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Revision as of 10:41, 12 August 2017

Multiple Choice Section

Problem 1

A twin prime pair is a set of two primes $(p, q)$ such that $q$ is $2$ greater than $p$. What is the arithmetic mean of the two primes in the smallest twin prime pair?

$\mathrm{(A)}\, 4$

Solution

Problem 2

Find $a + b$ if $a$ and $b$ satisfy $3a + 7b = 1977$ and $5a + b = 2007$.

$\mathrm{(A)}\, 488\quad\mathrm{(B)}\, 498$

Solution

Problem 3

An abundant number is a natural number that's proper divisors sum is greater than the number. Which one of the following natural numbers is an abundant number?

$\mathrm{(A)}\, 14\quad\mathrm{(B)}\, 28\quad\mathrm{(C)}\, 56$

Solution

Problem 4

Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.

$\mathrm{(A)}\,\frac{1}{8}\quad\mathrm{(B)}\,\frac{3}{16}\quad\mathrm{(C)}\,\frac{3}{8}\quad\mathrm{(D)}\,\frac{1}{2}$

Solution

Problem 5

Compute the sum of all twenty-one terms of the geometric series \[1 + 2 + 4 + 8 + \ldots + 1048576\].

$\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161$

Solution

Problem 6

Find the units digit of the sum

\[\sum_{i=1}^{100}(i!)^{2}\]

$\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9$

Solution

Problem 7

An equilateral triangle with side length $1$ has the same area as a square with side length $s$. Find $s$.

$\mathrm{(A)}\,\frac{\sqrt[4]{3}}{2}\quad\mathrm{(B)}\,\frac{\sqrt[4]{3}}{\sqrt{2}}\quad\mathrm{(C)}\,1\quad\mathrm{(D)}\,\frac{3}{4}\quad\mathrm{(E)}\,\frac{4}{3}\quad\mathrm{(F)}\,\sqrt{3}\quad\mathrm{(G)}\,\frac{\sqrt6}{2}$

Solution

Problem 8

Joe is right at the middle of a train tunnel and he realizes that a train is coming. The train travels at a speed of 50 miles per hour, and Joe can run at a speed of 10 miles per hour. Joe hears the train whistle when the train is a half mile from the point where it will enter the tunnel. At that point in time, Joe can run toward the train and just exit the tunnel as the train meets him. Instead, Joe runs away from the train when he hears the whistle. How many seconds does he have to spare (before the train is upon him) when he gets to the tunnel entrance?

$\mathrm{(A)}\,7.2\quad\mathrm{(B)}\,14.4\quad\mathrm{(C)}\,36\quad\mathrm{(D)}\,10\quad\mathrm{(E)}\,12\quad\mathrm{(F)}\,2.4\quad\mathrm{(G)}\,25.2\quad\mathrm{(H)}\,123456789$

Solution

Problem 9

Suppose that $m$ and $n$ are positive integers such that $m < n$, the geometric mean of $m$ and $n$ is greater than $2007$, and the arithmetic mean of $m$ and $n$ is less than $2007$. How many pairs $(m, n)$ satisfy these conditions?

$\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,2007$

Solution

Problem 10

My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only $4$ years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha’s age and the mean of my grandparents’ ages?

$\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,2\quad\mathrm{(D)}\,3\quad\mathrm{(E)}\,4\quad\mathrm{(F)}\,5\quad\mathrm{(G)}\,6\quad\mathrm{(H)}\,7\quad\mathrm{(I)}\,8\quad\mathrm{(J)}\,2007$

Solution

Problem 11

Consider the "tower of power" $2^{2^{2^{\cdot^{\cdot^\cdot^{2}}}}}$ (Error compiling LaTeX. Unknown error_msg), where there are 2007 twos including the base. What is the last (units digit) of this number?

$\text{(A) }0\qquad \text{(B) }1\qquad \text{(C) }2\qquad \text{(D) }3\qquad \text{(E) }4\qquad \text{(F) }5\qquad \text{(G) }6\qquad \text{(H) }7\qquad \text{(I) }8\qquad \text{(J) }9\qquad \text{(K) }2007\qquad$

Solution

Problem 12

My Frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team.

