Found problems: 67
DMM Individual Rounds, 2003
[b]p1.[/b] If Suzie has $6$ coins worth $23$ cents, how many nickels does she have?
[b]p2.[/b] Let $a * b = (a - b)/(a + b)$. If $8 * (2 * x) = 4/3$, what is $x$?
[b]p3.[/b] How many digits does $x = 100000025^2 - 99999975^2$ have when written in decimal form?
[b]p4.[/b] A paperboy’s route covers $8$ consecutive houses along a road. He does not necessarily deliver to all the houses every day, but he always traverses the road in the same direction, and he takes care never to skip over $2$ consecutive houses. How many possible routes can he take?
[b]p5.[/b] A regular $12$-gon is inscribed in a circle of radius $5$. What is the sum of the squares of the distances from any one fixed vertex to all the others?
[b]p6.[/b] In triangle $ABC$, let $D, E$ be points on $AB$, $AC$, respectively, and let $BE$ and $CD$ meet at point $P$. If the areas of triangles $ADE$, $BPD$, and $CEP$ are $5$, $8$, and $3$, respectively, find the area of triangle ABC.
[b]p7.[/b] Bob has $11$ socks in his drawer: $3$ different matched pairs, and $5$ socks that don’t match with any others. Suppose he pulls socks from the drawer one at a time until he manages to get a matched pair. What is the probability he will need to draw exactly $9$ socks?
[b]p8.[/b] Consider the unit cube $ABCDEFGH$. The triangle bound to $A$ is the triangle formed by the $3$ vertices of the cube adjacent to $A$ (and similarly for the other vertices of the cube). Suppose we slice a knife through each of the $8$ triangles bound to vertices of the cube. What is the volume of the remaining solid that contains the former center of the cube?
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DMM Individual Rounds, 2007 Tie
[b]p1.[/b] Let $p_b(m)$ be the sum of digits of $m$ when $m$ is written in base $b$. (So, for example, $p_2(5) = 2$). Let $f(0) = 2007^{2007}$, and for $n \ge 0$ let $f(n + 1) = p_7(f(n))$. What is $f(10^{10000})$?
[b]p2.[/b] Compute:
$$\sum^{\infty}_{n=1}\frac{(-1)^{n+1}4n}{n^4 - 8n^2 + 4}.$$
[b]p3.[/b] $ABCDEFGH$ is an octagon whose eight interior angles all have the same measure. The lengths of the eight sides of this octagon are, in some order, $$2, 2\sqrt2, 4, 4\sqrt2, 6, 7, 7, \,\,\, and \,\,\, 8.$$
Find the area of $ABCDEFGH$.
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DMM Individual Rounds, 1999
[b]p1.[/b] Function $f$ is defined by $f (x) = ax+b$ for some real values $a, b > 0$. If $f (f (x)) = 9x + 5$ for all $x$, find $b$.
[b]p2.[/b] At some point during a game, Will Avery has made $1/3$ of his shots. When he shoots once and makes a basket, his average increases to $2/5$. Find his average (expressed as a fraction) after a second additional basket.
[b]p3.[/b] A dealer has a deck of $1999$ cards. He takes the top card off and “ducks” it, that is, places it on the bottom of the deck. He deals the second card onto the table. He ducks the third card, deals the fourth card, ducks the fifth card, deals the sixth card, and so forth, continuing until he has only one card left; he then ducks the last card with itself and deals it. Some of the cards (like the second and fourth cards) are not ducked at all before being dealt, while others are ducked multiple times. The question is: what is the average number of ducks per card?
[b]p4.[/b] Point $P$ lies outside circle $O$. Perpendicular lines $\ell$ and m intersect at $P$. Line $\ell$ is tangent to circle $O$ at a point $6$ units from $P$. Line $m$ crosses circle $O$ at a point $4$ units from $P$. Find the radius of circle $O$.
[b]p5.[/b] Define $f(n)$ by $$f(n) = \begin{cases} n/2 \,\,\,\text{if} \,\,\, n\,\,\,is\,\,\, even \\
(n + 1023)/2\,\,\, \text{if} \,\,\, n\,\,\,is\,\,\, odd \end{cases}$$
Find the least positive integer $n$ such that $f(f(f(f(f(n))))) = n.$
[b]p6.[/b] Write $\sqrt{10001}$ to the sixth decimal place, rounding down.
[b]p7.[/b] Define $(a_n)$ recursively by $a_1 = 1$, $a_n = 20 \cos (a_{n-1}^o)$. As $n$ tends to infinity, $(a_n)$ tends to $18.9195...$. Define $(b_n)$ recursively by $b_1 = 1$, $b_n =\sqrt{800 + 800 \cos (b_{n-1}^o)}$. As $n$ tends to infinity, $(b_n)$ tends to $x$. Calculate $x$ to three decimal places.
