HMMT November 2011

Find all ordered pairs of real numbers $(x, y)$ such that $x^{2} y=3$ and $x+x y=4$.

Let $A B C$ be a triangle, and let $D, E$, and $F$ be the midpoints of sides $B C, C A$, and $A B$, respectively. Let the angle bisectors of $\angle F D E$ and $\angle F B D$ meet at $P$. Given that $\angle B A C=37^{\circ}$ and $\angle C B A=85^{\circ}$, determine the degree measure of $\angle B P D$.

Alberto, Bernardo, and Carlos are collectively listening to three different songs. Each is simultaneously listening to exactly two songs, and each song is being listened to by exactly two people. In how many ways can this occur?

Determine the remainder when

$$2^{\frac{1 \cdot 2}{2}}+2^{\frac{2 \cdot 3}{2}}+\cdots+2^{\frac{2011 \cdot 2012}{2}}$$

is divided by 7 .

Find all real values of $x$ for which

$$\frac{1}{\sqrt{x}+\sqrt{x-2}}+\frac{1}{\sqrt{x+2}+\sqrt{x}}=\frac{1}{4}$$

Five people of heights $65,66,67,68$, and 69 inches stand facing forwards in a line. How many orders are there for them to line up, if no person can stand immediately before or after someone who is exactly 1 inch taller or exactly 1 inch shorter than himself?

Determine the number of angles $\theta$ between 0 and $2 \pi$, other than integer multiples of $\pi / 2$, such that the quantities $\sin \theta, \cos \theta$, and $\tan \theta$ form a geometric sequence in some order.

Find the number of integers $x$ such that the following three conditions all hold:

• $x$ is a multiple of 5
• $121<x<1331$
• When $x$ is written as an integer in base 11 with no leading 0s (i.e. no 0s at the very left), its rightmost digit is strictly greater than its leftmost digit.

Let $P$ and $Q$ be points on line $l$ with $P Q=12$. Two circles, $\omega$ and $\Omega$, are both tangent to $l$ at $P$ and are externally tangent to each other. A line through $Q$ intersects $\omega$ at $A$ and $B$, with $A$ closer to $Q$ than $B$, such that $A B=10$. Similarly, another line through $Q$ intersects $\Omega$ at $C$ and $D$, with $C$ closer to $Q$ than $D$, such that $C D=7$. Find the ratio $A D / B C$.

Let $r_{1}, r_{2}, \ldots, r_{7}$ be the distinct complex roots of the polynomial $P(x)=x^{7}-7$. Let

$$K=\prod_{1 \leq i<j \leq 7}\left(r_{i}+r_{j}\right)$$

that is, the product of all numbers of the form $r_{i}+r_{j}$, where $i$ and $j$ are integers for which $1 \leq i<j \leq 7$. Determine the value of $K^{2}$.

Five of James' friends are sitting around a circular table to play a game of Fish. James chooses a place between two of his friends to pull up a chair and sit. Then, the six friends divide themselves into two disjoint teams, with each team consisting of three consecutive players at the table. If the order in which the three members of a team sit does not matter, how many possible (unordered) pairs of teams are possible?

In a game of Fish, R2 and R3 are each holding a positive number of cards so that they are collectively holding a total of 24 cards. Each player gives an integer estimate for the number of cards he is holding, such that each estimate is an integer between $80 \%$ of his actual number of cards and $120 \%$ of his actual number of cards, inclusive. Find the smallest possible sum of the two estimates.

Z gives one card to D.

He repeats steps 1-4 for each card dealt, including the last card. After all the cards have been dealt, what is the probability that each player has exactly 8 cards?

For any finite sequence of positive integers $\pi$, let $S(\pi)$ be the number of strictly increasing subsequences in $\pi$ with length 2 or more. For example, in the sequence $\pi=\{3,1,2,4\}$, there are five increasing sub-sequences: $\{3,4\},\{1,2\},\{1,4\},\{2,4\}$, and $\{1,2,4\}$, so $S(\pi)=5$. In an eight-player game of Fish, Joy is dealt six cards of distinct values, which she puts in a random order $\pi$ from left to right in her hand. Determine

$$\sum_{\pi} S(\pi)$$

where the sum is taken over all possible orders $\pi$ of the card values.

