Mock AIME 4 2006-2007 Problems

Problem 1

Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes $1, 10, 11, 12, \ldots$ but by the 1,000th digit he (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last three digits he wrote (the 998th, 999th, and 1000th digits, in that order).

Solution

Problem 2

Two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are chosen on the graph of $f(x) = \ln x$, with $0 < x_1 < x_2$. The points $C$ and $D$ trisect $\overline{AB}$, with $AC < CB$. Through $C$ a horizontal line is drawn to cut the curve at $E(x_3, y_3)$. Find $x_3$ if $x_1 = 1$ and $x_2 = 1000$.

Solution

Problem 3

Find the largest prime factor of the smallest positive integer $n$ such that $r_1, r_2, \ldots , r_{2006}$ are distinct integers such that the polynomial $(x-r_{1})(x-r_{2})\cdots (x-r_{2006})$ has exactly $n$ nonzero coefficients.

Solution

Problem 4

Points $A$, $B$, and $C$ are on the circumference of a unit circle so that the measure of $\widehat{AB}$ is $72^{\circ}$, the measure of $\widehat{BC}$ is $36^{\circ}$, and the measure of $\widehat{AC}$ is $108^\circ$. The area of the triangular shape bounded by $\widehat{BC}$ and line segments $\overline{AB}$ and $\overline{AC}$ can be written in the form $\frac{m}{n} \cdot \pi$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

Problem 5

How many 10-digit positive integers have all digits either 1 or 2, and have two consecutive 1's?

Solution

Problem 6

For how many positive integers $n < 1000$ does there exist a regular $n$-sided polygon such that the number of diagonals is a nonzero perfect square?

Solution

Problem 7

Find the remainder when $3^{3^{3^3}}$ is divided by 1000.

Solution

Problem 8

The number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_{10} \le 2007$ such that $a_i-i$ is even for $1\le i \le 10$ can be expressed as ${m \choose n}$ for some positive integers $m > n$. Compute the remainder when $m$ is divided by 1000.

Solution

Problem 9

Compute the smallest positive integer $k$ such that the fraction

$\frac{7k+100}{5k-3}$

is reducible.

Solution

Problem 10

Compute the remainder when

${2007 \choose 0} + {2007 \choose 3} + \cdots + {2007 \choose 2007}$

is divided by 1000.

Solution

Problem 11

Let $\triangle ABC$ be an equilateral triangle. Two points $D$ and $E$ are chosen on $\overline{AB}$ and $\overline{AC}$, respectively, such that $AD = CE$. Let $F$ be the intersection of $\overline{BE}$ and $\overline{CD}$. The area of $\triangle ABC$ is 13 and the area of $\triangle ACF$ is 3. If $\frac{CE}{EA}=\frac{p+\sqrt{q}}{r}$, where $p$, $q$, and $r$ are relatively prime positive integers, compute $p+q+r$.

Solution

Problem 12

The number of partitions of 2007 that have an even number of even parts can be expressed as $a^b$, where $a$ and $b$ are positive integers and $a$ is prime. Find the sum of the digits of $a + b$.

Solution

Problem 13

The sum

$\sum_{k=1}^{2007} \arctan\left(\frac{1}{k^2+k+1}\right)$

can be written in the form $\arctan\left(\frac{m}{n}\right)$, where $\gcd(m,n) = 1$. Compute the remainder when $m+n$ is divided by 100.

Solution

Problem 14

Let $x$ be the arithmetic mean of all positive integers $k<577$ such that

$k^4\equiv 144\pmod {577}$.

Find the greatest integer less than or equal to $x$.

Solution

Problem 15

Triangle $ABC$ has sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ of length 43, 13, and 48, respectively. Let $\omega$ be the circle circumscribed around $\triangle ABC$ and let $D$ be the intersection of $\omega$ and the perpendicular bisector of $\overline{AC}$ that is not on the same side of $\overline{AC}$ as $B$. The length of $\overline{AD}$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find the greatest integer less than or equal to $m + \sqrt{n}$.

Solution

See also

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