[b]p8.[/b] Let $mod_d (k)$ be the remainder of $k$ when divided by $d$. Find the number of positive integers $n$ satisfying $$mod_n(1999) = n^2 - 89n + 1999$$
[b]p9.[/b] Let $f(x) = x^3 + x$. Compute $$\sum^{10}_{k=1} \frac{1}{1 + f^{-1}(k - 1)^2 + f^{-1}(k - 1)f^{-1}(k) + f^{-1}(k)^2}$$
($f^{-1}$ is the inverse of $f$: $f (f^{-1}1 (x)) = f^{-1}1 (f (x)) = x$ for all $x$.)
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DMM Individual Rounds, 2019 Tie
[b]p1.[/b] Let $a(1), a(2), ..., a(n),...$ be an increasing sequence of positive integers satisfying $a(a(n)) = 3n$ for every positive integer $n$. Compute $a(2019)$.
[b]p2.[/b] Consider the function $f(12x - 7) = 18x^3 - 5x + 1$. Then, $f(x)$ can be expressed as $f(x) = ax^3 + bx^2 + cx + d$, for some real numbers $a, b, c$ and $d$. Find the value of $(a + c)(b + d)$.
[b]p3.[/b] Let $a, b$ be real numbers such that $\sqrt{5 + 2\sqrt6} = \sqrt{a} +\sqrt{b}$. Find the largest value of the quantity $$X = \dfrac{1}{a +\dfrac{1}{b+ \dfrac{1}{a+...}}}$$
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DMM Individual Rounds, 2020
[b]p1.[/b] Four witches are riding their brooms around a circle with circumference $10$ m. They are standing at the same spot, and then they all start to ride clockwise with the speed of $1$, $2$, $3$, and $4$ m/s, respectively. Assume that they stop at the time when every pair of witches has met for at least two times (the first position before they start counts as one time). What is the total distance all the four witches have travelled?
[b]p2.[/b] Suppose $A$ is an equilateral triangle, $O$ is its inscribed circle, and $B$ is another equilateral triangle inscribed in $O$. Denote the area of triangle $T$ as $[T]$. Evaluate $\frac{[A]}{[B]}$.
[b]p3. [/b]Tim has bought a lot of candies for Halloween, but unfortunately, he forgot the exact number of candies he has. He only remembers that it's an even number less than $2020$. As Tim tries to put the candies into his unlimited supply of boxes, he finds that there will be $1$ candy left if he puts seven in each box, $6$ left if he puts eleven in each box, and $3$ left if he puts thirteen in each box. Given the above information, find the total number of candies Tim has bought.
[b]p4.[/b] Let $f(n)$ be a function defined on positive integers n such that $f(1) = 0$, and $f(p) = 1$ for all prime numbers $p$, and $$f(mn) = nf(m) + mf(n)$$ for all positive integers $m$ and $n$. Let $$n = 277945762500 = 2^23^35^57^7$$ Compute the value of $\frac{f(n)}{n}$ .
[b]p5.[/b] Compute the only positive integer value of $\frac{404}{r^2-4}$ , where $r$ is a rational number.
[b]p6.[/b] Let $a = 3 +\sqrt{10}$ . If $$\prod^{\infty}_{k=1} \left( 1 + \frac{5a + 1}{a^k + a} \right)= m +\sqrt{n},$$
where $m$ and $n$ are integers, find $10m + n$.
[b]p7.[/b] Charlie is watching a spider in the center of a hexagonal web of side length $4$. The web also consists of threads that form equilateral triangles of side length $1$ that perfectly tile the hexagon. Each minute, the spider moves unit distance along one thread. If $\frac{m}{n}$ is the probability, in lowest terms, that after four minutes the spider is either at the edge of her web or in the center, find the value of $m + n$.
[b]p8.[/b] Let $ABC$ be a triangle with $AB = 10$; $AC = 12$, and $\omega$ its circumcircle. Let $F$ and $G$ be points on $\overline{AC}$ such that $AF = 2$, $FG = 6$, and $GC = 4$, and let $\overrightarrow{BF}$ and $\overrightarrow{BG}$ intersect $\omega$ at $D$ and $E$, respectively. Given that $AC$ and $DE$ are parallel, what is the square of the length of $BC$?
[b]p9.[/b] Two blue devils and $4$ angels go trick-or-treating. They randomly split up into $3$ non-empty groups. Let $p$ be the probability that in at least one of these groups, the number of angels is nonzero and no more than the number of devils in that group. If $p = \frac{m}{n}$ in lowest terms, compute $m + n$.
[b]p10.[/b] We know that$$2^{22000} = \underbrace{4569878...229376}_{6623\,\,\, digits}.$$ For how many positive integers $n < 22000$ is it also true that the first digit of $2^n$ is $4$?
