Found problems: 67
DMM Team Rounds, 2015
[b]p1.[/b] Let $U = \{-2, 0, 1\}$ and $N = \{1, 2, 3, 4, 5\}$. Let $f$ be a function that maps $U$ to $N$. For any $x \in U$, $x + f(x) + xf(x)$ is an odd number. How many $f$ satisfy the above statement?
[b]p2.[/b] Around a circle are written all of the positive integers from $ 1$ to $n$, $n \ge 2$ in such a way that any two adjacent integers have at least one digit in common in their decimal expressions. Find the smallest $n$ for which this is possible.
[b]p3.[/b] Michael loses things, especially his room key. If in a day of the week he has $n$ classes he loses his key with probability $n/5$. After he loses his key during the day he replaces it before he goes to sleep so the next day he will have a key. During the weekend(Saturday and Sunday) Michael studies all day and does not leave his room, therefore he does not lose his key. Given that on Monday he has 1 class, on Tuesday and Thursday he has $2$ classes and that on Wednesday and Friday he has $3$ classes, what is the probability that loses his key at least once during a week?
[b]p4.[/b] Given two concentric circles one with radius $8$ and the other $5$. What is the probability that the distance between two randomly chosen points on the circles, one from each circle, is greater than $7$ ?
[b]p5.[/b] We say that a positive integer $n$ is lucky if $n^2$ can be written as the sum of $n$ consecutive positive integers. Find the number of lucky numbers strictly less than $2015$.
[b]p6.[/b] Let $A = \{3^x + 3^y + 3^z|x, y, z \ge 0, x, y, z \in Z, x < y < z\}$. Arrange the set $A$ in increasing order. Then what is the $50$th number? (Express the answer in the form $3^x + 3^y + 3^z$).
[b]p7.[/b] Justin and Oscar found $2015$ sticks on the table. I know what you are thinking, that is very curious. They decided to play a game with them. The game is, each player in turn must remove from the table some sticks, provided that the player removes at least one stick and at most half of the sticks on the table. The player who leaves just one stick on the table loses the game. Justin goes first and he realizes he has a winning strategy. How many sticks does he have to take off to guarantee that he will win?
[b]p8.[/b] Let $(x, y, z)$ with $x \ge y \ge z \ge 0$ be integers such that $\frac{x^3+y^3+z^3}{3} = xyz + 21$. Find $x$.
[b]p9.[/b] Let $p < q < r < s$ be prime numbers such that $$1 - \frac{1}{p} -\frac{1}{q} -\frac{1}{r}- \frac{1}{s}= \frac{1}{pqrs}.$$ Find $p + q + r + s$.
[b]p10.[/b] In ”island-land”, there are $10$ islands. Alex falls out of a plane onto one of the islands, with equal probability of landing on any island. That night, the Chocolate King visits Alex in his sleep and tells him that there is a mountain of chocolate on one of the islands, with equal probability of being on each island. However, Alex has become very fat from eating chocolate his whole life, so he can’t swim to any of the other islands. Luckily, there is a teleporter on each island. Each teleporter will teleport Alex to exactly one other teleporter (possibly itself) and each teleporter gets teleported to by exactly one teleporter. The configuration of the teleporters is chosen uniformly at random from all possible configurations of teleporters satisfying these criteria. What is the probability that Alex can get his chocolate?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Team Rounds, 2007
[b]p1.[/b] If $x + z = v$, $w + z = 2v$, $z - w = 2y$, and $y \ne 0$, compute the value of $$\left(x + y +\frac{x}{y} \right)^{101}.$$
[b]p2. [/b]Every minute, a snail picks one cardinal direction (either north, south, east, or west) with equal probability and moves one inch in that direction. What is the probability that after four minutes the snail is more than three inches away from where it started?
[b]p3.[/b] What is the probability that a point chosen randomly from the interior of a cube is closer to the cube’s center than it is to any of the cube’s eight vertices?
[b]p4.[/b] Let $ABCD$ be a rectangle where $AB = 4$ and $BC = 3$. Inscribe circles within triangles $ABC$ and $ACD$. What is the distance between the centers of these two circles?
[b]p5.[/b] $C$ is a circle centered at the origin that is tangent to the line $x - y\sqrt3 = 4$. Find the radius of $C$.
[b]p6.[/b] I have a fair $100$-sided die that has the numbers $ 1$ through $100$ on its sides. What is the probability that if I roll this die three times that the number on the first roll will be greater than or equal to the sum of the two numbers on the second and third rolls?
[b]p7. [/b] List all solutions $(x, y, z)$ of the following system of equations with x, y, and z positive real numbers:
$$x^2 + y^2 = 16$$
$$x^2 + z^2 = 4 + xz$$
$$y^2 + z^2 = 4 + yz\sqrt3$$
[b]p8.[/b] $A_1A_2A_3A_4A_5A_6A_7$ is a regular heptagon ($7$ sided-figure) centered at the origin where $A_1 =
(\sqrt[91]{6}, 0)$. $B_1B_2B_3... B_{13}$ is a regular triskaidecagon ($13$ sided-figure) centered at the origin where $B_1 =(0,\sqrt[91]{41})$. Compute the product of all lengths $A_iB_j$ , where $i$ ranges between $1$ and $7$, inclusive, and $j$ ranges between $1$ and $13$, inclusive.
[b]p9.[/b] How many three-digit integers are there such that one digit of the integer is exactly two times a digit of the integer that is in a different place than the first? (For example, $100$, $122$, and $124$ should be included in the count, but $42$ and $130$ should not.)
[b]p10.[/b] Let $\alpha$ and $\beta$ be the solutions of the quadratic equation $$x^2 - 1154x + 1 = 0.$$ Find $\sqrt[4]{\alpha}+\sqrt[4]{\beta}$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Team Rounds, 2021
[b]p1. [/b] In basketball, teams can score $1, 2$, or $3$ points each time. Suppose that Duke basketball have scored $8$ points so far. What is the total number of possible ways (ordered) that they have scored?
For example, $(1, 2, 2, 2, 1)$,$(1, 1, 2, 2, 2)$ are two different ways.
[b]p2.[/b] All the positive integers that are coprime to $2021$ are grouped in increasing order, such that the nth group contains $2n - 1$ numbers. Hence the first three groups are $\{1\}$, $\{2, 3, 4\}$, $\{5, 6, 7, 8, 9\}$. Suppose that $2022$ belongs to the $k$th group. Find $k$.
[b]p3.[/b] Let $A = (0, 0)$ and $B = (3, 0)$ be points in the Cartesian plane. If $R$ is the set of all points $X$ such that $\angle AXB \ge 60^o$ (all angles are between $0^o$ and $180^o$), find the integer that is closest to the area of $R$.
[b]p4.[/b] What is the smallest positive integer greater than $9$ such that when its left-most digit is erased, the resulting number is one twenty-ninth of the original number?
[b]p5. [/b] Jonathan is operating a projector in the cartesian plane. He sets up $2$ infinitely long mirrors represented by the lines $y = \tan(15^o)x$ and $y = 0$, and he places the projector at $(1, 0)$ pointed perpendicularly to the $x$-axis in the positive $y$ direction. Jonathan furthermore places a screen on one of the mirrors such that light from the projector reflects off the mirrors a total of three times before hitting the screen. Suppose that the coordinates of the screen is $(a, b)$. Find $10a^2 + 5b^2$.