$\text{(A) }\frac{2}{3}\qquad \text{(B) }1\qquad \text{(C) }\frac{3}{2}\qquad \text{(D) }\frac{8}{5}\qquad \text{(E) }\frac{5}{8}\qquad \text{(F) }2\qquad\\ \\ \text{(G) }0\qquad \text{(H) }\frac{5}{2}\qquad \text{(I) }\frac{2}{5}\qquad \text{(J) }\frac{3}{4}\qquad \text{(K) }\frac{4}{3}\qquad \text{(L) }2007\qquad$

Solution

Problem 13

What is the smallest positive integer $k$ such that the number ${{2k}\choose k}$ ends in two zeros?

$\text{(A) } 3 \quad \text{(B) } 4 \quad \text{(C) } 5 \quad \text{(D) } 6 \quad \text{(E) } 7 \quad \text{(F) } 8 \quad \text{(G) } 9 \quad \text{(H) } 10 \quad \text{(I) } 11 \quad \text{(J) } 12 \quad \text{(K) } 13 \quad \text{(L) } 14 \quad \text{(M) } 2007\quad$

Solution

Problem 14

Let $\phi(n)$ be the number of positive integers $k< n$ which are relatively prime to $n$. For how many distinct values of $n$ is $\phi(n)=12$?

$\text{(A) } 0 \quad \text{(B) } 1 \quad \text{(C) } 2 \quad \text{(D) } 3 \quad \text{(E) } 4 \quad \text{(F) } 5 \quad \text{(G) } 6 \quad \text{(H) } 7 \quad \text{(I) } 8 \quad \text{(J) } 9 \quad \text{(K) } 10 \quad \text{(L) } 11 \quad \text{(M) } 12\quad \text{(N) } 13\quad$

Solution

Problem 15

Form a pentagon by taking a square of side length 1 and an equilateral triangle of side length 1, and placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the pentagon, passing through three of its vertices, so that the circle passes through exactly one of the vertices of the equilateral triangle, and through exactly two vertices of the square. What is the radius of the circle?

$\text{(A) }\frac{2}{3}\qquad \text{(B) }\frac{3}{4}\qquad \text{(C) }1\qquad \text{(D) }\frac{5}{4}\qquad \text{(E) }\frac{4}{3}\qquad \text{(F) }\frac{\sqrt{2}}{2}\qquad \text{(G) }\frac{\sqrt{3}}{2}\qquad \text{(H) }\sqrt{2}\qquad \$ (Error compiling LaTeX. Unknown error_msg)

$\text{(I) }\sqrt{3}\qquad \text{(J) }\frac{1+\sqrt{3}}{2}\qquad \text{(K) }\frac{2+\sqrt{6}}{2}\qquad \text{(L) }\frac{7}{6}\qquad \text{(M) }\frac{2+\sqrt{6}}{4}\qquad \text{(N) }\frac{4}{5}\qquad \text{(O) }2007\qquad$


Solution

Problem 16

How many lattice points lie within or on the border of the circle in the $xy$-plane defined by the equation \[x^2+y^2=100\]

$\text{(A) }1\qquad \text{(B) }2\qquad \text{(C) }4\qquad \text{(D) }5\qquad \text{(E) }41\qquad \text{(F) }42\qquad \text{(G) }69\qquad \text{(H) }76\qquad \text{(I) }130\qquad \\ \text{(J) }133\qquad \text{(K) }233\qquad \text{(L) }311\qquad \text{(M) }317\qquad \text{(N) }420\qquad \text{(O) }520\qquad \text{(P) }2007$


Solution

Problem 17

If $x$ and $y$ are acute angles such that $x+y=\frac{\pi}{4}$ and $\tan{y}=\frac{1}{6}$, find the value of $\tan{x}$.