Let $A B C$ be an equilateral triangle with $A B=3$. Circle $\omega$ with diameter 1 is drawn inside the triangle such that it is tangent to sides $A B$ and $A C$. Let $P$ be a point on $\omega$ and $Q$ be a point on segment $B C$. Find the minimum possible length of the segment $P Q$.

Let $X Y Z$ be a triangle with $\angle X Y Z=40^{\circ}$ and $\angle Y Z X=60^{\circ}$. A circle $\Gamma$, centered at the point $I$, lies inside triangle $X Y Z$ and is tangent to all three sides of the triangle. Let $A$ be the point of tangency of $\Gamma$ with $Y Z$, and let ray $\overrightarrow{X I}$ intersect side $Y Z$ at $B$. Determine the measure of $\angle A I B$.

Points $D, E, F$ lie on circle $O$ such that the line tangent to $O$ at $D$ intersects ray $\overrightarrow{E F}$ at $P$. Given that $P D=4, P F=2$, and $\angle F P D=60^{\circ}$, determine the area of circle $O$.

Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. Let $D$ be the foot of the altitude from $A$ to $B C$. The inscribed circles of triangles $A B D$ and $A C D$ are tangent to $A D$ at $P$ and $Q$, respectively, and are tangent to $B C$ at $X$ and $Y$, respectively. Let $P X$ and $Q Y$ meet at $Z$. Determine the area of triangle $X Y Z$.

Let $\Omega$ be a circle of radius 8 centered at point $O$, and let $M$ be a point on $\Omega$. Let $S$ be the set of points $P$ such that $P$ is contained within $\Omega$, or such that there exists some rectangle $A B C D$ containing $P$ whose center is on $\Omega$ with $A B=4, B C=5$, and $B C \| O M$. Find the area of $S$.

Find the number of positive integers $x$ less than 100 for which

$$3^{x}+5^{x}+7^{x}+11^{x}+13^{x}+17^{x}+19^{x}$$

is prime.

Determine the set of all real numbers $p$ for which the polynomial $Q(x)=x^{3}+p x^{2}-p x-1$ has three distinct real roots.

Find the sum of the coefficients of the polynomial $P(x)=x^{4}-29 x^{3}+a x^{2}+b x+c$, given that $P(5)=11, P(11)=17$, and $P(17)=23$.

Determine the number of quadratic polynomials $P(x)=p_{1} x^{2}+p_{2} x-p_{3}$, where $p_{1}, p_{2}, p_{3}$ are not necessarily distinct (positive) prime numbers less than 50 , whose roots are distinct rational numbers.

Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four two-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the adjacent square in the subgrid that is 90 degrees away in the clockwise direction, without changing the orientation of the C. Given that two colorings are the considered same if and only if one can be obtained from the other by a series of quadrant operations, determine the number of distinct colorings of the Cs.

C	C	C	C
C	C	C	C
C	C	C	C
C	C	C	C

Ten Cs are written in a row. Some Cs are upper-case and some are lower-case, and each is written in one of two colors, green and yellow. It is given that there is at least one lower-case C, at least one green C , and at least one C that is both upper-case and yellow. Furthermore, no lower-case C can be followed by an upper-case C , and no yellow C can be followed by a green C . In how many ways can the Cs be written?

Julia is learning how to write the letter C. She has 6 differently-colored crayons, and wants to write Cc Cc Cc Cc Cc . In how many ways can she write the ten Cs , in such a way that each upper case C is a different color, each lower case C is a different color, and in each pair the upper case C and lower case C are different colors?