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DMM Individual Rounds, 2002 Tie
[b]p1.[/b] Suppose $a$, $b$ and $c$ are integers such that $c$ divides $a^n + b^n$ for all integers, $n \ge 1$. If the greatest common divisor of $a$ and $b$ is $7$, what is the largest possible value of $c$?
[b]p2.[/b] Consider a sequence of points $\{P_1, P_2,...\}$ on a circle w with the property that $\overline{P_{i+1}P_{i+2}}$ is parallel to the tangent line through $P_i$ for each $i \ge 1$. If $P_5 = P_1$, what is the largest possible angle formed by $P_1P_3P_2$?
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DMM Individual Rounds, 1999 Tie
[b]p1A.[/b] Compute
$$1 + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + ...$$
$$1 - \frac{1}{2^3} + \frac{1}{3^3} - \frac{1}{4^3} + \frac{1}{5^3} - ...$$
[b]p1B.[/b] Real values $a$ and $b$ satisfy $ab = 1$, and both numbers have decimal expansions which repeat every five digits:
$$ a = 0.(a_1)(a_2)(a_3)(a_4)(a_5)(a_1)(a_2)(a_3)(a_4)(a_5)...$$
and
$$ b = 1.(b_1)(b_2)(b_3)(b_4)(b_5)(b_1)(b_2)(b_3)(b_4)(b_5)...$$
If $a_5 = 1$, find $b_5$.
[b]p2.[/b] $P(x) = x^4 - 3x^3 + 4x^2 - 9x + 5$. $Q(x)$ is a $3$rd-degree polynomial whose graph intersects the graph of $P(x)$ at $x = 1$, $2$, $5$, and $10$. Compute $Q(0)$.
[b]p3.[/b] Distinct real values $x_1$, $x_2$, $x_3$, $x_4 $all satisfy $ ||x - 3| - 5| = 1.34953$. Find $x_1 + x_2 + x_3 + x_4$.
[b]p4.[/b] Triangle $ABC$ has sides $AB = 8$, $BC = 10$, and $CA = 11$. Let $L$ be the locus of points in the interior of triangle $ABC$ which are within one unit of either $A$, $B$, or $C$. Find the area of $L$.
[b]p5.[/b] Triangles $ABC$ and $ADE$ are equilateral, and $AD$ is an altitude of $ABC$. The area of the intersection of these triangles is $3$. Find the area of the larger triangle $ABC$.
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DMM Team Rounds, 1998
[b][b]p1.[/b][/b] Find the perimeter of a regular hexagon with apothem $3$.
[b]p2.[/b] Concentric circles of radius $1$ and r are drawn on a circular dartboard of radius $5$. The probability that a randomly thrown dart lands between the two circles is $0.12$. Find $r$.
[b]p3.[/b] Find all ordered pairs of integers $(x, y)$ with $0 \le x \le 100$, $0 \le y \le 100$ satisfying $$xy = (x - 22) (y + 15) .$$
[b]p4.[/b] Points $A_1$,$A_2$,$...$,$A_{12}$ are evenly spaced around a circle of radius $1$, but not necessarily in order. Given that chords $A_1A_2$, $A_3A_4$, and $A_5A_6$ have length $2$ and chords $A_7A_8$ and $A_9A_{10}$ have length $2 sin (\pi / 12)$, find all possible lengths for chord $A_{11}A_{12}$.
[b]p5.[/b] Let $a$ be the number of digits of $2^{1998}$, and let $b$ be the number of digits in $5^{1998}$. Find $a + b$.
[b]p6.[/b] Find the volume of the solid in $R^3$ defined by the equations
$$x^2 + y^2 \le 2$$
$$x + y + |z| \le 3.$$
[b]p7.[/b] Positive integer $n$ is such that $3n$ has $28$ positive divisors and $4n$ has $36$ positive divisors. Find the number of positive divisors of $n$.
[b]p8.[/b] Define functions $f$ and $g$ by $f (x) = x +\sqrt{x}$ and $g (x) = x + 1/4$. Compute $$g(f(g(f(g(f(g(f(3)))))))).$$
(Your answer must be in the form $a + b \sqrt{ c}$ where $a$, $b$, and $c$ are rational.)
[b]p9.[/b] Sequence $(a_1, a_2,...)$ is defined recursively by $a_1 = 0$, $a_2 = 100$, and $a_n = 2a_{n-1}-a_{n-2}-3$. Find the greatest term in the sequence $(a_1, a_2,...)$.
[b]p10.[/b] Points $X = (3/5, 0)$ and $Y = (0, 4/5)$ are located on a Cartesian coordinate system. Consider all line segments which (like $\overline{XY}$ ) are of length 1 and have one endpoint on each axis. Find the coordinates of the unique point $P$ on $\overline{XY}$ such that none of these line segments (except $\overline{XY}$ itself) pass through $P$.
