[b]p13.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $1$. Let the unit circles centered at $A$, $B$, and $C$ be $\Omega_A$, $\Omega_B$, and $\Omega_C$, respectively. Then, let $\Omega_A$ and $\Omega_C$ intersect again at point $D$, and $\Omega_B$ and $\Omega_C$ intersect again at point $E$. Line $BD$ intersects $\Omega_B$ at point $F$ where $F$ lies between $B$ and $D$, and line $AE$ intersects $\Omega_A$ at $G$ where $G$ lies between $A$ and $E$. $BD$ and $AE$ intersect at $H$. Finally, let $CH$ and $FG$ intersect at $I$. Compute $IH$. [b]p14.[/b] Suppose Bob randomly fills in a $45 \times 45$ grid with the numbers from $1$ to $2025$, using each number exactly once. For each of the $45$ rows, he writes down the largest number in the row. Of these $45$ numbers, he writes down the second largest number. The probability that this final number is equal to $2023$ can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Compute the value of $p$. [b]p15.[/b] $f$ is a bijective function from the set $\{0, 1, 2, ..., 11\}$ to $\{0, 1, 2, ... , 11\}$, with the property that whenever $a$ divides $b$, $f(a)$ divides $f(b)$. How many such $f$ are there? [i]A bijective function maps each element in its domain to a distinct element in its range. [/i] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A chocolate bar in the shape of an equilateral triangle with side of the length $n$, consists of triangular chips with sides of the length $1$, parallel to sides of the bar. Two players take turns eating up the chocolate. Each player breaks off a triangular piece (along one of the lines), eats it up and passes leftovers to the other player (as long as bar contains more than one chip, the player is not allowed to eat it completely). A player who has no move or leaves exactly one chip to the opponent, loses. For each $n$, find who has a winning strategy.

[b]p1.[/b] Zion, RJ, Cam, and Tre decide to start learning languages. The four most popular languages that Duke offers are Spanish, French, Latin, and Korean. If each friend wants to learn exactly three of these four languages, how many ways can they pick courses such that they all attend at least one course together? [b]p2. [/b] Suppose we wrote the integers between $0001$ and $2019$ on a blackboard as such: $$000100020003 · · · 20182019.$$ How many $0$’s did we write? [b]p3.[/b] Duke’s basketball team has made $x$ three-pointers, $y$ two-pointers, and $z$ one-point free throws, where $x, y, z$ are whole numbers. Given that $3|x$, $5|y$, and $7|z$, find the greatest number of points that Duke’s basketball team could not have scored. [b]p4.[/b] Find the minimum value of $x^2 + 2xy + 3y^2 + 4x + 8y + 12$, given that $x$ and $y$ are real numbers. Note: calculus is not required to solve this problem. [b]p5.[/b] Circles $C_1, C_2$ have radii $1, 2$ and are centered at $O_1, O_2$, respectively. They intersect at points $ A$ and $ B$, and convex quadrilateral $O_1AO_2B$ is cyclic. Find the length of $AB$. Express your answer as $x/\sqrt{y}$ , where $x, y$ are integers and $y$ is square-free. [b]p6.[/b] An infinite geometric sequence $\{a_n\}$ has sum $\sum_{n=0}^{\infty} a_n = 3$. Compute the maximum possible value of the sum $\sum_{n=0}^{\infty} a_{3n} $. [b]p7.[/b] Let there be a sequence of numbers $x_1, x_2, x_3,...$ such that for all $i$, $$x_i = \frac{49}{7^{\frac{i}{1010}} + 49}.$$ Find the largest value of $n$ such that $$\left\lfloor \sum_{i=1}{n} x_i \right\rfloor \le 2019.$$ [b]p8.[/b] Let $X$ be a $9$-digit integer that includes all the digits $1$ through $9$ exactly once, such that any $2$-digit number formed from adjacent digits of $X$ is divisible by $7$ or $13$. Find all possible values of $X$. [b]p9.[/b] Two $2025$-digit numbers, $428\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}571$ and $571\underbrace{\hbox{99... 99}}_{\hbox{2019 \,\, 9's}}428$ , form the legs of a right triangle. Find the sum of the digits in the hypotenuse. [b]p10.[/b] Suppose that the side lengths of $\vartriangle ABC$ are positive integers and the perimeter of the triangle is $35$. Let $G$ the centroid and $I$ be the incenter of the triangle. Given that $\angle GIC = 90^o$ , what is the length of $AB$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Some of the people attending a meeting greet each other. Let $n$ be the number of people who greet an odd number of people. Prove that $n$ is even.