[b]p6.[/b] Dr Kraines has a cube of size $5 \times 5 \times 5$, which is made from $5^3$ unit cubes. He then decides to choose $m$ unit cubes that have an outside face such that any two different cubes don’t share a common vertex. What is the maximum value of $m$?
[b]p7.[/b] Let $a_n = \tan^{-1}(n)$ for all positive integers $n$. Suppose that $$\sum_{k=4}^{\infty}(-1)^{\lfloor \frac{k}{2} \rfloor +1} \tan(2a_k)$$ is equals to $a/b$ , where $a, b$ are relatively prime. Find $a + b$.
[b]p8.[/b] Rishabh needs to settle some debts. He owes $90$ people and he must pay \$ $(101050 + n)$ to the $n$th person where $1 \le n \le 90$. Rishabh can withdraw from his account as many coins of values \$ $2021$ and \$ $x$ for some fixed positive integer $x$ as is necessary to pay these debts. Find the sum of the four least values of $x$ so that there exists a person to whom Rishabh is unable to pay the exact amount owed using coins.
[b]p9.[/b] A frog starts at $(1, 1)$. Every second, if the frog is at point $(x, y)$, it moves to $(x + 1, y)$ with probability $\frac{x}{x+y}$ and moves to $(x, y + 1)$ with probability $\frac{y}{x+y}$ . The frog stops moving when its $y$ coordinate is $10$. Suppose the probability that when the frog stops its $x$-coordinate is strictly less than $16$, is given by $m/n$ where $m, n$ are positive integers that are relatively prime. Find $m + n.$
[b]p10.[/b] In the triangle $ABC$, $AB = 585$, $BC = 520$, $CA = 455$. Define $X, Y$ to be points on the segment $BC$. Let $Z \ne A$ be the intersection of $AY$ with the circumcircle of $ABC$. Suppose that $XZ$ is parallel to $AC$ and the circumcircle of $XYZ$ is tangent to the circumcircle of $ABC$ at $Z$. Find the length of $XY$ .
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Team Rounds, 2011
[b]p1.[/b] How many primes $p < 100$ satisfy $p = a^2 + b^2$ for some positive integers $a$ and $b$?
[b]p2. [/b] For $a < b < c$, there exists exactly one Pythagorean triple such that $a + b + c = 2000$. Find $a + c - b$.
[b]p3.[/b] Five points lie on the surface of a sphere of radius $ 1$ such that the distance between any two points is at least $\sqrt2$. Find the maximum volume enclosed by these five points.
[b]p4.[/b] $ABCDEF$ is a convex hexagon with $AB = BC = CD = DE = EF = FA = 5$ and $AC = CE = EA = 6$. Find the area of $ABCDEF$.
[b]p5.[/b] Joe and Wanda are playing a game of chance. Each player rolls a fair $11$-sided die, whose sides are labeled with numbers $1, 2, ... , 11$. Let the result of the Joe’s roll be $X$, and the result of Wanda’s roll be $Y$ . Joe wins if $XY$ has remainder $ 1$ when divided by $11$, and Wanda wins otherwise. What is the probability that Joe wins?
[b]p6.[/b] Vivek picks a number and then plays a game. At each step of the game, he takes the current number and replaces it with a new number according to the following rule: if the current number $n$ is divisible by $3$, he replaces $n$ with $\frac{n}{3} + 2$, and otherwise he replaces $n$ with $\lfloor 3 \log_3 n \rfloor$. If he starts with the number $3^{2011}$, what number will he have after $2011$ steps?
Note that $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.
[b]p7.[/b] Define a sequence an of positive real numbers with a$_1 = 1$, and $$a_{n+1} =\frac{4a^2_n - 1}{-2 + \frac{4a^2_n -1}{-2+ \frac{4a^2_n -1}{-2+...}}}.$$
What is $a_{2011}$?
[b]p8.[/b] A set $S$ of positive integers is called good if for any $x, y \in S$ either $x = y$ or $|x - y| \ge 3$. How many subsets of $\{1, 2, 3, ..., 13\}$ are good? Include the empty set in your count.
[b]p9.[/b] Find all pairs of positive integers $(a, b)$ with $a \le b$ such that $10 \cdot lcm \, (a, b) = a^2 + b^2$. Note that $lcm \,(m, n)$ denotes the least common multiple of $m$ and $n$.
[b]p10.[/b] For a natural number $n$, $g(n)$ denotes the largest odd divisor of $n$. Find $$g(1) + g(2) + g(3) + ... + g(2^{2011})$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Devil Rounds, 2005
[b]p1.[/b] Let $a$ and $b$ be complex numbers such that $a^3 + b^3 = -17$ and $a + b = 1$. What is the value of $ab$?
[b]p2.[/b] Let $AEFB$ be a right trapezoid, with $\angle AEF = \angle EAB = 90^o$. The two diagonals $EB$ and $AF$ intersect at point $D$, and $C$ is a point on $AE$ such that $AE \perp DC$. If $AB = 8$ and $EF = 17$, what is the lenght of $CD$?
[b]p3.[/b] How many three-digit numbers $abc$ (where each of $a$, $b$, and $c$ represents a single digit, $a \ne 0$) are there such that the six-digit number $abcabc$ is divisible by $2$, $3$, $5$, $7$, $11$, or $13$?
[b]p4.[/b] Let $S$ be the sum of all numbers of the form $\frac{1}{n}$ where $n$ is a postive integer and $\frac{1}{n}$ terminales in base $b$, a positive integer. If $S$ is $\frac{15}{8}$, what is the smallest such $b$?
[b]p5.[/b] Sysyphus is having an birthday party and he has a square cake that is to be cut into $25$ square pieces. Zeus gets to make the first straight cut and messes up badly. What is the largest number of pieces Zeus can ruin (cut across)? Diagram?
[b]p6.[/b] Given $(9x^2 - y^2)(9x^2 + 6xy + y^2) = 16$ and $3x + y = 2$. Find $x^y$.
[b]p7.[/b] What is the prime factorization of the smallest integer $N$ such that $\frac{N}{2}$ is a perfect square, $\frac{N}{3}$ is a perfect cube, $\frac{N}{5}$ is a perfect fifth power?
[b]p8.[/b] What is the maximum number of pieces that an spherical watermelon can be divided into with four straight planar cuts?
[b]p9.[/b] How many ordered triples of integers $(x,y,z)$ are there such that $0 \le x, y, z \le 100$ and $$(x - y)^2 + (y - z)^2 + (z - x)^2 \ge (x + y - 2z) + (y + z - 2x)^2 + (z + x - 2y)^2.$$
[b]p10.[/b] Find all real solutions to $(2x - 4)^2 + (4x - 2)^3 = (4x + 2x - 6)^3$.
[b]p11.[/b] Let $f$ be a function that takes integers to integers that also has $$f(x)=\begin{cases} x - 5\,\, if \,\, x \ge 50 \\ f (f (x + 12)) \,\, if \,\, x < 50 \end{cases}$$ Evaluate $f (2) + f (39) + f (58).$
[b]p12.[/b] If two real numbers are chosen at random (i.e. uniform distribution) from the interval $[0,1]$, what is the probability that theit difference will be less than $\frac35$?
