2010 AIME II

Let $N$ be the greatest integer multiple of $36$ all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by $1000.$

A point $P$ is chosen at random in the interior of a unit square $S.$ Let $d(P)$ denote the distance from $P$ to the closest side of $S.$ The probability that $\frac{1}{5} \le d(P) \le \frac{1}{3}$ is equal to $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Let $K$ be the product of all factors $(b - a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1 \le a \lt b \le 20.$ Find the greatest positive integer $n$ such that $2^n$ divides $K.$

Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $400$ feet or less to the new gate be a fraction $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Positive numbers $x,$ $y,$ and $z$ satisfy $xyz = 10^{81}$ and $(\log_{10} x)(\log_{10} yz)$ $+ (\log_{10} y)(\log_{10} z) = 468.$ Find $\small \sqrt{(\log_{10} x)^2 + (\log_{10} y)^2 + (\log_{10} z)^2}.$

Find the smallest positive integer $n$ with the property that the polynomial $x^4 - nx + 63$ can be written as a product of two nonconstant polynomials with integer coefficients.

Let $P(z) = z^3 + az^2 + bz + c,$ where $a,$ $b,$ and $c$ are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w + 3i,$ $w + 9i,$ and $2w - 4,$ where $i^2 = -1.$ Find $|a + b + c|.$

Let $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that have the following properties:

• $\mathcal{A} \cup \mathcal{B}$ $\small = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\},$

• $\mathcal{A} \cap \mathcal{B} = \emptyset,$

• the number of elements of $\mathcal{A}$ is not an element of $\mathcal{A},$

• the number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}.$

Find $N.$

Let $ABCDEF$ be a regular hexagon. Let $G,$ $H,$ $I,$ $J,$ $K,$ and $L$ be the midpoints of sides $AB,$ $BC,$ $CD,$ $DE,$ $EF,$ and $AF,$ respectively. The segments $AH,$ $BI,$ $CJ,$ $DK,$ $EL,$ and $FG$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of $ABCDEF$ be expressed as a fraction $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0) = 2010.$

Define a *T-grid* to be a $3 \times 3$ matrix which satisfies the following two properties:

$(1)$ Exactly five of the entries are $1$’s, and the remaining four entries are $0$’s.

$(2)$ Among the eight rows, columns, and long diagonals (the long diagonals are $\{a_{13}, a_{22}, a_{31}\}$ and $\{a_{11}, a_{22}, a_{33}\}$), no more than one of the eight has all three entries equal.

Find the number of distinct *T-grids*.

Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $8 : 7.$ Find the minimum possible value of their common perimeter.

The $52$ cards in a deck are numbered $1,$ $2,$ $\ldots,$ $52.$ Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked. The two persons with lower numbered cards form a team, and the two persons with higher numbered cards form another team. Let $p(a)$ be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards $a$ and $a + 9,$ and Dylan picks the other of these two cards. The minimum value of $p(a)$ for which $p(a) \ge \frac{1}{2}$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

In right triangle $ABC$ with the right angle at $C,$ $\angle BAC \lt 45^\circ$ and $AB = 4.$ Point $P$ on $\overline{AB}$ has the properties that $\angle APC = 2\angle ACP$ and $CP = 1.$ The ratio $\frac{AP}{BP}$ can be represented in the form $p + q\sqrt{r},$ where $p,$ $q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime. Find $p + q + r.$

In triangle $ABC,$ $AC = 13,$ $BC = 14,$ and $AB = 15.$ Points $M$ and $D$ lie on $\overline{AC}$ with $AM = MC$ and $\angle ABD = \angle DBC.$ Points $N$ and $E$ lie on $\overline{AB}$ with $AN = NB$ and $\angle ACE = \angle ECB.$ Let $P$ be the other point of intersection of the circumcircles of $\triangle AMN$ and $\triangle ADE.$ Ray $AP$ meets $\overline{BC}$ at $Q.$ The ratio $\frac{BQ}{CQ}$ can be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m - n.$