Let $G, A_{1}, A_{2}, A_{3}, A_{4}, B_{1}, B_{2}, B_{3}, B_{4}, B_{5}$ be ten points on a circle such that $G A_{1} A_{2} A_{3} A_{4}$ is a regular pentagon and $G B_{1} B_{2} B_{3} B_{4} B_{5}$ is a regular hexagon, and $B_{1}$ lies on minor arc $G A_{1}$. Let $B_{5} B_{3}$ intersect $B_{1} A_{2}$ at $G_{1}$, and let $B_{5} A_{3}$ intersect $G B_{3}$ at $G_{2}$. Determine the degree measure of $\angle G G_{2} G_{1}$.

Let $A B C$ be a triangle with $A B=9, B C=10$, and $C A=17$. Let $B^{\prime}$ be the reflection of the point $B$ over the line $C A$. Let $G$ be the centroid of triangle $A B C$, and let $G^{\prime}$ be the centroid of triangle $A B^{\prime} C$. Determine the length of segment $G G^{\prime}$.

Let $G_{1} G_{2} G_{3}$ be a triangle with $G_{1} G_{2}=7, G_{2} G_{3}=13$, and $G_{3} G_{1}=15$. Let $G_{4}$ be a point outside triangle $G_{1} G_{2} G_{3}$ so that ray $\overrightarrow{G_{1} G_{4}}$ cuts through the interior of the triangle, $G_{3} G_{4}=G_{4} G_{2}$, and $\angle G_{3} G_{1} G_{4}=30^{\circ}$. Let $G_{3} G_{4}$ and $G_{1} G_{2}$ meet at $G_{5}$. Determine the length of segment $G_{2} G_{5}$.

Determine the remainder when $1+2+\cdots+2014$ is divided by 2012 .

Let $A B C D$ be a rectangle with $A B=6$ and $B C=4$. Let $E$ be the point on $B C$ with $B E=3$, and let $F$ be the point on segment $A E$ such that $F$ lies halfway between the segments $A B$ and $C D$. If $G$ is the point of intersection of $D F$ and $B C$, find $B G$.

Let $x$ be a real number such that $2^{x}=3$. Determine the value of $4^{3 x+2}$.

Determine which of the following numbers is smallest in value: $54 \sqrt{3}, 144,108 \sqrt{6}-108 \sqrt{2}$.

Charlie folds an $\frac{17}{2}$-inch by 11-inch piece of paper in half twice, each time along a straight line parallel to one of the paper's edges. What is the smallest possible perimeter of the piece after two such folds?

To survive the coming Cambridge winter, Chim Tu doesn't wear one T-shirt, but instead wears up to FOUR T-shirts, all in different colors. An outfit consists of three or more T-shirts, put on one on top of the other in some order, such that two outfits are distinct if the sets of T-shirts used are different or the sets of T-shirts used are the same but the order in which they are worn is different. Given that Chim Tu changes his outfit every three days, and otherwise never wears the same outfit twice, how many days of winter can Chim Tu survive? (Needless to say, he only has four t-shirts.)

How many ordered triples of positive integers $(a, b, c)$ are there for which $a^{4} b^{2} c=54000$ ?

Let $a, b, c$ be not necessarily distinct integers between 1 and 2011, inclusive. Find the smallest possible value of $\frac{a b+c}{a+b+c}$.

Unit circle $\Omega$ has points $X, Y, Z$ on its circumference so that $X Y Z$ is an equilateral triangle. Let $W$ be a point other than $X$ in the plane such that triangle $W Y Z$ is also equilateral. Determine the area of the region inside triangle $W Y Z$ that lies outside circle $\Omega$.

Determine the number of integers $D$ such that whenever $a$ and $b$ are both real numbers with $-1 / 4<a, b<1 / 4$, then $\left|a^{2}-D b^{2}\right|<1$.

For positive integers $m, n$, let $\operatorname{gcd}(m, n)$ denote the largest positive integer that is a factor of both $m$ and $n$. Compute

$$\sum_{n=1}^{91} \operatorname{gcd}(n, 91) .$$

Joe has written 5 questions of different difficulties for a test with problems numbered 1 though 5 . He wants to make sure that problem $i$ is harder than problem $j$ whenever $i-j \geq 3$. In how many ways can he order the problems for his test?