[b]p13.[/b] Let $a$, $b$, and $c$ be positive integers, not all even, such that $a < b$, $b = c - 2$, and $a^2 + b^2 = c^2$. What is the smallest possible value for $c$?
[b]p14.[/b] Let $ABCD$ be a quadrilateral whose diagonals intersect at $O$. If $BO = 8$, $OD = 8$, $AO = 16$, $OC = 4$, and $AB = 16$, then find $AD$.
[b]p15.[/b] Let $P_0$ be a regular icosahedron with an edge length of $17$ units. For each nonnegative integer $n$, recursively construct $P_{n+1}$ from Pn by performing the following procedure on each face of $P_n$: glue a regular tetrahedron to that face such that three of the vertices of the tetrahedron are the midpoints of the three adjacent edges of the face, and the last vertex extends outside of $P_n$. Express the number of square units in the surface area of $P_{17}$ in the form $$\frac{u^v\cdot w \sqrt{x}}{y^z}$$ , where $u, v, w, x, y$, and $z$ are integers, all greater than or equal to $2$, that satisfy the following conditions: the only perfect square that evenly divides $x$ is $1$, the GCD of $u$ and y is $1$, and neither $u$ nor $y$ divides $w$. Answers written in any other form will not be considered correct!
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Team Rounds, 2017
[b]p1.[/b] What is the maximum possible value of $m$ such that there exist $m$ integers $a_1, a_2, ..., a_m$ where all the decimal representations of $a_1!, a_2!, ..., a_m!$ end with the same amount of zeros?
[b]p2.[/b] Let $f : R \to R$ be a function such that $f(x) + f(y^2) = f(x^2 + y)$, for all $x, y \in R$. Find the sum of all possible $f(-2017)$.
[b]p3. [/b] What is the sum of prime factors of $1000027$?
[b]p4.[/b] Let $$\frac{1}{2!} +\frac{2}{3!} + ... +\frac{2016}{2017!} =\frac{n}{m},$$ where $n, m$ are relatively prime. Find $(m - n)$.
[b]p5.[/b] Determine the number of ordered pairs of real numbers $(x, y)$ such that $\sqrt[3]{3 - x^3 - y^3} =\sqrt{2 - x^2 - y^2}$
[b]p6.[/b] Triangle $\vartriangle ABC$ has $\angle B = 120^o$, $AB = 1$. Find the largest real number $x$ such that $CA - CB > x$ for all possible triangles $\vartriangle ABC$.
[b]p7. [/b]Jung and Remy are playing a game with an unfair coin. The coin has a probability of $p$ where its outcome is heads. Each round, Jung and Remy take turns to flip the coin, starting with Jung in round $ 1$. Whoever gets heads first wins the game. Given that Jung has the probability of $8/15$ , what is the value of $p$?
[b]p8.[/b] Consider a circle with $7$ equally spaced points marked on it. Each point is $ 1$ unit distance away from its neighbors and labelled $0,1,2,...,6$ in that order counterclockwise. Feng is to jump around the circle, starting at the point $0$ and making six jumps counterclockwise with distinct lengths $a_1, a_2, ..., a_6$ in a way such that he will land on all other six nonzero points afterwards. Let $s$ denote the maximum value of $a_i$. What is the minimum possible value of $s$?
[b]p9. [/b]Justin has a $4 \times 4 \times 4$ colorless cube that is made of $64$ unit-cubes. He then colors $m$ unit-cubes such that none of them belong to the same column or row of the original cube. What is the largest possible value of $m$?
[b]p10.[/b] Yikai wants to know Liang’s secret code which is a $6$-digit integer $x$. Furthermore, let $d(n)$ denote the digital sum of a positive integer $n$. For instance, $d(14) = 5$ and $d(3) = 3$. It is given that $$x + d(x) + d(d(x)) + d(d(d(x))) = 999868.$$ Please find $x$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Individual Rounds, 2012
[b]p1.[/b] Vivek has three letters to send out. Unfortunately, he forgets which letter is which after sealing the envelopes and before putting on the addresses. He puts the addresses on at random sends out the letters anyways. What are the chances that none of the three recipients get their intended letter?
[b]p2.[/b] David is a horrible bowler. Luckily, Logan and Christy let him use bumpers. The bowling lane is $2$ meters wide, and David's ball travels a total distance of $24$ meters. How many times did David's bowling ball hit the bumpers, if he threw it from the middle of the lane at a $60^o$ degree angle to the horizontal?
[b]p3.[/b] Find $\gcd \,(212106, 106212)$.
[b]p4.[/b] Michael has two fair dice, one six-sided (with sides marked $1$ through $6$) and one eight-sided (with sides marked $1-8$). Michael play a game with Alex: Alex calls out a number, and then Michael rolls the dice. If the sum of the dice is equal to Alex's number, Michael gives Alex the amount of the sum. Otherwise Alex wins nothing. What number should Alex call to maximize his expected gain of money?
[b]p5.[/b] Suppose that $x$ is a real number with $\log_5 \sin x + \log_5 \cos x = -1$. Find $$|\sin^2 x \cos x + \cos^2 x \sin x|.$$
[b]p6.[/b] What is the volume of the largest sphere that FIts inside a regular tetrahedron of side length $6$?
[b]p7.[/b] An ant is wandering on the edges of a cube. At every second, the ant randomly chooses one of the three edges incident at one vertex and walks along that edge, arriving at the other vertex at the end of the second. What is the probability that the ant is at its starting vertex after exactly $6$ seconds?
[b]p8.[/b] Determine the smallest positive integer $k$ such that there exist $m, n$ non-negative integers with $m > 1$ satisfying $$k = 2^{2m+1} - n^2.$$
[b]p9.[/b] For $A,B \subset Z$ with $A,B \ne \emptyset$, define $A + B = \{a + b|a \in A, b \in B\}$. Determine the least $n$ such that there exist sets $A,B$ with $|A| = |B| = n$ and $A + B = \{0, 1, 2,..., 2012\}$.
[b]p10.[/b] For positive integers $n \ge 1$, let $\tau (n)$ and $\sigma (n)$ be, respectively, the number of and sum of the positive integer divisors of $n$ (including $1$ and $n$). For example, $\tau (1) = \sigma (1) = 1$ and $\tau (6) = 4$, $\sigma (6) = 12$. Find the number of positive integers $n \le 100$ such that
$$\sigma (n) \le (\sqrt{n} - 1)^2 +\tau (n)\sqrt{n}.$$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Individual Rounds, 1998
[b]p1.[/b] Find the greatest integer $n$ such that $n \log_{10} 4$ does not exceed $\log_{10} 1998$.
[b]p2.[/b] Rectangle $ABCD$ has sides $AB = CD = 12/5$, $BC = DA = 5$. Point $P$ is on $AD$ with $\angle BPC = 90^o$. Compute $BP + PC$.
[b]p3.[/b] Compute the number of sequences of four decimal digits $(a, b, c, d)$ (each between $0$ and $9$ inclusive) containing no adjacent repeated digits. (That is, each digit is distinct from the digits directly before and directly after it.)
