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Suppose That Y 34x and Xy Yx Art of Problem Solving

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2010 AIME I (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. 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 right answers; i.due east., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • two • 3 • 4 • 5 • half dozen • 7 • eight • 9 • 10 • 11 • 12 • thirteen • 14 • xv

Problem one

Maya lists all the positive divisors of $2010^2$. She and so randomly selects two singled-out divisors from this list. Permit $p$ be the probability that exactly one of the selected divisors is a perfect foursquare. The probability $p$ can exist expressed in the class $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

Problem 2

Find the residuum when $9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}$ is divided past $1000$.

Solution

Problem iii

Suppose that $y = \frac34x$ and $x^y = y^x$. The quantity $x + y$ can be expressed as a rational number $\frac {r}{s}$, where $r$ and $s$ are relatively prime positive integers. Find $r + s$.

Solution

Problem 4

Jackie and Phil have 2 fair coins and a third coin that comes upward heads with probability $\frac47$. Jackie flips the iii coins, and so Phil flips the 3 coins. Let $\frac {m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

Problem 5

Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2010$, and $a^2 - b^2 + c^2 - d^2 = 2010$. Find the number of possible values of $a$.

Solution

Problem half-dozen

Allow $P(x)$ be a quadratic polynomial with existent coefficients satisfying $x^2 - 2x + 2 \le P(x) \le 2x^2 - 4x + 3$ for all real numbers $x$, and suppose $P(11) = 181$. Observe $P(16)$.

Solution

Problem seven

Define an ordered triple $(A, B, C)$ of sets to be $\textit{minimally intersecting}$ if $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. For example, $(\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $N$ be the number of minimally intersecting ordered triples of sets for which each set is a subset of $\{1,2,3,4,5,6,7\}$. Find the rest when $N$ is divided by $1000$.

Notation: $|S|$ represents the number of elements in the set $S$.

Solution

Trouble 8

For a real number $a$, let $\lfloor a \rfloor$ denote the greatest integer less than or equal to $a$. Permit $\mathcal{R}$ denote the region in the coordinate plane consisting of points $(x,y)$ such that $\lfloor x \rfloor ^2 + \lfloor y \rfloor ^2 = 25$. The region $\mathcal{R}$ is completely independent in a disk of radius $r$ (a deejay is the spousal relationship of a circle and its interior). The minimum value of $r$ can exist written equally $\frac {\sqrt {m}}{n}$, where $m$ and $n$ are integers and $m$ is not divisible by the foursquare of any prime. Discover $m + n$.

Solution

Problem nine

Let $(a,b,c)$ be a existent solution of the organisation of equations $x^3 - xyz = 2$, $y^3 - xyz = 6$, $z^3 - xyz = 20$. The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Notice $m + n$.

Solution

Problem x

Let $N$ exist the number of means to write $2010$ in the course $2010 = a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0$, where the $a_i$'southward are integers, and $0 \le a_i \le 99$. An example of such a representation is $1\cdot 10^3 + 3\cdot 10^2 + 67\cdot 10^1 + 40\cdot 10^0$. Find $N$.

Solution

Problem 11

Let $\mathcal{R}$ be the region consisting of the set of points in the coordinate aeroplane that satisfy both $|8 - x| + y \le 10$ and $3y - x \ge 15$. When $\mathcal{R}$ is revolved around the line whose equation is $3y - x = 15$, the book of the resulting solid is $\frac {m\pi}{n\sqrt {p}}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is non divisible past the square of whatever prime number. Find $m + n + p$.

Solution

Problem 12

Let $m \ge 3$ exist an integer and let $S = \{3,4,5,\ldots,m\}$. Find the smallest value of $m$ such that for every segmentation of $S$ into two subsets, at to the lowest degree i of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.

Note: a partition of $S$ is a pair of sets $A$, $B$ such that $A \cap B = \emptyset$, $A \cup B = S$.

Solution

Problem 13

Rectangle $ABCD$ and a semicircle with diameter $AB$ are coplanar and have nonoverlapping interiors. Let $\mathcal{R}$ announce the region enclosed by the semicircle and the rectangle. Line $\ell$ meets the semicircle, segment $AB$, and segment $CD$ at distinct points $N$, $U$, and $T$, respectively. Line $\ell$ divides region $\mathcal{R}$ into two regions with areas in the ratio $1: 2$. Suppose that $AU = 84$, $AN = 126$, and $UB = 168$. And then $DA$ can be represented as $m\sqrt {n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the foursquare of any prime. Notice $m + n$.

Solution

Problem 14

For each positive integer $n,$ let $f(n) = \sum_{k = 1}^{100} \lfloor \log_{10} (kn) \rfloor$. Detect the largest value of $n$ for which $f(n) \le 300$.

Note: $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.

Solution

Trouble xv

In $\triangle{ABC}$ with $AB = 12$, $BC = 13$, and $AC = 15$, permit $M$ exist a point on $\overline{AC}$ such that the incircles of $\triangle{ABM}$ and $\triangle{BCM}$ have equal radii. Then $\frac{AM}{CM} = \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

Solution

See also

2010 AIME I (Problems • Answer Central • Resources)
Preceded past
2009 AIME Two Bug 		Followed by
2010 AIME II Issues
1 • ii • 3 • 4 • v • 6 • seven • 8 • 9 • ten • 11 • 12 • 13 • xiv • 15
All AIME Problems and Solutions
  • American Invitational Mathematics Exam
  • American Invitational Mathematics Exam
  • AIME Problems and Solutions
  • Mathematics competition resources

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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Source: https://artofproblemsolving.com/wiki/index.php/2010_AIME_I_Problems

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