Tac is dressing his cat to go outside. He has four indistinguishable socks, four indistinguishable shoes, and 4 indistinguishable show-shoes. In a hurry, Tac randomly pulls pieces of clothing out of a door and tries to put them on a random one of his cat's legs; however, Tac never tries to put more than one of each type of clothing on each leg of his cat. What is the probability that, after Tac is done, the snow-shoe on each of his cat's legs is on top of the shoe, which is on top of the sock?

Let $A M O L$ be a quadrilateral with $A M=10, M O=11$, and $O L=12$. Given that the perpendicular bisectors of sides $A M$ and $O L$ intersect at the midpoint of segment $A O$, find the length of side $L A$.

For positive integers $n$, let $L(n)$ be the largest factor of $n$ other than $n$ itself. Determine the number of ordered pairs of composite positive integers $(m, n)$ for which $L(m) L(n)=80$.

A small fish is holding 17 cards, labeled 1 through 17 , which he shuffles into a random order. Then, he notices that although the cards are not currently sorted in ascending order, he can sort them into ascending order by removing one card and putting it back in a different position (at the beginning, between some two cards, or at the end). In how many possible orders could his cards currently be?

For a positive integer $n$, let $p(n)$ denote the product of the positive integer factors of $n$. Determine the number of factors $n$ of 2310 for which $p(n)$ is a perfect square.

Consider a cube $A B C D E F G H$, where $A B C D$ and $E F G H$ are faces, and segments $A E, B F, C G, D H$ are edges of the cube. Let $P$ be the center of face $E F G H$, and let $O$ be the center of the cube. Given that $A G=1$, determine the area of triangle $A O P$.

Let $A B C D$ be a rectangle with $A B=3$ and $B C=7$. Let $W$ be a point on segment $A B$ such that $A W=1$. Let $X, Y, Z$ be points on segments $B C, C D, D A$, respectively, so that quadrilateral $W X Y Z$ is a rectangle, and $B X<X C$. Determine the length of segment $B X$.

The UEFA Champions League playoffs is a 16 -team soccer tournament in which Spanish teams always win against non-Spanish teams. In each of 4 rounds, each remaining team is randomly paired against one other team; the winner advances to the next round, and the loser is permanently knocked out of the tournament. If 3 of the 16 teams are Spanish, what is the probability that there are 2 Spanish teams in the final round?

Let $P(x)=x^{4}+2 x^{3}-13 x^{2}-14 x+24$ be a polynomial with roots $r_{1}, r_{2}, r_{3}, r_{4}$. Let $Q$ be the quartic polynomial with roots $r_{1}^{2}, r_{2}^{2}, r_{3}^{2}, r_{4}^{2}$, such that the coefficient of the $x^{4}$ term of $Q$ is 1 . Simplify the quotient $Q\left(x^{2}\right) / P(x)$, leaving your answer in terms of $x$. (You may assume that $x$ is not equal to any of $\left.r_{1}, r_{2}, r_{3}, r_{4}\right)$.

Let $A B C$ be a triangle with $A B=23, B C=24$, and $C A=27$. Let $D$ be the point on segment $A C$ such that the incircles of triangles $B A D$ and $B C D$ are tangent. Determine the ratio $C D / D A$.

Let $N=\overline{5 A B 37 C 2}$, where $A, B, C$ are digits between 0 and 9 , inclusive, and $N$ is a 7 -digit positive integer. If $N$ is divisible by 792 , determine all possible ordered triples $(A, B, C)$.

Three not necessarily distinct positive integers between 1 and 99, inclusive, are written in a row on a blackboard. Then, the numbers, without including any leading zeros, are concatenated to form a new integer $N$. For example, if the integers written, in order, are 25, 6, and 12, then $N=25612$ (and not $N=250612$ ). Determine the number of possible values of $N$.

Let $X Y Z$ be an equilateral triangle, and let $K, L, M$ be points on sides $X Y, Y Z, Z X$, respectively, such that $X K / K Y=B, Y L / L Z=1 / C$, and $Z M / M X=1$. Determine the ratio of the area of triangle $K L M$ to the area of triangle $X Y Z$.