Juan and Pedro play alternately on the given grid. Each one in turn traces $1$ to $5$ routes different from the ones outlined above, that join $A$ with $B$, moving only to the right and upwards on the grid lines. Juan starts playing. The one who traces a route that passes through $C$ or $D$ loses. Prove that one of them can win regardless of how the other plays. [img]https://cdn.artofproblemsolving.com/attachments/2/7/6a24ca9c4c1c710bd41e44bfcab3d3b61b6d4f.png[/img]

A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings.

Determine the smallest positive integer n for that there are positive integers $x_1, x_2,. . . , x_n$ so that each natural number from 1001 to 2021 inclusive can be written as sum of one or more different terms $x_i$ (i = 1, 2,..., n).

What minimal number of numbers from the set $\{1,2,...,1982\}$ should be deleted to provide the property: [i]none of the remained numbers equals to the product of two other remained numbers[/i]?

On a party with $99$ guests, hosts Ann and Bob play a game (the hosts are not regarded as guests). There are $99$ chairs arranged in a circle; initially, all guests hang around those chairs. The hosts take turns alternately. By a turn, a host orders any standing guest to sit on an unoccupied chair $c$. If some chair adjacent to $c$ is already occupied, the same host orders one guest on such chair to stand up (if both chairs adjacent to $c$ are occupied, the host chooses exactly one of them). All orders are carried out immediately. Ann makes the first move; her goal is to fulfill, after some move of hers, that at least $k$ chairs are occupied. Determine the largest $k$ for which Ann can reach the goal, regardless of Bob's play.

Prove that the edges of a finite simple planar graph (with no loops, multiple edges) may be oriented in such a way that at most three fourths of the total number of dges of any cycle share the same orientation. Moreover, show that this is the best global bound possible. Comment: The actual problem in the TST asked to prove that the edges can be $2$-colored so that the same conclusion holds. Under this circumstances, the problem is wrong and a counterexample was found in the contest by Marius Tiba.

