Difference between revisions of "2010 AIME II Problems"

Latest revision as of 20:58, 10 August 2020

2010 AIME II (Answer Key) Printable version \| AoPS Contest Collections • PDF
Instructions This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$ , inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

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 $1000$ .

Solution

Problem 2

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 $m+n$ .

Solution

Problem 3

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

Solution

Problem 4

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 $m+n$ .

Solution

Problem 5

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 $\sqrt {(\log_{10}x)^2 + (\log_{10}y)^2 + (\log_{10}z)^2}$ .

Solution

Problem 6

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.

Solution

Problem 7

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 $|a+b+c|$ .

Solution

Problem 8

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

$\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$ ,
$\mathcal{A} \cap \mathcal{B} = \emptyset$ ,
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 $N$ .

Solution

Problem 9

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 $\overline{AH}$ , $\overline{BI}$ , $\overline{CJ}$ , $\overline{DK}$ , $\overline{EL}$ , and $\overline{FG}$ 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 $m + n$ .

Solution

Problem 10

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

Solution

Problem 11

Define a T-grid to be a $3\times3$ matrix which satisfies the following two properties:

Exactly five of the entries are $1$ 's, and the remaining four entries are $0$ 's.
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.

Solution

Problem 12

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.

Solution

Problem 13

The $52$ cards in a deck are numbered $1, 2, \cdots, 52$ . Alex, Blair, Corey, and Dylan each pick a card from the deck randomly and without replacement. The two people with lower numbered cards form a team, and the two people 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 $m+n$ .

Solution

Problem 14

Triangle $ABC$ with right angle at $C$ , $\angle BAC < 45^\circ$ and $AB = 4$ . Point $P$ on $\overline{AB}$ is chosen such 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 $p$ , $q$ , $r$ are positive integers and $r$ is not divisible by the square of any prime. Find $p+q+r$ .

Solution

Problem 15

In triangle $ABC$ , $AC=13$ , $BC=14$ , and $AB=15$ . Points $M$ and $D$ lie on $AC$ with $AM=MC$ and $\angle ABD = \angle DBC$ . Points $N$ and $E$ lie on $AB$ with $AN=NB$ and $\angle ACE = \angle ECB$ . Let $P$ be the point, other than $A$ , of intersection of the circumcircles of $\triangle AMN$ and $\triangle ADE$ . Ray $AP$ meets $BC$ 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 $m-n$ .