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DMM Team Rounds, 1999
[b]p1.[/b] The least prime factor of $a$ is $3$, the least prime factor of $b$ is $7$. Find the least prime factor of $a + b$.
[b]p2.[/b] In a Cartesian coordinate system, the two tangent lines from $P = (39, 52)$ meet the circle defined by $x^2 + y^2 = 625$ at points $Q$ and $R$. Find the length $QR$.
[b]p3.[/b] For a positive integer $n$, there is a sequence $(a_0, a_1, a_2,..., a_n)$ of real values such that $a_0 = 11$ and $(a_k + a_{k+1}) (a_k - a_{k+1}) = 5$ for every $k$ with $0 \le k \le n-1$. Find the maximum possible value of $n$. (Be careful that your answer isn’t off by one!)
[b]p4.[/b] Persons $A$ and $B$ stand at point $P$ on line $\ell$. Point $Q$ lies at a distance of $10$ from point $P$ in the direction perpendicular to $\ell$. Both persons intially face towards $Q$. Person $A$ walks forward and to the left at an angle of $25^o$ with $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the right, and continues walking. Person $B$ walks forward and to the right at an angle of $55^o$ with line $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the left, and continues walking. Their paths cross at point $R$. Find the distance $PR$.
[b]p5.[/b] Compute
$$\frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}.$$
[b]p6.[/b] There is a unique real value $A$ such that for all $x$ with $1 < x < 3$ and $x \ne 2$, $$\left| \frac{A}{x^2-x - 2} +\frac{1}{x^2 - 6x + 8} \right|< 1999.$$
Compute $A$.
[b]p7.[/b] Nine poles of height $1, 2,..., 9$ are placed in a line in random order. A pole is called [i]dominant [/i] if it is taller than the pole immediately to the left of it, or if it is the pole farthest to the left. Count the number of possible orderings in which there are exactly $2$ dominant poles.
[b]p8.[/b] $\tan (11x) = \tan (34^o)$ and $\tan (19x) = \tan (21^o)$. Compute $\tan (5x)$.
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DMM Team Rounds, 2012
[b]p1.[/b] Let $2^k$ be the largest power of $2$ dividing $30! = 30 \cdot 29 \cdot 28 ... 2 \cdot 1$. Find $k$.
[b]p2.[/b] Let $d(n)$ be the total number of digits needed to write all the numbers from $1$ to $n$ in base $10$, for example, $d(5) = 5$ and $d(20) = 31$. Find $d(2012)$.
[b]p3.[/b] Jim and TongTong play a game. Jim flips $10$ coins and TongTong flips $11$ coins, whoever gets the most heads wins. If they get the same number of heads, there is a tie. What is the probability that TongTong wins?
[b]p4.[/b] There are a certain number of potatoes in a pile. When separated into mounds of three, two remain. When divided into mounds of four, three remain. When divided into mounds of five, one remain. It is clear there are at least $150$ potatoes in the pile. What is the least number of potatoes there can be in the pile?
[b]p5.[/b] Call an ordered triple of sets $(A, B, C)$ nice if $|A \cap B| = |B \cap C| = |C \cap A| = 2$ and $|A \cap B \cap C| = 0$. How many ordered triples of subsets of $\{1, 2, · · · , 9\}$ are nice?
[b]p6.[/b] Brett has an $ n \times n \times n$ cube (where $n$ is an integer) which he dips into blue paint. He then cuts the cube into a bunch of $ 1 \times 1 \times 1$ cubes, and notices that the number of un-painted cubes (which is positive) evenly divides the number of painted cubes. What is the largest possible side length of Brett’s original cube?
Note that $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.
[b]p7.[/b] Choose two real numbers $x$ and $y$ uniformly at random from the interval $[0, 1]$. What is the probability that $x$ is closer to $1/4$ than $y$ is to $1/2$?
[b]p8. [/b] In triangle $ABC$, we have $\angle BAC = 20^o$ and $AB = AC$. $D$ is a point on segment $AB$ such that $AD = BC$. What is $\angle ADC$, in degree.
[b]p9.[/b] Let $a, b, c, d$ be real numbers such that $ab + c + d = 2012$, $bc + d + a = 2010$, $cd + a + b = 2013$, $da + b + c = 2009$. Find $d$.
[b]p10. [/b]Let $\theta \in [0, 2\pi)$ such that $\cos \theta = 2/3$. Find $\sum_{n=0}^{\infty}\frac{1}{2^n}\cos(n \theta)$
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DMM Team Rounds, 2005
[b]p1.[/b] Find the sum of the seventeenth powers of the seventeen roots of the seventeeth degree polynomial equation $x^{17} - 17x + 17 = 0$.