[b]p1.[/b] Evaluate $6^4 +5^4 +3^4 +2^4$. [b]p2.[/b] What digit is most frequent between $1$ and $1000$ inclusive? [b]p3.[/b] Let $n = gcd \, (2^2 \cdot 3^3 \cdot 4^4,2^4 \cdot 3^3 \cdot 4^2)$. Find the number of positive integer factors of $n$. [b]p4.[/b] Suppose $p$ and $q$ are prime numbers such that $13p +5q = 91$. Find $p +q$. [b]p5.[/b] Let $x = (5^3 -5)(4^3 -4)(3^3 -3)(2^3 -2)(1^3 -1)$. Evaluate $2018^x$ . [b]p6.[/b] Liszt the lister lists all $24$ four-digit integers that contain each of the digits $1,2,3,4$ exactly once in increasing order. What is the sum of the $20$th and $18$th numbers on Liszt’s list? [b]p7.[/b] Square $ABCD$ has center $O$. Suppose $M$ is the midpoint of $AB$ and $OM +1 =OA$. Find the area of square $ABCD$. [b]p8.[/b] How many positive $4$-digit integers have at most $3$ distinct digits? [b]p9.[/b] Find the sumof all distinct integers obtained by placing $+$ and $-$ signs in the following spaces $$2\_3\_4\_5$$ [b]p10.[/b] In triangle $ABC$, $\angle A = 2\angle B$. Let $I$ be the intersection of the angle bisectors of $B$ and $C$. Given that $AB = 12$, $BC = 14$,and $C A = 9$, find $AI$ . [b]p11.[/b] You have a $3\times 3\times 3$ cube in front of you. You are given a knife to cut the cube and you are allowed to move the pieces after each cut before cutting it again. What is the minimumnumber of cuts you need tomake in order to cut the cube into $27$ $1\times 1\times 1$ cubes? p12. How many ways can you choose $3$ distinct numbers fromthe set $\{1,2,3,...,20\}$ to create a geometric sequence? [b]p13.[/b] Find the sum of all multiples of $12$ that are less than $10^4$ and contain only $0$ and $4$ as digits. [b]p14.[/b] What is the smallest positive integer that has a different number of digits in each base from $2$ to $5$? [b]p15.[/b] Given $3$ real numbers $(a,b,c)$ such that $$\frac{a}{b +c}=\frac{b}{3a+3c}=\frac{c}{a+3b},$$ find all possible values of $\frac{a +b}{c}$. [b]p16.[/b] Let S be the set of lattice points $(x, y, z)$ in $R^3$ satisfying $0 \le x, y, z \le 2$. How many distinct triangles exist with all three vertices in $S$? [b]p17.[/b] Let $\oplus$ be an operator such that for any $2$ real numbers $a$ and $b$, $a \oplus b = 20ab -4a -4b +1$. Evaluate $$\frac{1}{10} \oplus \frac19 \oplus \frac18 \oplus \frac17 \oplus \frac16 \oplus \frac15 \oplus \frac14 \oplus \frac13 \oplus \frac12 \oplus 1.$$ [b]p18.[/b] A function $f :N \to N$ satisfies $f ( f (x)) = x$ and $f (2f (2x +16)) = f \left(\frac{1}{x+8} \right)$ for all positive integers $x$. Find $f (2018)$. [b]p19.[/b] There exists an integer divisor $d$ of $240100490001$ such that $490000 < d < 491000$. Find $d$. [b]p20.[/b] Let $a$ and $b$ be not necessarily distinct positive integers chosen independently and uniformly at random from the set $\{1,2, 3, ... ,511,512\}$. Let $x = \frac{a}{b}$ . Find the probability that $(-1)^x$ is a real number. [b]p21[/b]. In $\vartriangle ABC$ we have $AB = 4$, $BC = 6$, and $\angle ABC = 135^o$. $\angle ABC$ is trisected by rays $B_1$ and $B_2$. Ray $B_1$ intersects side $C A$ at point $F$, and ray $B_2$ intersects side $C A$ at point $G$. What is the area of $\vartriangle BFG$? [b]p22.[/b] A level number is a number which can be expressed as $x \cdot \lfloor x \rfloor \cdot \lceil x \rceil$ where $x$ is a real number. Find the number of positive integers less than or equal to $1000$ which are also level numbers. [b]p23.[/b] Triangle $\vartriangle ABC$ has sidelengths $AB = 13$, $BC = 14$, $C A = 15$ and circumcenter $O$. Let $D$ be the intersection of $AO$ and $BC$. Compute $BD/DC$. [b]p24.[/b] Let $f (x) = x^4 -3x^3 +2x^2 +5x -4$ be a quartic polynomial with roots $a,b,c,d$. Compute $$\left(a+1 +\frac{1}{a} \right)\left(b+1 +\frac{1}{b} \right)\left(c+1 +\frac{1}{c} \right)\left(d+1 +\frac{1}{d} \right).$$ [b]p25.[/b] Triangle $\vartriangle ABC$ has centroid $G$ and circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 2018$, $BD =20$, and $CD = 18$, find the area of triangle $\vartriangle DOG$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An odd positive integer $n$ is called pretty if there exists at least one permutation $a_1, a_2,..., a_n$, of $1,2,...,n$, such that all $n$ sums $a_1-a_2+a_3-...+a_n$, $a_2-a_3+a_4-...+a_1$,..., $a_n-a_1+a_2-...+a_{n-1}$ are positive. Find all pretty integers.