$\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad \text{(B) }\frac{35\sqrt{2}-6}{71}\qquad \text{(C) }\frac{35\sqrt{3}+12}{33}\qquad \text{(D) }\frac{37\sqrt{3}+24}{33}\qquad \text{(E) }1\qquad \text{(F) }\frac{5}{7}\qquad \text{(G) }\frac{3}{7}\qquad \text{(H) }6\qquad \$ (Error compiling LaTeX. Unknown error_msg)

$\text{(I) }\frac{1}{6}\qquad \text{(J) }\frac{1}{2}\qquad \text{(K) }\frac{6}{7}\qquad \text{(L) }\frac{4}{7}\qquad \text{(M) }\sqrt{3}\qquad \text{(N) }\frac{\sqrt{3}}{3}\qquad \text{(O) }\frac{5}{6}\qquad \text{(P) }\frac{2}{3}\qquad \text{(Q) }\frac{1}{2007}\qquad\$ (Error compiling LaTeX. Unknown error_msg)


Solution

Problem 18

Suppose that $x^3+px^2+qx+r$ is a cubic with a double root at $a$ and another root at b, where $a$ and $b$ are real numbers. If $p=-6$ and $q=9$, what is $r$?

$\text{(A) }0\qquad \text{(B) }4\qquad \text{(C) }108\qquad \text{(D) It could be }0 \text{ or } 4\qquad \text{(E) It could be }0 \text{ or } 108 \$ (Error compiling LaTeX. Unknown error_msg)

$\text{(F) }18\qquad \text{(G) }-4\qquad \text{(H) }-108\qquad \text{(I) It could be } 0 \text{ or } -4 \$ (Error compiling LaTeX. Unknown error_msg)

$\text{(J) It could be } 0 \text{ or } {-108} \qquad \text{(K) It could be } 4 \text{ or } {-4}\qquad \text{(L) There is no such value of } r\qquad \$ (Error compiling LaTeX. Unknown error_msg)

$\text{(M) } 1 \qquad \text{(N) } {-2} \qquad \text{(O) It could be } 4 \text{ or } -4 \qquad \text{(P) It could be } 0 \text{ or } -2 \qquad \$ (Error compiling LaTeX. Unknown error_msg)

$\text{(Q) It could be } 2007 \text{ or a yippy dog} \qquad \text{(R) } 2007 \$ (Error compiling LaTeX. Unknown error_msg)


Solution

Problem 19

One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!" Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to clop it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin. If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins $\textit{ten gold coins.}$ What is the expected number of gold coins Jason wins at this game?

$\textbf{(A) }0\qquad \textbf{(B) }\dfrac1{10}\qquad \textbf{(C) }\dfrac18\qquad \textbf{(D) }\dfrac15\qquad \textbf{(E) }\dfrac14\qquad \textbf{(F) }\dfrac13\qquad \textbf{(G) }\dfrac25\qquad \$ (Error compiling LaTeX. Unknown error_msg) $\textbf{(H) }\dfrac12\qquad \textbf{(I) }\dfrac35\qquad \textbf{(J) }\dfrac23\qquad \textbf{(K) }\dfrac45\qquad \textbf{(L) }1\qquad \textbf{(M) }\dfrac54\qquad \$ (Error compiling LaTeX. Unknown error_msg) $\textbf{(N) }\dfrac43\qquad \textbf{(O) }\dfrac32\qquad \textbf{(P) }2\qquad \textbf{(Q) }3\qquad \textbf{(R) }4\qquad \textbf{(S) }2007$


Solution

Problem 20

Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$

$\text{(A) } 1\qquad \text{(B) } 2\qquad \text{(C) } 3\qquad \text{(D) } 4\qquad \text{(E) } 5\qquad \text{(F) } 6\qquad \text{(G) } 7\qquad \text{(H) } 8\qquad \$ (Error compiling LaTeX. Unknown error_msg) $\text{(I) } 9\qquad \text{(J) } 10\qquad \text{(K) } 11\qquad \text{(L) } 12\qquad \text{(M) } 13\qquad \text{(N) } 14\qquad \text{(O) } 15\qquad \text{(P) } 16\qquad \$ (Error compiling LaTeX. Unknown error_msg) $\text{(Q) } 55\qquad \text{(R) } 63\qquad \text{(S) } 64\qquad \text{(T) } 2007\qquad$


Solution

Problem 21

James writes down fifteen 1's in a row and randomly writes + or - between each pair of consecutive 1's. One such example is \[1+1+1-1-1+1-1+1-1+1-1-1-1+1+1.\] What is the probability that the value of the expression James wrote down is $7$?