Solution

@@ Line 1: / Line 1: @@
 {{AIME Problems|year=2010|n=II}}
-'''NOTE: THESE ARE THE PROBLEMS FROM THE AIME I. THE PROBLEMS WILL BE UPDATED SHORTLY.'''
 == Problem 1 ==
-Maya lists all the positive divisors of <math>2010^2</math>. She then randomly selects two distinct divisors from this list. Let <math>p</math> be the probability that exactly one of the selected divisors is a perfect square. The probability <math>p</math> can be expressed in the form <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
+Let <math>N</math> be the greatest integer multiple of <math>36</math> all of whose digits are even and no two of whose digits are the same. Find the remainder when <math>N</math> is divided by <math>1000</math>.
 [[2010 AIME II Problems/Problem 1|Solution]]
 == Problem 2 ==
-Find the remainder when <math>9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}</math> is divided by <math>1000</math>.
+A point <math>P</math> is chosen at random in the interior of a unit square <math>S</math>. Let <math>d(P)</math> denote the distance from <math>P</math> to the closest side of <math>S</math>. The probability that <math>\frac{1}{5}\le d(P)\le\frac{1}{3}</math> is equal to <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 [[2010 AIME II Problems/Problem 2|Solution]]
 == Problem 3 ==
-Suppose that <math>y = \frac34x</math> and <math>x^y = y^x</math>. The quantity <math>x + y</math> can be expressed as a rational number <math>\frac {r}{s}</math>, where <math>r</math> and <math>s</math> are relatively prime positive integers. Find <math>r + s</math>.
+Let <math>K</math> be the product of all factors <math>(b-a)</math> (not necessarily distinct) where <math>a</math> and <math>b</math> are integers satisfying <math>1\le a < b \le 20</math>. Find the greatest positive integer <math>n</math> such that <math>2^n</math> divides <math>K</math>.
 [[2010 AIME II Problems/Problem 3|Solution]]
 == Problem 4 ==
-Jackie and Phil have two fair coins and a third coin that comes up heads with probability <math>\frac47</math>. Jackie flips the three coins, and then Phil flips the three coins. Let <math>\frac {m}{n}</math> be the probability that Jackie gets the same number of heads as Phil, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
+Dave arrives at an airport which has twelve gates arranged in a straight line with exactly <math>100</math> 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 <math>400</math> feet or less to the new gate be a fraction <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 [[2010 AIME II Problems/Problem 4|Solution]]
 == Problem 5 ==
-Positive integers <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> satisfy <math>a > b > c > d</math>, <math>a + b + c + d = 2010</math>, and <math>a^2 - b^2 + c^2 - d^2 = 2010</math>. Find the number of possible values of <math>a</math>.
+Positive numbers <math>x</math>, <math>y</math>, and <math>z</math> satisfy <math>xyz = 10^{81}</math> and <math>(\log_{10}x)(\log_{10} yz) + (\log_{10}y) (\log_{10}z) = 468</math>. Find <math>\sqrt {(\log_{10}x)^2 + (\log_{10}y)^2 + (\log_{10}z)^2}</math>.
 [[2010 AIME II Problems/Problem 5|Solution]]
 == Problem 6 ==
-Let <math>P(x)</math> be a quadratic polynomial with real coefficients satisfying <math>x^2 - 2x + 2 \le P(x) \le 2x^2 - 4x + 3</math> for all real  numbers <math>x</math>, and suppose <math>P(11) = 181</math>. Find <math>P(16)</math>.
+Find the smallest positive integer <math>n</math> with the property that the polynomial <math>x^4 - nx + 63</math> can be written as a product of two nonconstant polynomials with integer coefficients.
 [[2010 AIME II Problems/Problem 6|Solution]]
 == Problem 7 ==
-Define an ordered triple <math>(A, B, C)</math> of sets to be minimally intersecting if <math>|A \cap B| = |B \cap C| = |C \cap A| = 1</math> and <math>A \cap B \cap C = \emptyset</math>. For example, <math>(\{1,2\},\{2,3\},\{1,3,4\})</math> is a minimally intersecting triple. Let <math>N</math> be the number of minimally intersecting ordered triples of sets for which each set is a subset of <math>\{1,2,3,4,5,6,7\}</math>. Find the remainder when <math>N</math> is divided by <math>1000</math>.
+Let <math>P(z)=z^3+az^2+bz+c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are real. There exists a complex number <math>w</math> such that the three roots of <math>P(z)</math> are <math>w+3i</math>, <math>w+9i</math>, and <math>2w-4</math>, where <math>i^2=-1</math>. Find <math>|a+b+c|</math>.