Each of the $8$ vertices of a given cube is given a value $1$ or $-1$. Each of the $6$ faces is given the value of product of its four vertices. Let $A$ be the sum of all the $14$ values. Which are the possible values of $A$?

A grid consists of $2n$ vertical and $2n$ horizontal lines, each group disposed at equal distances. The lines are all painted in red and black, such that exactly $n$ vertical and $n$ horizontal lines are red. Find the smallest $n$ such that for any painting satisfying the above condition, there is a square formed by the intersection of two vertical and two horizontal lines, all of the same colour.

Let $n > 1$ be an integer. In a circular arrangement of $n$ lamps $L_0, \ldots, L_{n-1},$ each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \ldots .$ If $L_{j-1}$ ($j$ is taken mod $n$) is ON then $Step_j$ changes the state of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If $L_{j-1}$ is OFF then $Step_j$ does not change anything at all. Show that: (i) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again, (ii) If $n$ has the form $2^k$ then all the lamps are ON after $n^2-1$ steps, (iii) If $n$ has the form $2^k + 1$ then all lamps are ON after $n^2 - n + 1$ steps.

There is a complete graph $G$ with $4027$ vertices drawn on the whiteboard. Ivan paints all the edges by red or blue colour. Find all ordered pairs $(r, b)$ such that Ivan can paint the edges so that every vertex is connected to exactly $r$ red edges and $b$ blue edges.

Can the squares of a $1990 \times 1990$ chessboard be colored black or white so that half the squares in each row and column are black and cells symmetric with respect to the center are of opposite color?

The numbers $\frac 32$, $\frac 43$ and $\frac 65$ are intially written on the blackboard. A move consists of erasing one of the numbers from the blackboard, call it $a$, and replacing it with $bc-b-c+2$, where $b,c$ are the other two numbers currently written on the blackboard. Is it possible that $\frac{1000}{999}$ would eventually appear on the blackboard? What about $\frac{113}{108}$? [i] (Andrei Bâra)[/i]

Let $A$ and $B$ be two subsets of $S = \{1, 2, . . . , 2000\}$ with $|A| \cdot |B| \geq 3999$. For a set $X$ , let $X-X$ denotes the set $\{s-t | s, t \in X, s \not = t\}$. Prove that $(A-A) \cap (B-B)$ is nonempty.

In a country, mathematicians chose an $\alpha> 2$ and issued coins in denominations of 1 ruble, as well as $\alpha ^k$ rubles for each positive integer k. $\alpha$ was chosen so that the value of each coins, except the smallest, was irrational. Is it possible that any natural number of rubles can be formed with at most 6 of each denomination of coins?

The [i]giraffe[/i] is a chess piece that moves $4$ squares in one direction and then a box in a perpendicular direction. What is the smallest value of $n$ such that the giraffe that starts from a corner on an $n \times n$ board can visit all the squares of said board?

For each $n$ find the number of ways one can put the numbers $\{1,2,3,...,n\}$ numbers on the circle, such that if for any $4$ numbers $a,b,c,d$ where $n|a+b-c-d$. The segments joining $a,b$ and $c,d$ do not meet inside the circle. (Two ways are said to be identical , if one can be obtained from rotaiting the other)

How many ordered triplets $(a, b, c)$ of positive integers such that $30a + 50b + 70c \leq 343$.

In every $1\times1$ square of an $m\times n$ table we have drawn one of two diagonals. Prove that there is a path including these diagonals either from left side to the right side, or from the upper side to the lower side.

In the table are written the positive integers $1, 2,3,...,2018$. John and Mary have the ability to make together the following move: [i]They select two of the written numbers in the table, let $a,b$ and they replace them with the numbers $5a-2b$ and $3a-4b$.[/i] John claims that after a finite number of such moves, it is possible to triple all the numbers in the table, e.g. have the numbers: $3, 6, 9,...,6054$. Mary thinks a while and replies that this is not possible. Who of them is right?