[b]p4.[/b] Solve for $t$, $-\pi/4 \le t \le \pi/4 $:
$$\sin^3 t + \sin^2 t \cos t + \sin t \cos^2 t + \cos^3 t =\frac{\sqrt6}{2}$$
[b]p5.[/b] Find all integers $n$ such that $n - 3$ divides $n^2 + 2$.
[b]p6.[/b] Find the maximum number of bishops that can occupy an $8 \times 8$ chessboard so that no two of the bishops attack each other. (Bishops can attack an arbitrary number of squares in any diagonal direction.)
[b]p7.[/b] Points $A, B, C$, and $D$ are on a Cartesian coordinate system with $A = (0, 1)$, $B = (1, 1)$, $C = (1,-1)$, and $D = (-1, 0)$. Compute the minimum possible value of $PA + PB + PC + PD$ over all points $P$.
[b]p8.[/b] Find the number of distinct real values of $x$ which satisfy
$$(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)+(1^2 \cdot 3^2\cdot 5^2\cdot 7^2\cdot 9^2)/2^{10} = 0.$$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Individual Rounds, 2017 Tie
[b]p1.[/b] Find the sum of all $3$-digit positive integers $\overline{abc}$ that satisfy $$\overline{abc} = {n \choose a}+{n \choose b}+ {n \choose c}$$ for some $n \le 10$.
[b]p2.[/b] Feng and Trung play a game. Feng chooses an integer $p$ from $1$ to $90$, and Trung tries to guess it. In each round, Trung asks Feng two yes-or-no questions about $p$. Feng must answer one question truthfully and one question untruthfully. After $15$ rounds, Trung concludes there are n possible values for $p$. What is the least possible value of $n$, assuming Feng chooses the best strategy to prevent Trung from guessing correctly?
[b]p3.[/b] A hypercube $H_n$ is an $n$-dimensional analogue of a cube. Its vertices are all the points $(x_1, .., x_n)$ that satisfy $x_i = 0$ or $1$ for all $1 \le i \le n$ and its edges are all segments that connect two adjacent vertices. (Two vertices are adjacent if their coordinates differ at exactly one $x_i$ . For example, $(0,0,0,0)$ and $(0,0,0,1)$ are adjacent on $H_4$.) Let $\phi (H_n)$ be the number of cubes formed by the edges and vertices of $H_n$. Find $\phi (H_4) + \phi (H_5)$.
[b]p4.[/b] Denote the legs of a right triangle as $a$ and $b$, the radius of the circumscribed circle as $R$ and the radius of the inscribed circle as $r$. Find $\frac{a+b}{R+r}$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Team Rounds, 2022
[b]p1.[/b] The serpent of fire and the serpent of ice play a game. Since the serpent of ice loves the lucky number $6$, he will roll a fair $6$-sided die with faces numbered $1$ through $6$. The serpent of fire will pay him $\log_{10} x$, where $x$ is the number he rolls. The serpent of ice rolls the die $6$ times. His expected total amount of winnings across the $6$ rounds is $k$. Find $10^k$.
[b]p2.[/b] Let $a = \log_3 5$, $b = \log_3 4$, $c = - \log_3 20$, evaluate $\frac{a^2+b^2}{a^2+b^2+ab} +\frac{b^2+c^2}{b^2+c^2+bc} +\frac{c^2+a^2}{c^2+a^2+ca}$.
[b]p3.[/b] Let $\vartriangle ABC$ be an isosceles obtuse triangle with $AB = AC$ and circumcenter $O$. The circle with diameter $AO$ meets $BC$ at points $X, Y$ , where X is closer to $B$. Suppose $XB = Y C = 4$, $XY = 6$, and the area of $\vartriangle ABC$ is $m\sqrt{n}$ for positive integers $m$ and $n$, where $n$ does not contain any square factors. Find $m + n$.
[b]p4.[/b] Alice is not sure what to have for dinner, so she uses a fair $6$-sided die to decide. She keeps rolling, and if she gets all the even numbers (i.e. getting all of $2, 4, 6$) before getting any odd number, she will reward herself with McDonald’s. Find the probability that Alice could have McDonald’s for dinner.
[b]p5.[/b] How many distinct ways are there to split $50$ apples, $50$ oranges, $50$ bananas into two boxes, such that the products of the number of apples, oranges, and bananas in each box are nonzero and equal?
[b]p6.[/b] Sujay and Rishabh are taking turns marking lattice points within a square board in the Cartesian plane with opposite vertices $(1, 1)$,$(n, n)$ for some constant $n$. Sujay loses when the two-point pattern $P$ below shows up:[img]https://cdn.artofproblemsolving.com/attachments/1/9/d1fe285294d4146afc0c7a2180b15586b04643.png[/img]
That is, Sujay loses when there exists a pair of points $(x, y)$ and $(x + 2, y + 1)$. He and Rishabh stop marking points when the pattern $P$ appears on the board. If Rishabh goes first, let $S$ be the set of all integers $3 \le n \le 100$ such that Rishabh has a strategy to always trick Sujay into being the one who creates $P$. Find the sum of all elements of $S$.
[b]p7.[/b] Let $a$ be the shortest distance between the origin $(0, 0)$ and the graph of $y^3 = x(6y -x^2)-8$. Find $\lfloor a^2 \rfloor $. ($\lfloor x\rfloor $ is the largest integer not exceeding $x$)
[b]p8.[/b] Find all real solutions to the following equation:
$$2\sqrt2x^2 + x -\sqrt{1 - x^2 } -\sqrt2 = 0.$$
[b]p9.[/b] Given the expression $S = (x^4 - x)(x^2 - x^3)$ for $x = \cos \frac{2\pi}{5 }+ i\sin \frac{2\pi}{5 }$, find the value of $S^2$
.
[b]p10.[/b] In a $32$ team single-elimination rock-paper-scissors tournament, the teams are numbered from $1$ to $32$. Each team is guaranteed (through incredible rock-paper-scissors skill) to win any match against a team with a higher number than it, and therefore will lose to any team with a lower number. Each round, teams who have not lost yet are randomly paired with other teams, and the losers of each match are eliminated. After the $5$ rounds of the tournament, the team that won all $5$ rounds is ranked $1$st, the team that lost the 5th round is ranked $2$nd, and the two teams that lost the $4$th round play each other for $3$rd and $4$th place. What is the probability that the teams numbered $1, 2, 3$, and $4$ are ranked $1$st, 2nd, 3rd, and 4th respectively? If the probability is $\frac{m}{n}$ for relatively prime integers $m$ and $n$, find $m$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Individual Rounds, 2015
[b]p1.[/b] Find the minimum value of $x^4 +2x^3 +3x^2 +2x+2$, where x can be any real number.
[b]p2.[/b] A type of digit-lock has $5$ digits, each digit chosen from $\{1,2, 3, 4, 5\}$. How many different passwords are there that have an odd number of $1$'s?
[b]p3.[/b] Tony is a really good Ping Pong player, or at least that is what he claims. For him, ping pong balls are very important and he can feel very easily when a ping pong ball is good and when it is not. The Ping Pong club just ordered new balls. They usually order form either PPB company or MIO company. Tony knows that PPB balls have $80\%$ chance to be good balls and MIO balls have $50\%$ chance to be good balls. I know you are thinking why would anyone order MIO balls, but they are way cheaper than PPB balls. When the box full with balls arrives (huge number of balls), Tony tries the first ball in the box and realizes it is a good ball. Given that the Ping Pong club usually orders half of the time from PPB and half of the time from MIO, what is the probability that the second ball is a good ball?