[b]p2.[/b] Four identical spherical cows, each of radius $17$ meters, are arranged in a tetrahedral pyramid (their centers are the vertices of a regular tetrahedron, and each one is tangent to the other three). The pyramid of cows is put on the ground, with three of them laying on it. What is the distance between the ground and the top of the topmost cow?
[b]p3.[/b] If $a_n$ is the last digit of $\sum^{n}_{i=1} i$, what would the value of $\sum^{1000}_{i=1}a_i$ be?
[b]p4.[/b] If there are $15$ teams to play in a tournament, $2$ teams per game, in how many ways can the tournament be organized if each team is to participate in exactly $5$ games against dierent opponents?
[b]p5.[/b] For $n = 20$ and $k = 6$, calculate $$2^k {n \choose 0}{n \choose k}- 2^{k-1}{n \choose 1}{{n - 1} \choose {k - 1}} + 2^{k-2}{n \choose 2}{{n - 2} \choose {k - 2}} +...+ (-1)^k {n \choose k}{{n - k} \choose 0}$$ where ${n \choose k}$ is the number of ways to choose $k$ things from a set of $n$.
[b]p6.[/b] Given a function $f(x) = ax^2 + b$, with a$, b$ real numbers such that $$f(f(f(x))) = -128x^8 + \frac{128}{3}x^6 - \frac{16}{22}x^2 +\frac{23}{102}$$ , find $b^a$.
[b]p7.[/b] Simplify the following fraction $$\frac{(2^3-1)(3^3-1)...(100^3-1)}{(2^3+1)(3^3+1)...(100^3+1)}$$
[b]p8.[/b] Simplify the following expression
$$\frac{\sqrt{3 + \sqrt5} + \sqrt{3 - \sqrt5}}{\sqrt{3 - \sqrt8}} -\frac{4}{ \sqrt{8 - 2\sqrt{15}}}$$
[b]p9.[/b] Suppose that $p(x)$ is a polynomial of degree $100$ such that $p(k) = k2^{k-1}$ , $k =1, 2, 3 ,... , 100$. What is the value of $p(101)$ ?
[b]p10. [/b] Find all $17$ real solutions $(w, x, y, z)$ to the following system of equalities:
$$ 2w + w^2x = x$$
$$ 2x + x^2y=y $$
$$ 2y + y^2z=z $$
$$ -2z+z^2w=w $$
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DMM Team Rounds, 2002
[b]p1.[/b] What is the last digit of
$$1! + 2! + ... + 10!$$
where $n!$ is defined to equal $1 \cdot 2 \cdot ... \cdot n$?
[b]p2.[/b] What pair of positive real numbers, $(x, y)$, satisfies
$$x^2y^2 = 144$$
$$(x - y)^3 = 64?$$
[b]p3.[/b] Paul rolls a standard $6$-sided die, and records the results. What is the probability that he rolls a $1$ ten times before he rolls a $6$ twice?
[b]p4.[/b] A train is approaching a $1$ kilometer long tunnel at a constant $40$ km/hr. It so happens that if Roger, who is inside, runs towards either end of the tunnel at a contant $10$ km/hr, he will reach that end at the exact same time as the train. How far from the center of the tunnel is Roger?
[b]p5.[/b] Let $ABC$ be a triangle with $A$ being a right angle. Let $w$ be a circle tangent to $\overline{AB}$ at $A$ and tangent to $\overline{BC}$ at some point $D$. Suppose $w$ intersects $\overline{AC}$ again at $E$ and that $\overline{CE} = 3$, $\overline{CD} = 6$. Compute $\overline{BD}$.
[b]p6.[/b] In how many ways can $1000$ be written as a sum of consecutive integers?
[b]p7.[/b] Let $ABC$ be an isosceles triangle with $\overline{AB} = \overline{AC} = 10$ and $\overline{BC} = 6$. Let $M$ be the midpoint of $\overline{AB}$, and let $\ell$ be the line through $A$ parallel to $\overline{BC}$. If $\ell$ intersects the circle through $A$, $C$ and $M$ at $D$, then what is the length of $\overline{AD}$?
[b]p8.[/b] How many ordered triples of pairwise relatively prime, positive integers, $\{a, b, c\}$, have the property that $a + b$ is a multiple of $c$, $b + c$ is a multiple of $a$, and $a + c$ is a multiple of $b$?
[b]p9.[/b] Consider a hexagon inscribed in a circle of radius $r$. If the hexagon has two sides of length $2$, two sides of length $7$, and two sides of length $11$, what is $r$?
[b]p10.[/b] Evaluate
$$\sum^{\infty}_{i=0} \sum^{\infty}_{j=0} \frac{\left( (-1)^i + (-1)^j\right) \cos (i) \sin (j)}{i!j!} ,$$
where angles are measured in degrees, and $0!$ is defined to equal $1$.