$\text{(A) }0\qquad \text{(B) }\frac{6435 }{2^{14}}\qquad \text{(C) }\frac{6435 }{2^{13}}\qquad \text{(D) }\frac{429}{2^{12}}\qquad \text{(E) }\frac{429}{2^{11}}\qquad \text{(F) }\frac{429}{2^{10}}\qquad \text{(G) }\frac{1}{15}\qquad \text{(H) }\frac{1}{31}\qquad$

$\text{(I) }\frac{1}{30}\qquad \text{(J) }\frac{1}{29}\qquad \text{(K) }\frac{6435 }{2^{15}}\qquad \text{(L) }\frac{6435 }{2^{14}}\qquad \text{(M) }\frac{6435 }{2^{13}}\qquad \text{(N) }\frac{1}{2^{7}}\qquad \text{(O) }\frac{1}{2^{14}}\qquad \text{(P) }\frac{1}{2^{15}}\qquad$

$\text{(P) }\frac{2007}{2^{14}}\qquad \text{(P) }\frac{2007}{2^{15}}\qquad \text{(P) }\frac{2007}{2^{2007}}\qquad \text{(P) }\frac{1}{2007}\qquad \text{(P) }\frac{-2007}{2^{14}}\qquad$


Solution

Problem 22

Find the value of $c$ such that the system of equations \[|x+y|=2007 \\ |x-y|=c\] has exactly two solutions $(x,y)$ in real numbers.


$\text{(A) } 0 \quad \text{(B) } 1 \quad \text{(C) } 2 \quad \text{(D) } 3 \quad \text{(E) } 4 \quad \text{(F) } 5 \quad \text{(G) } 6 \quad \text{(H) } 7 \quad \text{(I) } 8 \quad \text{(J) } 9 \quad \text{(K) } 10 \quad \text{(L) } 11 \quad \text{(M) } 12 \quad$

$\text{(N) } 13 \quad \text{(O) } 14 \quad \text{(P) } 15 \quad \text{(Q) } 16 \quad \text{(R) } 17 \quad \text{(S) } 18 \quad \text{(T) } 223 \quad \text{(U) } 678 \quad \text{(V) } 2007 \quad$


Solution

Problem 23

Find the product of the non-real roots of the equation \[(x^2-3)^2+5(x^2-3)+6=0\]

$\text{(A) } 0\quad \text{(B) } 1\quad \text{(C) } -1\quad \text{(D) } 2\quad \text{(E) } -2\quad \text{(F) } 3\quad \text{(G) } -3\quad \text{(H) } 4\quad \text{(I) } -4\quad$

$\text{(J) } 5\quad \text{(K) } -5\quad \text{(L) } 6\quad \text{(M) } -6\quad \text{(N) } 3+2i\quad \text{(O) } 3-2i\quad$

$\text{(P) } \frac{-3+i\sqrt{3}}{2}\quad \text{(Q) } 8\quad \text{(R) } -8\qquad \text{(S) } 12\quad \text{(T) } -12\quad \text{(U) } 42\quad$

$\text{(V) Ying} \quad \text{(W) } 207$


Solution

Problem 24

Let $N$ be the smallest positive integer such that $2008N$ is a perfect square and $2007N$ is a perfect cube. Find the remainder when $N$ is divided by $25$.