Determine the positive real value of $x$ for which

$$\sqrt{2+A C+2 C x}+\sqrt{A C-2+2 A x}=\sqrt{2(A+C) x+2 A C} .$$

In-Young generates a string of $B$ zeroes and ones using the following method:

• First, she flips a fair coin. If it lands heads, her first digit will be a 0 , and if it lands tails, her first digit will be a 1.
• For each subsequent bit, she flips an unfair coin, which lands heads with probability $A$. If the coin lands heads, she writes down the number (zero or one) different from previous digit, while if the coin lands tails, she writes down the previous digit again.

What is the expected value of the number of zeroes in her string?

Determine the value of

$$\sum_{k=1}^{2011} \frac{k-1}{k!(2011-k)!} .$$

Let $A B C$ be a triangle with $A B=4, B C=8$, and $C A=5$. Let $M$ be the midpoint of $B C$, and let $D$ be the point on the circumcircle of $A B C$ so that segment $A D$ intersects the interior of $A B C$, and $\angle B A D=\angle C A M$. Let $A D$ intersect side $B C$ at $X$. Compute the ratio $A X / A D$.

Let $S$ be a set of consecutive positive integers such that for any integer $n$ in $S$, the sum of the digits of $n$ is not a multiple of 11. Determine the largest possible number of elements of $S$.

Each square in a $3 \times 10$ grid is colored black or white. Let $N$ be the number of ways this can be done in such a way that no five squares in an ' X ' configuration (as shown by the black squares below) are all white or all black. Determine $\sqrt{N}$.

Guts Round

Find all real numbers $x$ satisfying

$$x^{9}+\frac{9}{8} x^{6}+\frac{27}{64} x^{3}-x+\frac{219}{512}=0$$

Let $A B C$ be a triangle with $A B=5, B C=8$, and $C A=7$. Let $\Gamma$ be a circle internally tangent to the circumcircle of $A B C$ at $A$ which is also tangent to segment $B C$. $\Gamma$ intersects $A B$ and $A C$ at points $D$ and $E$, respectively. Determine the length of segment $D E$.

The integer 843301 is prime. The primorial of a prime number $p$, denoted $p \#$, is defined to be the product of all prime numbers less than or equal to $p$. Determine the number of digits in 843301#. Your score will be

$$\left.\max \left\{\left\lvert 60\left(\frac{1}{3}-\left|\ln \left(\frac{A}{d}\right)\right|\right)\right.\right\rfloor, 0\right\},$$

where $A$ is your answer and $d$ is the actual answer.

Let $G$ be the number of Google hits of "guts round" at 10:31PM on October 31, 2011. Let $B$ be the number of Bing hits of "guts round" at the same time. Determine $B / G$. Your score will be

$$\max \left(0,\left\lfloor 20\left(1-\frac{20|a-k|}{k}\right)\right\rfloor\right),$$

where $k$ is the actual answer and $a$ is your answer.

Order any subset of the following twentieth century mathematical achievements chronologically, from earliest to most recent. If you correctly place at least six of the events in order, your score will be $2(n-5)$, where $n$ is the number of events in your sequence; otherwise, your score will be zero. Note: if you order any number of events with one error, your score will be zero. A). Axioms for Set Theory published by Zermelo B). Category Theory introduced by Mac Lane and Eilenberg C). Collatz Conjecture proposed D). Erdos number defined by Goffman E). First United States delegation sent to International Mathematical Olympiad F). Four Color Theorem proven with computer assistance by Appel and Haken G). Harvard-MIT Math Tournament founded H). Hierarchy of grammars described by Chomsky I). Hilbert Problems stated J). Incompleteness Theorems published by Godel K). Million dollar prize for Millennium Problems offered by Clay Mathematics Institute L). Minimum number of shuffles needed to randomize a deck of cards established by Diaconis M). Nash Equilibrium introduced in doctoral dissertation N). Proof of Fermat's Last Theorem completed by Wiles O). Quicksort algorithm invented by Hoare

Write your answer as a list of letters, without any commas or parentheses.