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DMM Individual Rounds, 2009
[b]p1.[/b] Let $p > 5$ be a prime. It is known that the average of all of the prime numbers that are at least $5$ and at most $p$ is $12$. Find $p$.
[b]p2.[/b] The numbers $1, 2,..., n$ are written down in random order. What is the probability that $n-1$ and $n$ are written next to each other? (Give your answer in term of $n$.)
[b]p3.[/b] The Duke Blue Devils are playing a basketball game at home against the UNC Tar Heels. The Tar Heels score $N$ points and the Blue Devils score $M$ points, where $1 < M,N < 100$. The first digit of $N$ is $a$ and the second digit of $N$ is $b$. It is known that $N = a+b^2$. The first digit of $M$ is $b$ and the second digit of $M$ is $a$. By how many points do the Blue Devils win?
[b]p4.[/b] Let $P(x)$ be a polynomial with integer coefficients. It is known that $P(x)$ gives a remainder of $1$ upon polynomial division by $x + 1$ and a remainder of $2$ upon polynomial division by $x + 2$. Find the remainder when $P(x)$ is divided by $(x + 1)(x + 2)$.
[b]p5.[/b] Dracula starts at the point $(0,9)$ in the plane. Dracula has to pick up buckets of blood from three rivers, in the following order: the Red River, which is the line $y = 10$; the Maroon River, which is the line $y = 0$; and the Slightly Crimson River, which is the line $x = 10$. After visiting all three rivers, Dracula must then bring the buckets of blood to a castle located at $(8,5)$. What is the shortest distance that Dracula can walk to accomplish this goal?
[b]p6.[/b] Thirteen hungry zombies are sitting at a circular table at a restaurant. They have five identical plates of zombie food. Each plate is either in front of a zombie or between two zombies. If a plate is in front of a zombie, that zombie and both of its neighbors can reach the plate. If a plate is between two zombies, only those two zombies may reach it. In how many ways can we arrange the plates of food around the circle so that each zombie can reach exactly one plate of food? (All zombies are distinct.)
[b]p7.[/b] Let $R_I$ , $R_{II}$ ,$R_{III}$ ,$R_{IV}$ be areas of the elliptical region $$\frac{(x - 10)^2}{10}+ \frac{(y-31)^2}{31} \le 2009$$ that lie in the first, second, third, and fourth quadrants, respectively. Find $R_I -R_{II} +R_{III} -R_{IV}$ .
[b]p8.[/b] Let $r_1, r_2, r_3$ be the three (not necessarily distinct) solutions to the equation $x^3+4x^2-ax+1 = 0$. If $a$ can be any real number, find the minimum possible value of
$$\left(r_1 +\frac{1}{r_1} \right)^2+ \left(r_2 +\frac{1}{r_2} \right)^2+ \left(r_3 +\frac{1}{r_3} \right)^2$$
[b]p9.[/b] Let $n$ be a positive integer. There exist positive integers $1 = a_1 < a_2 <... < a_n = 2009$ such that the average of any $n - 1$ of elements of $\{a_1, a_2,..., a_n\}$ is a positive integer. Find the maximum possible value of $n$.
[b]p10.[/b] Let $A(0) = (2, 7, 8)$ be an ordered triple. For each $n$, construct $A(n)$ from $A(n - 1)$ by replacing the $k$th position in $A(n - 1)$ by the average (arithmetic mean) of all entries in $A(n - 1)$, where $k \equiv n$ (mod $3$) and $1 \le k \le 3$. For example, $A(1) = \left( \frac{17}{3} , 7, 8 \right)$ and $A(2) = \left( \frac{17}{3} , \frac{62}{9}, 8\right)$. It is known that all entries converge to the same number $N$. Find the value of $N$.
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DMM Individual Rounds, 1998 Tie
[b]p1A[/b] Positive reals $x$, $y$, and $z$ are such that $x/y +y/x = 7$ and $y/z +z/y = 7$. There are two possible values for $z/x + x/z;$ find the greater value.
[b]p1B[/b] Real values $x$ and $y$ are such that $x+y = 2$ and $x^3+y^3 = 3$. Find $x^2+y^2$.
[b]p2[/b] Set $A = \{5, 6, 8, 13, 20, 22, 33, 42\}$. Let $\sum S$ denote the sum of the members of $S$; then $\sum A = 149$. Find the number of (not necessarily proper) subsets $B$ of $A$ for which $\sum B \ge 75$.
[b]p3[/b] $99$ dots are evenly spaced around a circle. Call two of these dots ”close” if they have $0$, $1$, or $2$ dots between them on the circle. We wish to color all $99$ dots so that any two dots which are close are colored differently. How many such colorings are possible using no more than $4$ different colors?
[b]p4[/b] Given a $9 \times 9$ grid of points, count the number of nondegenerate squares that can be drawn whose vertices are in the grid and whose center is the middle point of the grid.
