer eld. The two players start on two different corners of the rectangle separated by $100$ feet, then run parallel along the long edges of the eld, passing a soccer ball back and forth between them. Assume that the ball travels at a constant speed of $50$ feet per second, both players run at a constant speed of $30$ feet per second, and the players lead each other perfectly and pass the ball as soon as they receive it, how far has the ball travelled by the time it reaches the other end of the eld? [b]p3.[/b] A trapezoid $ABCD$ has $AB$ and $CD$ both perpendicular to $AD$ and $BC =AB + AD$. If $AB = 26$, what is $\frac{CD^2}{AD+CD}$ ? [b]p4.[/b] A hydrophobic, hungry, and lazy mouse is at $(0, 0)$, a piece of cheese at $(26, 26)$, and a circular lake of radius $5\sqrt2$ is centered at $(13, 13)$. What is the length of the shortest path that the mouse can take to reach the cheese that also does not also pass through the lake? [b]p5.[/b] Let $a, b$, and $c$ be real numbers such that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 3$. If $a^5 + b^5 + c^5\ne 0$, compute $\frac{(a^3+b^3+c^3)(a^4+b^4+c^4)}{a^5+b^5+c^5}$. [b]p6. [/b] Let $S$ be the number of points with integer coordinates that lie on the line segment with endpoints $\left( 2^{2^2}, 4^{4^4}\right)$ and $\left(4^{4^4}, 0\right)$. Compute $\log_2 (S - 1)$. [b]p7.[/b] For a positive integer $n$ let $f(n)$ be the sum of the digits of $n$. Calculate $$f(f(f(2^{2006})))$$ [b]p8.[/b] If $a_1, a_2, a_3, a_4$ are roots of $x^4 - 2006x^3 + 11x + 11 = 0$, find $|a^3_1 + a^3_2 + a^3_3 + a^3_4|$. [b]p9.[/b] A triangle $ABC$ has $M$ and $N$ on sides $BC$ and $AC$, respectively, such that $AM$ and $BN$ intersect at $P$ and the areas of triangles $ANP$, $APB$, and $PMB$ are $5$, $10$, and $8$ respectively. If $R$ and $S$ are the midpoints of $MC$ and $NC$, respectively, compute the area of triangle $CRS$. [b]p10.[/b] Jack's calculator has a strange button labelled ''PS.'' If Jack's calculator is displaying the positive integer $n$, pressing PS will cause the calculator to divide $n$ by the largest power of $2$ that evenly divides $n$, and then adding 1 to the result and displaying that number. If Jack randomly chooses an integer $k$ between $ 1$ and $1023$, inclusive, and enters it on his calculator, then presses the PS button twice, what is the probability that the number that is displayed is a power of $2$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two types of pieces, bishops and rooks, are to be placed on a $10\times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to [i]attack[/i] a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on that diagonal; a rook $R$ is said to attack a piece $P$ if $R$ and $P$ are on the same row or column and there are no pieces between $R$ and $P$ on that row or column. A piece $P$ is [i]chocolate[/i] if no other piece $Q$ attacks $P$. What is the maximum number of chocolate pieces there may be, after placing some pieces on the chessboard? [i]Proposed by José Alejandro Reyes González[/i]

Each edge of a graph with $15$ vertices is colored either red or blue in such a way that no three vertices are pairwise connected with edges of the same color. Determine the largest possible number of edges in the graph.

Soma has a tower of $63$ bricks , consisting of $6$ levels. On the $k$-th level from the top, there are $2k-1$ bricks (where $k = 1, 2, 3, 4, 5, 6$), and every brick which is not on the lowest level lies on precisely $2$ smaller bricks (which lie one level below) - see the figure. Soma takes away $7$ bricks from the tower, one by one. He can only remove a brick if there is no brick lying on it. In how many ways can he do this, if the order of removals is considered as well? [img]https://cdn.artofproblemsolving.com/attachments/b/6/4b0ce36df21fba89708dd5897c43a077d86b5e.png[/img]

Farhad has made a machine. When the machine starts, it prints some special numbers. The property of this machine is that for every positive integer $n$, it prints exactly one of the numbers $n,2n,3n$. We know that the machine prints $2$. Prove that it doesn't print $13824$.

Let $s \geq 2$ and $n \geq k \geq 2$ be integes, and let $A$ be a subset of $\{1, 2, . . . , n\}^k$ of size at least $2sk^2n^{k-2}$ such that any two members of $A$ share some entry. Prove that there are an integer $p \leq k$ and $s+2$ members $A_1, A_2, . . . , A_{s+2}$ of $A$ such that $A_i$ and $A_j$ share the $p$-th entry alone, whenever $i$ and $j$ are distinct. [i]Miroslav Marinov, Bulgaria[/i]

Find all natural numbers n with the following property: there exists a permutation $(i_1,i_2,\ldots,i_n)$ of the numbers $1,2,\ldots,n$ such that, if on the circular table there are $n$ people seated and for all $k=1,2,\ldots,n$ the $k$-th person is moving $i_n$ places in the right, all people will sit on different places. [i]V. Drenski[/i]

Alice and Bob are going to play a game called extra tricky double rock paper scissors (ETDRPS). In ETDRPS, each player simultaneously selects [i]two [/i] moves, one for his or her right hand, and one for his or her left hand. Whereas Alice can play rock, paper, or scissors, Bob is only allowed to play rock or scissors. After revealing their moves, the players compare right hands and left hands separately. Alice wins if she wins [i]strictly [/i] more hands than Bob. Otherwise, Bob wins. For example, if Alice and Bob were to both play rock with their right hands and scissors with their left hands, then both hands would be tied, so Bob would win the game. However, if Alice were to instead play rock with both hands, then Alice would win the left hand. The right hand would still be tied, so Alice would win the game. Assuming both players play optimally, compute the probability that Alice will win the game.

Given a positive integer $n$, there are $n$ boxes $B_1,...,B_n$. The following procedure can be used to add balls. $$\text{(Procedure) Chosen two positive integers }n\geq i\geq j\geq 1\text{, we add one ball each to the boxes }B_k\text{ that }i\geq k\geq j.$$ For positive integers $x_1,...,x_n$ let $f(x_1,...,x_n)$ be the minimum amount of procedures to get all boxes have its amount of balls to be a multiple of 3, starting with $x_i$ balls for $B_i(i=1,...,n)$. Find the largest possible value of $f(x_1,...,x_n)$. (If $x_1,...,x_n$ are all multiples of 3, $f(x_1,...,x_n)=0$.)