[b]p4.[/b] What is the smallest positive integer that is one-ninth of its reverse?
[b]p5.[/b] When Michael wakes up in the morning he is usually late for class so he has to get dressed very quickly. He has to put on a short sleeved shirt, a sweater, pants, two socks and two shoes. People usually put the sweater on after they put the short sleeved shirt on, but Michael has a different style, so he can do it both ways. Given that he puts on a shoe on a foot after he put on a sock on that foot, in how many di erent orders can Michael get dressed?
[b]p6.[/b] The numbers $1, 2,..., 2015$ are written on a blackboard. At each step we choose two numbers and replace them with their nonnegative difference. We stop when we have only one number. How many possibilities are there for this last number?
[b]p7.[/b] Let $A = (a_1b_1a_2b_2... a_nb_n)_{34}$ and $B = (b_1b_2... b_n)_{34}$ be two numbers written in base $34$. If the sum of the base-$34$ digits of $A$ is congruent to $15$ (mod $77$) and the sum of the base $34$ digits of $B$ is congruent to $23$ (mod $77$). Then if $(a_1b_1a_2b_2... a_nb_n)_{34} \equiv x$ (mod $77$) and $0 \le x \le 76$, what is $x$? (you can write $x$ in base $10$)
[b]p8.[/b] What is the sum of the medians of all nonempty subsets of $\{1, 2,..., 9\}$?
[b]p9.[/b] Tony is moving on a straight line for $6$ minutes{classic Tony. Several finitely many observers are watching him because, let's face it, you can't really trust Tony. In fact, they must watch him very closely{ so closely that he must never remain unattended for any second. But since the observers are lazy, they only watch Tony uninterruptedly for exactly one minute, and during this minute, Tony covers exactly one meter. What is the sum of the minimal and maximal possible distance Tony can walk during the six minutes?
[b]p10.[/b] Find the number of nonnegative integer triplets $a, b, c$ that satisfy $$2^a3^b + 9 = c^2.$$
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Team Rounds, 2018
[b]p1. [/b] If $f(x) = 3x - 1$, what is $f^6(2) = (f \circ f \circ f \circ f \circ f \circ f)(2)$?
[b]p2.[/b] A frog starts at the origin of the $(x, y)$ plane and wants to go to $(6, 6)$. It can either jump to the right one unit or jump up one unit. How many ways are there for the frog to jump from the origin to $(6, 6)$ without passing through point $(2, 3)$?
[b]p3.[/b] Alfred, Bob, and Carl plan to meet at a café between noon and $2$ pm. Alfred and Bob will arrive at a random time between noon and $2$ pm. They will wait for $20$ minutes or until $2$ pm for all $3$ people to show up after which they will leave. Carl will arrive at the café at noon and leave at $1:30$ pm. What is the probability that all three will meet together?
[b]p4.[/b] Let triangle $ABC$ be isosceles with $AB = AC$. Let $BD$ be the altitude from $ B$ to $AC$, $E$ be the midpoint of $AB$, and $AF$ be the altitude from $ A$ to $BC$. If $AF = 8$ and the area of triangle $ACE$ is $ 8$, find the length of $CD$.
[b]p5.[/b] Find the sum of the unique prime factors of $(2018^2 - 121) \cdot (2018^2 - 9)$.
[b]p6.[/b] Compute the remainder when $3^{102} + 3^{101} + ... + 3^0$ is divided by $101$.
[b]p7.[/b] Take regular heptagon $DUKMATH$ with side length $ 3$. Find the value of $$\frac{1}{DK}+\frac{1}{DM}.$$
[b]p8.[/b] RJ’s favorite number is a positive integer less than $1000$. It has final digit of $3$ when written in base $5$ and final digit $4$ when written in base $6$. How many guesses do you need to be certain that you can guess RJ’s favorite number?
[b]p9.[/b] Let $f(a, b) = \frac{a^2+b^2}{ab-1}$ , where $a$ and $b$ are positive integers, $ab \ne 1$. Let $x$ be the maximum positive integer value of $f$, and let $y$ be the minimum positive integer value of f. What is $x - y$ ?
[b]p10.[/b] Haoyang has a circular cylinder container with height $50$ and radius $5$ that contains $5$ tennis balls, each with outer-radius $5$ and thickness $1$. Since Haoyang is very smart, he figures out that he can fit in more balls if he cuts each of the balls in half, then puts them in the container, so he is ”stacking” the halves. How many balls would he have to cut up to fill up the container?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Team Rounds, 2008
[b]p1.[/b] $ABCD$ is a convex quadrilateral such that $AB = 20$, $BC = 24$, $CD = 7$, $DA = 15$, and $\angle DAB$ is a right angle. What is the area of $ABCD$?
[b]p2.[/b] A triangular number is one that can be written in the form $1 + 2 +...·+n$ for some positive number $n$. $ 1$ is clearly both triangular and square. What is the next largest number that is both triangular and square?
[b]p3.[/b] Find the last (i.e. rightmost) three digits of $9^{2008}$.
[b]p4.[/b] When expressing numbers in a base $b \ge 11$, you use letters to represent digits greater than $9$. For example, $A$ represents $10$ and $B$ represents $11$, so that the number $110$ in base $10$ is $A0$ in base $11$. What is the smallest positive integer that has four digits when written in base $10$, has at least one letter in its base $12$ representation, and no letters in its base $16$ representation?
[b]p5.[/b] A fly starts from the point $(0, 16)$, then flies straight to the point $(8, 0)$, then straight to the point $(0, -4)$, then straight to the point $(-2, 0)$, and so on, spiraling to the origin, each time intersecting the coordinate axes at a point half as far from the origin as its previous intercept. If the fly flies at a constant speed of $2$ units per second, how many seconds will it take the fly to reach the origin?
[b]p6.[/b] A line segment is divided into two unequal lengths so that the ratio of the length of the short part to the length of the long part is the same as the ratio of the length of the long part to the length of the whole line segment. Let $D$ be this ratio. Compute $$D^{-1} + D^{[D^{-1}+D^{(D^{-1}+D^2)}]}.$$
[b]p7.[/b] Let $f(x) = 4x + 2$. Find the ordered pair of integers $(P, Q)$ such that their greatest common divisor is $1, P$ is positive, and for any two real numbers $a$ and $b$, the sentence:
“$P a + Qb \ge 0$”
is true if and only if the following sentence is true:
“For all real numbers x, if $|f(x) - 6| < b$, then $|x - 1| < a$.”
[b]p8.[/b] Call a rectangle “simple” if all four of its vertices have integers as both of their coordinates and has one vertex at the origin. How many simple rectangles are there whose area is less than or equal to $6$?
[b]p9.[/b] A square is divided into eight congruent triangles by the diagonals and the perpendicular bisectors of its sides. How many ways are there to color the triangles red and blue if two ways that are reflections or rotations of each other are considered the same?
[b]p10.[/b] In chess, a knight can move by jumping to any square whose center is $\sqrt5$ units away from the center of the square that it is currently on. For example, a knight on the square marked by the horse in the diagram below can move to any of the squares marked with an “X” and to no other squares. How many ways can a knight on the square marked by the horse in the diagram move to the square with a circle in exactly four moves?
