HMMT November 2008

Find the minimum of $x^{2}-2 x$ over all real numbers $x$.

How many diagonals does a regular undecagon (11-sided polygon) have?

How many numbers between 1 and $1,000,000$ are perfect squares but not perfect cubes?

Joe has a triangle with area $\sqrt{3}$. What's the smallest perimeter it could have?

We say " $s$ grows to $r$ " if there exists some integer $n>0$ such that $s^{n}=r$. Call a real number $r$ "sparse" if there are only finitely many real numbers $s$ that grow to $r$. Find all real numbers that are sparse.

Find all ordered pairs $(x, y)$ such that

$$(x-2 y)^{2}+(y-1)^{2}=0$$

How many integers between 2 and 100 inclusive cannot be written as $m \cdot n$, where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to 1 ? Note that there are 25 primes less than 100 .

Find the product of all real $x$ for which

$$2^{3 x+1}-17 \cdot 2^{2 x}+2^{x+3}=0$$

Find the largest positive integer $n$ such that $n^{3}+4 n^{2}-15 n-18$ is the cube of an integer.

A triangle has sides of length 9,40 , and 41 . What is its area?

Let $A B C$ be a triangle, and let $M$ be the midpoint of side $A B$. If $A B$ is 17 units long and $C M$ is 8 units long, find the maximum possible value of the area of $A B C$.

Let $D E F$ be a triangle and $H$ the foot of the altitude from $D$ to $E F$. If $D E=60, D F=35$, and $D H=21$, what is the difference between the minimum and the maximum possible values for the area of $D E F$ ?

Right triangle $X Y Z$, with hypotenuse $Y Z$, has an incircle of radius $\frac{3}{8}$ and one leg of length 3. Find the area of the triangle.

A triangle has altitudes of length 15,21 , and 35 . Find its area.

Explain how any unit fraction $\frac{1}{n}$ can be decomposed into other unit fractions.

(a) Write 1 as a sum of 4 distinct unit fractions.

Say that $\frac{a}{b}$ is a positive rational number in simplest form, with $a \neq 1$. Further, say that $n$ is an integer such that:

$$\frac{1}{n}>\frac{a}{b}>\frac{1}{n+1}$$

Show that when $\frac{a}{b}-\frac{1}{n+1}$ is written in simplest form, its numerator is smaller than $a$.

An aside: the sum of all the unit fractions

It is possible to show that, given any real M , there exists a positive integer $k$ large enough that:

$$\sum_{n=1}^{k} \frac{1}{n}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3} \ldots>M$$

Note that this statement means that the infinite harmonic series, $\sum_{n=1}^{\infty} \frac{1}{n}$, grows without bound, or diverges. For the specific example $\mathrm{M}=5$, find a value of $k$, not necessarily the smallest, such that the inequality holds. Justify your answer.

Now, using information from problems 4 and 5 , prove that the following method to decompose any positive rational number will always terminate: Step 1. Start with the fraction $\frac{a}{b}$. Let $t_{1}$ be the largest unit fraction $\frac{1}{n}$ which is less than or equal to $\frac{a}{b}$. Step 2. If we have already chosen $t_{1}$ through $t_{k}$, and if $t_{1}+t_{2}+\ldots+t_{k}$ is still less than $\frac{a}{b}$, then let $t_{k+1}$ be the largest unit fraction less than both $t_{k}$ and $\frac{a}{b}$. Step 3. If $t_{1}+\ldots+t_{k+1}$ equals $\frac{a}{b}$, the decomposition is found. Otherwise, repeat step 2 . Why does this method never result in an infinite sequence of $t_{i}$ ?

Find the sum of all solutions for $x$ :

$$\begin{aligned} x y & =1 \\ x+y & =3 \end{aligned}$$

Evaluate the sum

$$1-2+3-4+\cdots+2007-2008$$

What is the largest $x$ such that $x^{2}$ divides $24 \cdot 35 \cdot 46 \cdot 57$ ?

What is the smallest prime divisor of $5^{7^{10^{7^{10}}}}+1$ ?

What is the sum of all integers $x$ such that $|x+2| \leq 10$ ?

Sarah is deciding whether to visit Russia or Washington, DC for the holidays. She makes her decision by rolling a regular 6 -sided die. If she gets a 1 or 2 , she goes to DC . If she rolls a 3,4 , or 5 , she goes to Russia. If she rolls a 6 , she rolls again. What is the probability that she goes to DC?

Alice rolls two octahedral dice with the numbers $2,3,4,5,6,7,8,9$. What's the probability the two dice sum to 11 ?

Let $a_{0}=\frac{6}{7}$, and

$$a_{n+1}= \begin{cases}2 a_{n} & \text { if } a_{n}<\frac{1}{2} \\ 2 a_{n}-1 & \text { if } a_{n} \geq \frac{1}{2}\end{cases}$$

Find $a_{2008}$.

Find the sum of all positive integers $n$ such that $n$ divides $n^{2}+n+2$.

Al has a rectangle of integer side lengths $a$ and $b$, and area 1000 . What is the smallest perimeter it could have?

Solve the following system of equations for $w$.

$$\begin{aligned} & 2 w+x+y+z=1 \\ & w+2 x+y+z=2 \\ & w+x+2 y+z=2 \\ & w+x+y+2 z=1 \end{aligned}$$

Find the number of distinct primes dividing $1 \cdot 2 \cdot 3 \cdots 9 \cdot 10$.