$\text{(A) }0 \quad \text{(B) }1 \quad \text{(C) }2 \quad \text{(D) }3 \quad \text{(E) }4 \quad \text{(F) }5 \quad \text{(G) }6 \quad \text{(H) }7 \quad \text{(I) } 8\quad$

$\text{(J) }9 \quad \text{(K) }10 \quad \text{(L) }11 \quad \text{(M) }12 \quad \text{(N) }13 \quad \text{(O) }14 \quad \text{(P) }15 \quad \text{(Q) }16 \quad$

$\text{(R) }17 \quad \text{(S) }18 \quad \text{(T) }19 \quad \text{(U) }20 \quad \text{(V) }21 \quad \text{(W) }22 \quad \text{(X) }23$


Solution

Problem 25

Ted's favorite number is equal to \[1\cdot{2007\choose 1}+2\cdot {2007\choose 2}+3\cdot {2007\choose 3} + \cdots + 2007\cdot {2007 \choose 2007}\]

Find the remainder when Ted's favorite number is divided by 25.

$\text{(A) } 0\qquad \text{(B) } 1\qquad \text{(C) } 2\qquad \text{(D) } 3\qquad \text{(E) } 4\qquad \text{(F) } 5\qquad \text{(G) } 6\qquad \text{(H) } 7\qquad \text{(I) } 8\qquad$

$\text{(J) } 9\qquad \text{(K) } 10\qquad \text{(L) } 11\qquad \text{(M) } 12\qquad \text{(N) } 13\qquad \text{(O) } 14\qquad \text{(P) } 15\qquad \text{(Q) } 16\qquad$

$\text{(R) } 17\qquad \text{(S) } 18\qquad \text{(T) } 19\qquad \text{(U) } 20\qquad \text{(V) } 21\qquad \text{(A) } 22\qquad \text{(X) } 23\qquad \text{(Y) } 24$


Solution

Short Answer Section

Problem 26

Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of $370. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of $180. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday?


Solution

Problem 27

The face diagonal of a cube has length $4$. Find the value of n given that $n\sqrt2$ is the $\textit{volume}$ of the cube.


Solution

Problem 28

The space diagonal (interior diagonal) of a cube has length 6. Find the $\textit{surface area}$ of the cube.


Solution

Problem 29

Let $S$ be equal to the sum $1+2+3+\cdots+2007$. Find the remainder when $S$ is divided by $1000$.


Solution

Problem 30

While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers $a, b$, and $c$, and recalled that their product is $24$, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than $25$ with fewer than $6$ divisors. Help James by computing $a+b+c$.


Solution

Problem 31

Let $x$ be the length of one side of a triangle and let y be the height to that side. If $x+y=418$, find the maximum possible $\textit{integral value}$ of the area of the triangle.


Solution

Problem 32

When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio $2$ to $1$. How many integer values of k are there such that $0<k\leq 2007$ and the area between the parabola $y=k-x^2$ and the $x$-axis is an integer?

[asy] import graph; size(300); defaultpen(linewidth(0.8)+fontsize(10)); real k=1.5; real endp=sqrt(k); real f(real x) { return k-x^2; } path parabola=graph(f,-endp,endp)--cycle; filldraw(parabola, lightgray); draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0)); label("Region I", (0,2*k/5)); label("Box II", (51/64*endp,13/16*k)); label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2));[/asy]


Solution

Problem 33

How many $\textit{odd}$ four-digit integers have the property that their digits, read left to right, are in strictly decreasing order?


Solution

Problem 34

Let $a/b$ be the probability that a randomly selected divisor of $2007$ is a multiple of $3$. If $a$ and $b$ are relatively prime positive integers, find $a+b$.


Solution

Problem 35

Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighboring digits.


Solution

Problem 36

Let b be a real number randomly selected from the interval $[-17,17]$. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation $x^4+25b^2=(4b^2-10b)x^2$ has $\textit{at least}$ two distinct real solutions. Find the value of $m+n$.