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DMM Individual Rounds, 2011
[b]p1.[/b] Elsie M. is fixing a watch with three gears. Gear $A$ makes a full rotation every $5$ minutes, gear $B$ makes a full rotation every $8$ minutes, and gear $C$ makes a full rotation every $12$ minutes. The gears continue spinning until all three gears are in their original positions at the same time. How many minutes will it take for the gears to stop spinning?
[b]p2.[/b] Optimus has to pick $10$ distinct numbers from the set of positive integers $\{2, 3, 4,..., 29, 30\}$. Denote the numbers he picks by $\{a_1, a_2, ...,a_{10}\}$. What is the least possible value of $$d(a_1 ) + d(a_2) + ... + d(a_{10}),$$ where $d(n)$ denotes the number of positive integer divisors of $n$? For example, $d(33) = 4$ since $1$, $3$, $11$, and $33$ divide $33$.
[b]p3.[/b] Michael is given a large supply of both $1\times 3$ and $1\times 5$ dominoes and is asked to arrange some of them to form a $6\times 13$ rectangle with one corner square removed. What is the minimum number of $1\times 3$ dominoes that Michael can use?
[img]https://cdn.artofproblemsolving.com/attachments/6/6/c6a3ef7325ecee417e37ec9edb5374aceab9fd.png[/img]
[b]p4.[/b] Andy, Ben, and Chime are playing a game. The probabilities that each player wins the game are, respectively, the roots $a$, $b$, and $c$ of the polynomial $x^3 - x^2 + \frac{111}{400}x - \frac{9}{400} = 0$ with $a \le b \le c$. If they play the game twice, what is the probability of the same player winning twice?
[b]p5.[/b] TongTong is doodling in class and draws a $3 \times 3$ grid. She then decides to color some (that is, at least one) of the squares blue, such that no two $1 \times 1$ squares that share an edge or a corner are both colored blue. In how many ways may TongTong color some of the squares blue? TongTong cannot rotate or reflect the board.
[img]https://cdn.artofproblemsolving.com/attachments/6/0/4b4b95a67d51fda0f155657d8295b0791b3034.png[/img]
[b]p6.[/b] Given a positive integer $n$, we define $f(n)$ to be the smallest possible value of the expression $$| \square 1 \square 2 ... \square n|,$$ where we may place a $+$ or a $-$ sign in each box. So, for example, $f(3) = 0$, since $| + 1 + 2 - 3| = 0$. What is $f(1) + f(2) + ... + f(2011)$?
[b]p7.[/b] The Duke Men's Basketball team plays $11$ home games this season. For each game, the team has a $\frac34$ probability of winning, except for the UNC game, which Duke has a $\frac{9}{10}$ probability of winning. What is the probability that Duke wins an odd number of home games this season?
[b]p8.[/b] What is the sum of all integers $n$ such that $n^2 + 2n + 2$ divides $n^3 + 4n^2 + 4n - 14$?
[b]p9.[/b] Let $\{a_n\}^N_{n=1}$ be a finite sequence of increasing positive real numbers with $a_1 < 1$ such that
$$a_{n+1} = a_n \sqrt{1 - a^2_1}+ a_1\sqrt{1 - a^2_n}$$ and $a_{10} = 1/2$. What is $a_{20}$?
[b]p10.[/b] Three congruent circles are placed inside a unit square such that they do not overlap. What is the largest
possible radius of one of these circles?
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DMM Individual Rounds, 2013 (-14)
[b]p1.[/b] $p, q, r$ are prime numbers such that $p^q + 1 = r$. Find $p + q + r$.
[b]p2.[/b] $2014$ apples are distributed among a number of children such that each child gets a different number of apples. Every child gets at least one apple. What is the maximum possible number of children who receive apples?
[b]p3.[/b] Cathy has a jar containing jelly beans. At the beginning of each minute he takes jelly beans out of the jar. At the $n$-th minute, if $n$ is odd, he takes out $5$ jellies. If n is even he takes out $n$ jellies. After the $46$th minute there are only $4$ jellies in the jar. How many jellies were in the jar in the beginning?
[b]p4.[/b] David is traveling to Budapest from Paris without a cellphone and he needs to use a public payphone. He only has two coins with him. There are three pay-phones - one that never works, one that works half of the time, and one that always works. The first phone that David tries does not work. Assuming that he does not use the same phone again, what is the probability that the second phone that he uses will work?
[b]p5.[/b] Let $a, b, c, d$ be positive real numbers such that
$$a^2 + b^2 = 1$$
$$c^2 + d^2 = 1;$$
$$ad - bc =\frac17$$
Find $ac + bd$.