You have a $2 \times 3$ grid filled with integers between 1 and 9 . The numbers in each row and column are distinct, the first row sums to 23 , and the columns sum to 14,16 , and 17 respectively.

	14	16	17
23	$a$	$b$	$c$
	$x$	$y$	$z$

What is $x+2 y+3 z ?$

A cat is going up a stairwell with ten stairs. However, instead of walking up the stairs one at a time, the cat jumps, going either two or three stairs up at each step (though if necessary, it will just walk the last step). How many different ways can the cat go from the bottom to the top?

If $p$ and $q$ are positive integers and $\frac{2008}{2009}<\frac{p}{q}<\frac{2009}{2010}$, what is the minimum value of $p$ ?

Determine the last two digits of $17^{17}$, written in base 10 .

Find the coefficient of $x^{6}$ in the expansion of

$$(x+1)^{6} \cdot \sum_{i=0}^{6} x^{i}$$

Let $P$ be a polynomial with $P(1)=P(2)=\cdots=P(2007)=0$ and $P(0)=2009$ !. $P(x)$ has leading coefficient 1 and degree 2008. Find the largest root of $P(x)$.

You have a die with faces labelled 1 through 6 . On each face, you draw an arrow to an adjacent face, such that if you start on a face and follow the arrows, after 6 steps you will have passed through every face once and will be back on your starting face. How many ways are there to draw the arrows so that this is true?

Call a number overweight if it has at least three positive integer divisors (including 1 and the number), and call a number obese if it has at least four positive integer divisors (including 1 and the number). How many positive integers between 1 and 200 are overweight, but not obese?

Sandra the Maverick has 5 pairs of shoes in a drawer, each pair a different color. Every day for 5 days, Sandra takes two shoes out and throws them out the window. If they are the same color, she treats herself to a practice problem from a past HMMT. What is the expected value (average number) of practice problems she gets to do?

If $x$ and $y$ are real numbers such that $\frac{(x-4)^{2}}{4}+\frac{y^{2}}{9}=1$, find the largest possible value of $\frac{x^{2}}{4}+\frac{y^{2}}{9}$.

Let $f(x)=\frac{1}{1-x}$. Let $f^{k+1}(x)=f\left(f^{k}(x)\right)$, with $f^{1}(x)=f(x)$. What is $f^{2008}(2008)$ ?

Evaluate the sum

$$\cos \left(\frac{2 \pi}{18}\right)+\cos \left(\frac{4 \pi}{18}\right)+\cdots+\cos \left(\frac{34 \pi}{18}\right)$$

John M. is sitting at ( 0,0 ), looking across the aisle at his friends sitting at $(i, j)$ for each $1 \leq i \leq 10$ and $0 \leq j \leq 5$. Unfortunately, John can only see a friend if the line connecting them doesn't pass through any other friend. How many friends can John see?

$A B C D E$ is a regular pentagon inscribed in a circle of radius 1 . What is the area of the set of points inside the circle that are farther from $A$ than they are from any other vertex?

Johnny the grad student is typing all the integers from 1 to $\infty$, in order. The 2 on his computer is broken however, so he just skips any number with a 2 . What's the 2008th number he types?

Let $p(x)$ be the polynomial of degree 4 with roots $1,2,3,4$ and leading coefficient 1 . Let $q(x)$ be the polynomial of degree 4 with roots $1, \frac{1}{2}, \frac{1}{3}$, and $\frac{1}{4}$ and leading coefficient 1 . Find $\lim _{x \rightarrow 1} \frac{p(x)}{q(x)}$.

Alice has an equilateral triangle $A B C$ of area 1. Put $D$ on $B C, E$ on $C A$, and $F$ on $A B$, with $B D=D C, C E=2 E A$, and $2 A F=F B$. Note that $A D, B E$, and $C F$ pass through a single point $M$. What is the area of triangle $E M C$ ?

Find the sum of all primes $p$ for which there exists a prime $q$ such that $p^{2}+p q+q^{2}$ is a square.

Pirate ships Somy and Lia are having a tough time. At the end of the year, they are both one pillage short of the minimum required for maintaining membership in the Pirate Guild, so they decide to pillage each other to bring their counts up. Somy by tradition only pillages $28 \cdot 3^{k}$ coins for integers $k$, and Lia by tradition only pillages $82 \cdot 3^{j}$ coins for integers $j$. Note that each pillage can have a different $k$ or $j$. Soma and Lia work out a system where Somy pillages Lia $n$ times, Lia pillages Somy $n$ times, and after both sets of pillages Somy and Lia are financially even. What is the smallest $n$ can be?

The polynomial $a x^{2}-b x+c$ has two distinct roots $p$ and $q$, with $a, b$, and $c$ positive integers and with $0<p, q<1$. Find the minimum possible value of $a$.

Find $\max \{\operatorname{Perimeter}(T)\}$ for $T$ a triangle contained in a regular septagon (7-sided figure) of unit edge length. Write your answer $N$ to 2 places after the decimal. If the correct answer rounded to 2 decimal places is $A$, you will receive 0 points if $N<A$ and $\lfloor\max \{0,25-50 \cdot(N-A)\}\rfloor$ points otherwise.

How many numbers less than $1,000,000$ are the product of exactly 2 distinct primes? You will receive $\left\lfloor 25-50 \cdot\left|\frac{N}{A}-1\right|\right\rfloor$ points, if you submit $N$ and the correct answer is $A$.