-'''Note''': <math>|S|</math> represents the number of elements in the set <math>S</math>.
 [[2010 AIME II Problems/Problem 7|Solution]]
 == Problem 8 ==
-For a real number <math>a</math>, let <math>\lfloor a \rfloor</math> denominate the greatest integer less than or equal to <math>a</math>. Let <math>\mathcal{R}</math> denote the region in the coordinate plane consisting of points <math>(x,y)</math> such that <math>\lfloor x \rfloor ^2 + \lfloor y \rfloor ^2 = 25</math>. The region <math>\mathcal{R}</math> is completely contained in a disk of radius <math>r</math> (a disk is the union of a circle and its interior). The minimum value of <math>r</math> can be written as <math>\frac {\sqrt {m}}{n}</math>, where <math>m</math> and <math>n</math> are integers and <math>m</math> is not divisible by the square of any prime. Find <math>m + n</math>.
+Let <math>N</math> be the number of ordered pairs of nonempty sets <math>\mathcal{A}</math> and <math>\mathcal{B}</math> that have the following properties:
+<UL>
+<LI> <math>\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}</math>,</LI>
+<LI> <math>\mathcal{A} \cap \mathcal{B} = \emptyset</math>, </LI>
+<LI> The number of elements of <math>\mathcal{A}</math> is not an element of <math>\mathcal{A}</math>,</LI>
+<LI> The number of elements of <math>\mathcal{B}</math> is not an element of <math>\mathcal{B}</math>.
+</UL>
+Find <math>N</math>.
 [[2010 AIME II Problems/Problem 8|Solution]]
 == Problem 9 ==
-Let <math>(a,b,c)</math> be the real solution of the system of equations <math>x^3 - xyz = 2</math>, <math>y^3 - xyz = 6</math>, <math>z^3 - xyz = 20</math>. The greatest possible value of <math>a^3 + b^3 + c^3</math> can be written in the form <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
+Let <math>ABCDEF</math> be a regular hexagon. Let <math>G</math>, <math>H</math>, <math>I</math>, <math>J</math>, <math>K</math>, and <math>L</math> be the midpoints of sides <math>AB</math>, <math>BC</math>, <math>CD</math>, <math>DE</math>, <math>EF</math>, and <math>AF</math>, respectively. The segments <math>\overline{AH}</math>, <math>\overline{BI}</math>, <math>\overline{CJ}</math>, <math>\overline{DK}</math>, <math>\overline{EL}</math>, and <math>\overline{FG}</math> bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of <math>ABCDEF</math> be expressed as a fraction <math>\frac {m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
 [[2010 AIME II Problems/Problem 9|Solution]]
 == Problem 10 ==
-Let <math>N</math> be the number of ways to write <math>2010</math> in the form <math>2010 = a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0</math>, where the <math>a_i</math>'s are integers, and <math>0 \le a_i \le 99</math>. An example of such a representation is <math>1\cdot 10^3 + 3\cdot 10^2 + 67\cdot 10^1 + 40\cdot 10^0</math>. Find <math>N</math>.
+Find the number of second-degree polynomials <math>f(x)</math> with integer coefficients and integer zeros for which <math>f(0)=2010</math>.
 [[2010 AIME II Problems/Problem 10|Solution]]
 == Problem 11 ==
-Let <math>\mathcal{R}</math> be the region consisting of the set of points in the coordinate plane that satisfy both <math>|8 - x| + y \le 10</math> and <math>3y - x \ge 15</math>. When <math>\mathcal{R}</math> is revolved around the line whose equation is <math>3y - x = 15</math>, the volume of the resulting solid is <math>\frac {m\pi}{n\sqrt {p}}</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, <math>m</math> and <math>n</math> are relatively prime, and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>.
+Define a <i>T-grid</i> to be a <math>3\times3</math> matrix which satisfies the following two properties:
+<OL>
+<LI>Exactly five of the entries are <math>1</math>'s, and the remaining four entries are <math>0</math>'s.</LI>
+<LI>Among the eight rows, columns, and long diagonals (the long diagonals are <math>\{a_{13},a_{22},a_{31}\}</math> and <math>\{a_{11},a_{22},a_{33}\})</math>, no more than one of the eight has all three entries equal.</LI></OL>
+Find the number of distinct <i>T-grids</i>.
 [[2010 AIME II Problems/Problem 11|Solution]]
 == Problem 12 ==
-Let <math>M \ge 3</math> be an integer and let <math>S = \{3,4,5,\ldots,m\}</math>. Find the smallest value of <math>m</math> such that for every partition of <math>S</math> into two subsets, at least one of the subsets contains integers <math>a</math>, <math>b</math>, and <math>c</math> (not necessarily distinct) such that <math>ab = c</math>.