[b]p6.[/b] Three circles $C_A,C_B,C_C$ of radius $1$ are centered at points $A,B,C$ such that $A$ lies on $C_B$ and $C_C$, $B$ lies on $C_C$ and $C_A$, and $C$ lies on $C_A$ and $C_B$. Find the area of the region where $C_A$, $C_B$, and $C_C$ all overlap.
[b]p7.[/b] Two distinct numbers $a$ and $b$ are randomly and uniformly chosen from the set $\{3, 8, 16, 18, 24\}$. What is the probability that there exist integers $c$ and $d$ such that $ac + bd = 6$?
[b]p8.[/b] Let $S$ be the set of integers $1 \le N \le 2^{20}$ such that $N = 2^i + 2^j$ where $i, j$ are distinct integers. What is the probability that a randomly chosen element of $S$ will be divisible by $9$?
[b]p9.[/b] Given a two-pan balance, what is the minimum number of weights you must have to weigh any object that weighs an integer number of kilograms not exceeding $100$ kilograms?
[b]p10.[/b] Alex, Michael and Will write $2$-digit perfect squares $A,M,W$ on the board. They notice that the $6$-digit number $10000A + 100M +W$ is also a perfect square. Given that $A < W$, find the square root of the $6$-digit number.
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DMM Devil Rounds, 2011
[u]Round 1[/u]
[b]p1.[/b] The fractal T-shirt for this year's Duke Math Meet is so complicated that the printer broke trying to print it. Thus, we devised a method for manually assembling each shirt - starting with the full-size 'base' shirt, we paste a smaller shirt on top of it. And then we paste an even smaller shirt on top of that one. And so on, infinitely many times. (As you can imagine, it took a while to make all the shirts.) The completed T-shirt consists of the original 'base' shirt along with all of the shirts we pasted onto it. Now suppose the base shirt requires $2011$ $cm^2$ of fabric to make, and that each pasted-on shirt requires $4/5$ as much fabric as the previous one did. How many $cm^2$ of fabric in total are required to make one complete shirt?
[b]p2.[/b] A dog is allowed to roam a yard while attached to a $60$-meter leash. The leash is anchored to a $40$-meter by $20$-meter rectangular house at the midpoint of one of the long sides of the house. What is the total area of the yard that the dog can roam?
[b]p3.[/b] $10$ birds are chirping on a telephone wire. Bird $1$ chirps once per second, bird $2$ chirps once every $2$ seconds, and so on through bird $10$, which chirps every $10$ seconds. At time $t = 0$, each bird chirps. Define $f(t)$ to be the number of birds that chirp during the $t^{th}$ second. What is the smallest $t > 0$ such that $f(t)$ and $f(t + 1)$ are both at least $4$?
[u]Round 2[/u]
[b]p4.[/b] The answer to this problem is $3$ times the answer to problem 5 minus $4$ times the answer to problem 6 plus $1$.
[b]p5.[/b] The answer to this problem is the answer to problem 4 minus $4$ times the answer to problem 6 minus $1$.
[b]p6.[/b] The answer to this problem is the answer to problem 4 minus $2$ times the answer to problem 5.
[u]Round 3[/u]
[b]p7.[/b] Vivek and Daniel are playing a game. The game ends when one person wins $5$ rounds. The probability that either wins the first round is $1/2$. In each subsequent round the players have a probability of winning equal to the fraction of games that the player has lost. What is the probability that Vivek wins in six rounds?
[b]p8.[/b] What is the coefficient of $x^8y^7$ in $(1 + x^2 - 3xy + y^2)^{17}$?
[b]p9.[/b] Let $U(k)$ be the set of complex numbers $z$ such that $z^k = 1$. How many distinct elements are in the union of $U(1),U(2),...,U(10)$?
[u]Round 4[/u]
[b]p10.[/b] Evaluate $29 {30 \choose 0}+28{30 \choose 1}+27{30 \choose 2}+...+0{30 \choose 29}-{30\choose 30}$. You may leave your answer in exponential format.
[b]p11.[/b] What is the number of strings consisting of $2a$s, $3b$s and $4c$s such that $a$ is not immediately followed by $b$, $b$ is not immediately followed by $c$ and $c$ is not immediately followed by $a$?
[b]p12.[/b] Compute $\left(\sqrt3 + \tan (1^o)\right)\left(\sqrt3 + \tan (2^o)\right)...\left(\sqrt3 + \tan (29^o)\right)$.
[u]Round 5[/u]
[b]p13.[/b] Three massless legs are randomly nailed to the perimeter of a massive circular wooden table with uniform density. What is the probability that the table will not fall over when it is set on its legs?
[b]p14.[/b] Compute $$\sum^{2011}_{n=1}\frac{n + 4}{n(n + 1)(n + 2)(n + 3)}$$
[b]p15.[/b] Find a polynomial in two variables with integer coefficients whose range is the positive real numbers.
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