[img]https://cdn.artofproblemsolving.com/attachments/d/9/2ef9939642362182af12089f95836d4e294725.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Team Rounds, 2003
[b]p1.[/b] In a $3$-person race, how many different results are possible if ties are allowed?
[b]p2.[/b] An isosceles trapezoid has lengths $5$, $5$, $5$, and $8$. What is the sum of the lengths of its diagonals?
[b]p3.[/b] Let $f(x) = (1 + x + x^2)(1 + x^3 + x^6)(1 + x^9 + x^{18})...$. Compute $f(4/5)$.
[b]p4.[/b] Compute the largest prime factor of $3^{12} - 1$.
[b]p5.[/b] Taren wants to throw a frisbee to David, who starts running perpendicular to the initial line between them at rate $1$ m/s. Taren throws the frisbee at rate $2$ m/s at the same instant David begins to run. At what angle should Taren throw the frisbee?
[b]p6.[/b] The polynomial $p(x)$ leaves remainder $6$ when divided by $x-5$, and $5$ when divided by $x-6$. What is the remainder when $p(x)$ is divided by $(x - 5)(x - 6)$?
[b]p7.[/b] Find the sum of the cubes of the roots of $x^{10} + x^9 + ... + x + 1 = 0$.
[b]p8.[/b] A circle of radius $1$ is inscribed in a the parabola $y = x^2$. What is the $x$-coordinate of the intersection in the first quadrant?
[b]p9.[/b] You are stuck in a cave with $3$ tunnels. The first tunnel leads you back to your starting point in $5$ hours, and the second tunnel leads you back there in $7$ hours. The third tunnel leads you out of the cave in $4$ hours. What is the expected number of hours for you to exit the cave, assuming you choose a tunnel randomly each time you come across your point of origin?
[b]p10.[/b] What is the minimum distance between the line $y = 4x/7 + 1/5$ and any lattice point in the plane? (lattice points are points with integer coordinates)
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Individual Rounds, 2014
[b]p1.[/b] Trung has $2$ bells. One bell rings $6$ times per hour and the other bell rings $10$ times per hour. At the start of the hour both bells ring. After how much time will the bells ring again at the same time? Express your answer in hours.
[b]p2.[/b] In a soccer tournament there are $n$ teams participating. Each team plays every other team once. The matches can end in a win for one team or in a draw. If the match ends with a win, the winner gets $3$ points and the loser gets $0$. If the match ends in a draw, each team gets $1$ point. At the end of the tournament the total number of points of all the teams is $21$. Let $p$ be the number of points of the team in the first place. Find $n + p$.
[b]p3.[/b] What is the largest $3$ digit number $\overline{abc}$ such that $b \cdot \overline{ac} = c \cdot \overline{ab} + 50$?
[b]p4.[/b] Let s(n) be the number of quadruplets $(x, y, z, t)$ of positive integers with the property that $n = x + y + z + t$. Find the smallest $n$ such that $s(n) > 2014$.
[b]p5.[/b] Consider a decomposition of a $10 \times 10$ chessboard into p disjoint rectangles such that each rectangle contains an integral number of squares and each rectangle contains an equal number of white squares as black squares. Furthermore, each rectangle has different number of squares inside. What is the maximum of $p$?
[b]p6.[/b] If two points are selected at random from a straight line segment of length $\pi$, what is the probability that the distance between them is at least $\pi- 1$?
[b]p7.[/b] Find the length $n$ of the longest possible geometric progression $a_1, a_2,..,, a_n$ such that the $a_i$ are distinct positive integers between $100$ and $2014$ inclusive.
[b]p8.[/b] Feng is standing in front of a $100$ story building with two identical crystal balls. A crystal ball will break if dropped from a certain floor $m$ of the building or higher, but it will not break if it is dropped from a floor lower than $m$. What is the minimum number of times Feng needs to drop a ball in order to guarantee he determined $m$ by the time all the crystal balls break?
[b]p9.[/b] Let $A$ and $B$ be disjoint subsets of $\{1, 2,..., 10\}$ such that the product of the elements of $A$ is equal to the sum of the elements in $B$. Find how many such $A$ and $B$ exist.
[b]p10.[/b] During the semester, the students in a math class are divided into groups of four such that every two groups have exactly $2$ students in common and no two students are in all the groups together. Find the maximum number of such groups.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Devil Rounds, 2009
[b]p1.[/b] Find all positive integers $n$ such that $n^3 - 14n^2 + 64n - 93$ is prime.
[b]p2.[/b] Let $a, b, c$ be real numbers such that $0 \le a, b, c \le 1$. Find the maximum value of
$$\frac{a}{1 + bc}+\frac{b}{1 + ac}+\frac{c}{1 + ab}$$
[b]p3.[/b] Find the maximum value of the function $f(x, y, z) = 4x + 3y + 2z$ on the ellipsoid $16x^2 + 9y^2 + 4z^2 = 1$
[b]p4.[/b] Let $x_1,..., x_n$ be numbers such that $x_1+...+x_n = 2009$. Find the minimum value of $x^2_1+...+x^2_n$ (in term of $n$).
[b]p5.[/b] Find the number of odd integers between $1000$ and $9999$ that have at least 3 distinct digits.
[b]p6.[/b] Let $A_1,A_2,...,A_{2^n-1}$ be all the possible nonempty subsets of $\{1, 2, 3,..., n\}$. Find the maximum value of $a_1 + a_2 + ... + a_{2^n-1}$ where $a_i \in A_i$ for each $i = 1, 2,..., 2^n - 1$.
[b]p7.[/b] Find the rightmost digit when $41^{2009}$ is written in base $7$.
[b]p8.[/b] How many integral ordered triples $(x, y, z)$ satisfy the equation $x+y+z = 2009$, where $10 \le x < 31$, $100 < z < 310$ and $y \ge 0$.
[b]p9.[/b] Scooby has a fair six-sided die, labeled $1$ to $6$, and Shaggy has a fair twenty-sided die, labeled $1$ to $20$. During each turn, they both roll their own dice at the same time. They keep rolling the die until one of them rolls a 5. Find the probability that Scooby rolls a $5$ before Shaggy does.
[b]p10.[/b] Let $N = 1A323492110877$ where $A$ is a digit in the decimal expansion of $N$. Suppose $N$ is divisible by $7$. Find $A$.
[b]p11.[/b] Find all solutions $(x, y)$ of the equation $\tan^4(x+y)+\cot^4(x+y) = 1-2x-x^2$, where $-\frac{\pi}{2}
\le x; y \le \frac{\pi}{2}$
[b]p12.[/b] Find the remainder when $\sum^{50}_{k=1}k!(k^2 + k - 1)$ is divided by $1008$.
[b]p13.[/b] The devil set of a positive integer $n$, denoted $D(n)$, is defined as follows:
(1) For every positive integer $n$, $n \in D(n)$.
(2) If $n$ is divisible by $m$ and $m < n$, then for every element $a \in D(m)$, $a^3$ must be in $D(n)$.