+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 <math>8: 7</math>. Find the minimum possible value of their common perimeter.
-'''Note''': a partition of <math>S</math> is a pair of sets <math>A</math>, <math>B</math> such that <math>A \cap B = \emptyset</math>, <math>A \cup B = S</math>.
 [[2010 AIME II Problems/Problem 12|Solution]]
 == Problem 13 ==
-Rectangle <math>ABCD</math> and a semicircle with diameter <math>AB</math> are coplanar and have nonoverlapping interiors. Let <math>\mathcal{R}</math> denote the region enclosed by the semicircle and the rectangle. Line <math>\ell</math> meets the semicircle, segment <math>AB</math>, and segment <math>CD</math> at distinct points <math>N</math>, <math>U</math>, and <math>T</math>, respectively. Line <math>\ell</math> divides region <math>\mathcal{R}</math> into two regions with areas in the ratio <math>1: 2</math>. Suppose that <math>AU = 84</math>, <math>AN = 126</math>, and <math>UB = 168</math>. Then <math>DA</math> can be represented as <math>m\sqrt {n}</math>, where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m + n</math>.
+The <math>52</math> cards in a deck are numbered <math>1, 2, \cdots, 52</math>. Alex, Blair, Corey, and Dylan each pick a card from the deck randomly and without replacement. The two people with lower numbered cards form a team, and the two people with higher numbered cards form another team. Let <math>p(a)</math> be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards <math>a</math> and <math>a+9</math>, and Dylan picks the other of these two cards. The minimum value of <math>p(a)</math> for which <math>p(a)\ge\frac{1}{2}</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 [[2010 AIME II Problems/Problem 13|Solution]]
 == Problem 14 ==
-For each positive integer n, let <math>f(n) = \sum_{k = 1}^{100} \lfloor log_{10} (kn) \rfloor</math>. Find the largest value of n for which <math>f(n) \le 300</math>.
+Triangle <math>ABC</math> with right angle at <math>C</math>, <math>\angle BAC < 45^\circ</math> and <math>AB = 4</math>. Point <math>P</math> on <math>\overline{AB}</math> is chosen such that <math>\angle APC = 2\angle ACP</math> and <math>CP = 1</math>. The ratio <math>\frac{AP}{BP}</math> can be represented in the form <math>p + q\sqrt{r}</math>, where <math>p</math>, <math>q</math>, <math>r</math> are positive integers and <math>r</math> is not divisible by the square of any prime. Find <math>p+q+r</math>.
-'''Note:''' <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x</math>.
 [[2010 AIME II Problems/Problem 14|Solution]]
 == Problem 15 ==
-In <math>\triangle{ABC}</math> with <math>AB = 12</math>, <math>BC = 13</math>, and <math>AC = 15</math>, let <math>M</math> be a point on <math>\overline{AC}</math> such that the incircles of <math>\triangle{ABM}</math> and <math>\triangle{BCM}</math> have equal radii. Let <math>p</math> and <math>q</math> be positive relatively prime integers such that <math>\frac {AM}{CM} = \frac {p}{q}</math>. Find <math>p + q</math>.
+In triangle <math>ABC</math>, <math>AC=13</math>, <math>BC=14</math>, and <math>AB=15</math>. Points <math>M</math> and <math>D</math> lie on <math>AC</math> with <math>AM=MC</math> and <math>\angle ABD = \angle DBC</math>. Points <math>N</math> and <math>E</math> lie on <math>AB</math> with <math>AN=NB</math> and <math>\angle ACE = \angle ECB</math>. Let <math>P</math> be the point, other than <math>A</math>, of intersection of the circumcircles of <math>\triangle AMN</math> and <math>\triangle ADE</math>. Ray <math>AP</math> meets <math>BC</math> at <math>Q</math>. The ratio <math>\frac{BQ}{CQ}</math> can be written in the form <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m-n</math>.
 [[2010 AIME II Problems/Problem 15|Solution]]
 == See also ==
+{{AIME box|year=2010|n=II|before=[[2010 AIME I Problems]]|after=[[2011 AIME I Problems]]}}
+* [[American Invitational Mathematics Examination]]
 * [[American Invitational Mathematics Examination]]
 * [[AIME Problems and Solutions]]
 * [[Mathematics competition resources]]
+{{MAA Notice}}

2010 AIME II (Problems • Answer Key • Resources)
Preceded by 2010 AIME I Problems	Followed by 2011 AIME I Problems
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Difference between revisions of "2010 AIME II Problems"

Latest revision as of 20:58, 10 August 2020

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

See also