Furthermore, call a set $S$ scary if for any $a, b \in S$, $a < b$ implies that $b$ is divisible by $a$. What is the least positive integer $n$ such that $D(n)$ is scary and has at least $2009$ elements?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Individual Rounds, 2006 Tie
[b]p1.[/b] Suppose that $a$, $b$, and $c$ are positive integers such that not all of them are even, $a < b$, $a^2 + b^2 = c^2$, and $c - b = 289$. What is the smallest possible value for $c$?
[b]p2.[/b] If $a, b > 1$ and $a^2$ is $11$ in base $b$, what is the third digit from the right of $b^2$ in base $a$?
[b]p3.[/b] Find real numbers $a, b$ such that $x^2 - x - 1$ is a factor of $ax^{10} + bx^9 + 1$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Team Rounds, 2010
[b]p1.[/b] Find the smallest positive integer $N$ such that $N!$ is a multiple of $10^{2010}$.
[b]p2.[/b] An equilateral triangle $T$ is externally tangent to three mutually tangent unit circles, as shown in the diagram. Find the area of $T$.
[b]p3. [/b]The polynomial $p(x) = x^3 + ax^2 + bx + c$ has the property that the average of its roots, the product of its roots, and the sum of its coefficients are all equal. If $p(0) = 2$, find $b$.
[b]p4.[/b] A regular pentagon $P = A_1A_2A_3A_4A_5$ and a square $S = B_1B_2B_3B_4$ are both inscribed in the unit circle. For a given pentagon $P$ and square $S$, let $f(P, S)$ be the minimum length of the minor arcs AiBj , for $1 \le i \le 5$ and $1 \le j \le 4$. Find the maximum of $f(P, S)$ over all pairs of shapes.
[b]p5.[/b] Let $ a, b, c$ be three three-digit perfect squares that together contain each nonzero digit exactly once. Find the value of $a + b + c$.
[b]p6. [/b]There is a big circle $P$ of radius $2$. Two smaller circles $Q$ and $R$ are drawn tangent to the big circle $P$ and tangent to each other at the center of the big circle $P$. A fourth circle $S$ is drawn externally tangent to the smaller circles $Q$ and $R$ and internally tangent to the big circle $P$. Finally, a tiny fifth circle $T$ is drawn externally tangent to the $3$ smaller circles $Q, R, S$. What is the radius of the tiny circle $T$?
[b]p7.[/b] Let $P(x) = (1 +x)(1 +x^2)(1 +x^4)(1 +x^8)(...)$. This infinite product converges when $|x| < 1$.
Find $P\left( \frac{1}{2010}\right)$.
[b]p8.[/b] $P(x)$ is a polynomial of degree four with integer coefficients that satisfies $P(0) = 1$ and $P(\sqrt2 + \sqrt3) = 0$. Find $P(5)$.
[b]p9.[/b] Find all positive integers $n \ge 3$ such that both roots of the equation $$(n - 2)x^2 + (2n^2 - 13n + 38)x + 12n - 12 = 0$$ are integers.
[b]p10.[/b] Let $a, b, c, d, e, f$ be positive integers (not necessarily distinct) such that $$a^4 + b^4 + c^4 + d^4 + e^4 = f^4.$$ Find the largest positive integer $n$ such that $n$ is guaranteed to divide at least one of $a, b, c, d, e, f$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Individual Rounds, 2022
[b]p1.[/b] Sujay sees a shooting star go across the night sky, and took a picture of it. The shooting star consists of a star body, which is bounded by four quarter-circle arcs, and a triangular tail. Suppose $AB = 2$, $AC = 4$. Let the area of the shooting star be $X$. If $6X = a-b\pi$ for positive integers $a, b$, find $a + b$.
[img]https://cdn.artofproblemsolving.com/attachments/0/f/f9c9ff23416565760df225c133330e795b9076.png[/img]
[b]p2.[/b] Assuming that each distinct arrangement of the letters in $DISCUSSIONS$ is equally likely to occur, what is the probability that a random arrangement of the letters in $DISCUSSIONS$ has all the $S$’s together?
[b]p3.[/b] Evaluate
$$\frac{(1 + 2022)(1 + 2022^2)(1 + 2022^4) ... (1 + 2022^{2^{2022}})}{1 + 2022 + 2022^2 + ... + 2022^{2^{2023}-1}} .$$
[b]p4.[/b] Dr. Kraines has $27$ unit cubes, each of which has one side painted red while the other five are white. If he assembles his cubes into one $3 \times 3 \times 3$ cube by placing each unit cube in a random orientation, what is the probability that the entire surface of the cube will be white, with no red faces visible? If the answer is $2^a3^b5^c$ for integers $a$, $b$, $c$, find $|a + b + c|$.
[b]p5.[/b] Let S be a subset of $\{1, 2, 3, ... , 1000, 1001\}$ such that no two elements of $S$ have a difference of $4$ or $7$. What is the largest number of elements $S$ can have?
[b]p6.[/b] George writes the number $1$. At each iteration, he removes the number $x$ written and instead writes either $4x+1$ or $8x+1$. He does this until $x > 1000$, after which the game ends. What is the minimum possible value of the last number George writes?
[b]p7.[/b] List all positive integer ordered pairs $(a, b)$ satisfying $a^4 + 4b^4 = 281 \cdot 61$.
[b]p8.[/b] Karthik the farmer is trying to protect his crops from a wildfire. Karthik’s land is a $5 \times 6$ rectangle divided into $30$ smaller square plots. The $5$ plots on the left edge contain fire, the $5$ plots on the right edge contain blueberry trees, and the other $5 \times 4$ plots of land contain banana bushes. Fire will repeatedly spread to all squares with bushes or trees that share a side with a square with fire. How many ways can Karthik replace $5$ of his $20$ plots of banana bushes with firebreaks so that fire will not consume any of his prized blueberry trees?
[b]p9.[/b] Find $a_0 \in R$ such that the sequence $\{a_n\}^{\infty}_{n=0}$ defined by $a_{n+1} = -3a_n + 2^n$ is strictly increasing.
[b]p10.[/b] Jonathan is playing with his life savings. He lines up a penny, nickel, dime, quarter, and half-dollar from left to right. At each step, Jonathan takes the leftmost coin at position $1$ and uniformly chooses a position $2 \le k \le 5$. He then moves the coin to position $k$, shifting all coins at positions $2$ through $k$ leftward. What is the expected number of steps it takes for the half-dollar to leave and subsequently return to position $5$?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Individual Rounds, 2021 Tie
You are standing on one of the faces of a cube. Each turn, you randomly choose another face that shares an edge with the face you are standing on with equal probability, and move to that face. Let $F(n)$ the probability that you land on the starting face after $n$ turns. Supposed that $F(8) = \frac{43}{256}$ , and F(10) can be expressed as a simplified fraction $\frac{a}{b}$. Find $a+b$.
DMM Team Rounds, 2013 (-14)
[b]p1.[/b] Suppose $5$ bales of hay are weighted two at a time in all possible ways. The weights obtained are $110$, $112$, $113$, $114$, $115$, $116$, $117$, $118$, $120$, $121$. What is the difference between the heaviest and the lightest bale?
[b]p2.[/b] Paul and Paula are playing a game with dice. Each have an $8$-sided die, and they roll at the same time. If the number is the same they continue rolling; otherwise the one who rolled a higher number wins. What is the probability that the game lasts at most $3$ rounds?
[b]p3[/b]. Find the unique positive integer $n$ such that $\frac{n^3+5}{n^2-1}$ is an integer.
[b]p4.[/b] How many numbers have $6$ digits, some four of which are $2, 0, 1, 4$ (not necessarily consecutive or in that order) and have the sum of their digits equal to $9$?
[b]p5.[/b] The Duke School has $N$ students, where $N$ is at most $500$. Every year the school has three sports competitions: one in basketball, one in volleyball, and one in soccer. Students may participate in all three competitions. A basketball team has $5$ spots, a volleyball team has $6$ spots, and a soccer team has $11$ spots on the team. All students are encouraged to play, but $16$ people choose not to play basketball, $9$ choose not to play volleyball and $5$ choose not to play soccer. Miraculously, other than that all of the students who wanted to play could be divided evenly into teams of the appropriate size. How many players are there in the school?
[b]p6.[/b] Let $\{a_n\}_{n\ge 1}$ be a sequence of real numbers such that $a_1 = 0$ and $a_{n+1} =\frac{a_n-\sqrt3}{\sqrt3 a_n+1}$ . Find $a_1 + a_2 +.. + a_{2014}$.
[b]p7.[/b] A soldier is fighting a three-headed dragon. At any minute, the soldier swings her sword, at which point there are three outcomes: either the soldier misses and the dragon grows a new head, the soldier chops off one head that instantaneously regrows, or the soldier chops off two heads and none grow back. If the dragon has at least two heads, the soldier is equally likely to miss or chop off two heads. The dragon dies when it has no heads left, and it overpowers the soldier if it has at least five heads. What is the probability that the soldier wins
[b]p8.[/b] A rook moves alternating horizontally and vertically on an infinite chessboard. The rook moves one square horizontally (in either direction) at the first move, two squares vertically at the second, three horizontally at the third and so on. Let $S$ be the set of integers $n$ with the property that there exists a series of moves such that after the $n$-th move the rock is back where it started. Find the number of elements in the set $S \cap \{1, 2, ..., 2014\}$.
[b]p9.[/b] Find the largest integer $n$ such that the number of positive integer divisors of $n$ (including $1$ and $n$) is at least $\sqrt{n}$.
[b]p10.[/b] Suppose that $x, y$ are irrational numbers such that $xy$, $x^2 + y$, $y^2 + x$ are rational numbers. Find $x + y$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Individual Rounds, 2013(-14)Tie
[b]p1.[/b] A light beam shines from the origin into the unit square at an angle of $\theta$ to one of the sides such that $\tan \theta = \frac{13}{17}$ . The light beam is reflected by the sides of the square. How many times does the light beam hit a side of the square before hitting a vertex of the square?
[img]https://cdn.artofproblemsolving.com/attachments/5/7/1db0aad33ed9bf82bee3303c7fbbe0b7c2574f.png[/img]
[b]p2.[/b] Alex is given points $A_1,A_2,...,A_{150}$ in the plane such that no three are collinear and $A_1$, $A_2$, $...$, $A_{100}$ are the vertices of a convex polygon $P$ containing $A_{101}$, $A_{102}$, $ ...$, $A_{150}$ in its interior. He proceeds to draw edges $A_iA_j$ such that no two edges intersect (except possibly at their endpoints), eventually dividing $P$ up into triangles. How many triangles are there?
[img]https://cdn.artofproblemsolving.com/attachments/d/5/12c757077e87809837d16128b018895a8bcc94.png[/img]
[b]p3. [/b]The polynomial P(x) has the property that $P(1)$, $P(2)$, $P(3)$, $P(4)$, and $P(5)$ are equal to $1$, $2$, $3$, $4$,$5$ in some order. How many possibilities are there for the polynomial $P$, given that the degree of $P$ is strictly less than $4$?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Team Rounds, 2019
[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].
DMM Individual Rounds, 2019
[b]p1.[/b] Compute the value of $N$, where
$$N = 818^3 - 6 \cdot 818^2 \cdot 209 + 12 \cdot 818 \cdot 209^2 - 8 \cdot 209^3$$
[b]p2.[/b] Suppose $x \le 2019$ is a positive integer that is divisible by $2$ and $5$, but not $3$. If $7$ is one of the digits in $x$, how many possible values of $x$ are there?
[b]p3.[/b] Find all non-negative integer solutions $(a,b)$ to the equation $$b^2 + b + 1 = a^2.$$
[b]p4.[/b] Compute the remainder when $\sum^{2019}_{n=1} n^4$ is divided by $53$.
[b]p5.[/b] Let $ABC$ be an equilateral triangle and $CDEF$ a square such that $E$ lies on segment $AB$ and $F$ on segment $BC$. If the perimeter of the square is equal to $4$, what is the area of triangle $ABC$?
[img]https://cdn.artofproblemsolving.com/attachments/1/6/52d9ef7032c2fadd4f97d7c0ea051b3766b584.png[/img]
[b]p6.[/b] $$S = \frac{4}{1\times 2\times 3}+\frac{5}{2\times 3\times 4} +\frac{6}{3\times 4\times 5}+ ... +\frac{101}{98\times 99\times 100}$$
Let $T = \frac54 - S$. If $T = \frac{m}{n}$ , where $m$ and $n$ are relatively prime integers, find the value of
$m + n$.
[b]p7.[/b] Find the sum of $$\sum^{2019}_{i=0}\frac{2^i}{2^i + 2^{2019-i}}$$
[b]p8.[/b] Let $A$ and $B$ be two points in the Cartesian plane such that $A$ lies on the line $y = 12$, and $B$ lies on the line $y = 3$. Let $C_1$, $C_2$ be two distinct circles that intersect both $A$ and $B$ and are tangent to the $x$-axis at $P$ and $Q$, respectively. If $PQ = 420$, determine the length of $AB$.
[b]p9.[/b] Zion has an average $2$ out of $3$ hit rate for $2$-pointers and $1$ out of $3$ hit rate for $3$-pointers. In a recent basketball match, Zion scored $18$ points without missing a shot, and all the points came from $2$ or $3$-pointers. What is the probability that all his shots were $3$-pointers?
[b]p10.[/b] Let $S = \{1,2, 3,..., 2019\}$. Find the number of non-constant functions $f : S \to S$ such that
$$f(k) = f(f(k + 1)) \le f(k + 1) \,\,\,\, for \,\,\,\, all \,\,\,\, 1 \le k \le 2018.$$
Express your answer in the form ${m \choose n}$, where $m$ and $n$ are integers.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Individual Rounds, 2016 Tie
[b]p1.[/b] How many ordered triples of integers $(a, b, c)$ where $1 \le a, b, c \le 10$ are such that for every natural number, the equation $(a + n)x^2 + (b + 2n)x + c + n = 0$ has at least one real root?
[b]p2.[/b] Find the smallest integer $n$ such that we can cut a $n \times n$ grid into $5$ rectangles with distinct side lengths in $\{1, 2, 3..., 10\}$. Every value is used exactly once.
[b]p3.[/b] A plane is flying at constant altitude along a circle of radius $12$ miles with center at a point $A$.The speed of the aircraft is v. At some moment in time, a missile is fired at the aircraft from the point $A$, which has speed v and is guided so that its velocity vector always points towards the aircraft. How far does the missile travel before colliding with the aircraft?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].