Solution

Problem 37

Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three lengths of bamboo together, such that they meet at the same point, and each pair of bamboo rods meet at a right angle. Three more lengths of bamboo are then cut to connect the other ends of the first three rods. Rob then cuts out four triangular pieces of fabric: a blue piece, a red piece, a green piece, and a yellow piece. These triangular pieces of fabric just fill in the triangular spaces between the bamboo, making up the four faces of the tetrahedron. The areas in square feet of the red, yellow, and green pieces are $60, 20$, and $15$ respectively. If the blue piece is the largest of the four sides, find the number of square feet in its area.


Solution

Problem 38

Find the largest positive integer that is equal to the cube of the sum of its digits.


Solution

Problem 39

Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation $\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}$. Find $a+b$.


Solution

Problem 40

Let $S$ be the sum of all $x$ such that $1\leq x\leq 99$ and $\{x^2\}=\{x\}^2$. Compute $\lfloor S\rfloor$.


Solution

Problem 41

The sequence of digits $123456789101112131415161718192021\ldots$ is obtained by writing the positive integers in order. If the $10^{nth}$ digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2) = 2$ because the $100^{\text{th}}$ digit enters the sequence in the placement of the two-digit integer $55$. Find the value of $f(2007)$.

Solution

Problem 42

During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a $100$ foot by $100$ foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to the fence than to the flag pole can be written in simplest terms as $\dfrac{a-b\sqrt c}d$, where all four variables are positive integers, $c$ is a multple of no perfect square greater than $1$, a is coprime with $d$, and $b$ is coprime with $d$. Find the value of $a+b+c+d$.


Solution

Problem 43

Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers:

\begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*}

She notes that two of them have exactly $8$ positive divisors each. Find the common prime divisor of those two integers.

Solution

Problem 44

A positive integer $n$ between $1$ and $N=2007^{2007}$ inclusive is selected at random. If $a$ and $b$ are natural numbers such that $a/b$ is the probability that $N$ and $n^3-36n$ are relatively prime, find the value of $a+b$.

Solution

Problem 45

Find the sum of all positive integers $B$ such that $(111)_B=(aabbcc)_6$, where $a,b,c$ represent distinct base $6$ digits, $a\neq 0$.

Solution

Problem 46

Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$, find the modulo $2007$ residue of $m$.

Solution

Problem 47

Let $\{X_n\}$ and $\{Y_n\}$ be sequences defined as follows: $X_0=Y_0=X_1=Y_1=1$,

\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qquad(n=1,2,3\ldots),\\ Y_{n+1}&=3Y_n+4Y_{n-1}\qquad(n=1,2,3\ldots).\end{align*}

Let $k$ be the largest integer that satisfies all of the following conditions: $|X_i-k|\leq 2007$, for some positive integer $i$; $|Y_j-k|\leq 2007$, for some positive integer $j$; and $k<10^{2007}$. Find the remainder when $k$ is divided by $2007$.

Solution

Problem 48

Let a and b be relatively prime positive integers such that $a/b$ is the maximum possible value of $\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007}$, where, for $1\leq i\leq 2007, x_i$ is a nonnegative real number, and $x_1+x_2+x_3+\cdots+x_{2007}=\pi$. Find the value of $a+b$.

Solution

Problem 49

How many $7$-element subsets of $\{1, 2, 3,\ldots , 14\}$ are there, the sum of whose elements is divisible by $14$?

Solution

Problem 50

A block $Z$ is formed by gluing one face of a solid cube with side length $6$ onto one of the circular faces of a right circular cylinder with radius $10$ and height $3$ so that the centers of the square and circle coincide. If $V$ is the smallest convex region that contains $Z$, calculate $\lfloor\operatorname{vol}V\rfloor$ (the greatest integer less than or equal to the volume of $V$).

Solution

Ultimate Question

Problem 51

Find the highest point (largest possible $y$-coordinate) on the parabola \[y=-2x^2+ 28x+ 418\]

Problem 52

Let $T=TNFTPP$. Let $R$ be the region consisting of points $(x,y)$ of the Cartesian plane satisfying both $|x|-|y|\le T-500$ and $|y|\le T-500$. Find the area of region $R$.

Problem 53

Let $T=\text{TNFTPP}$. Three distinct positive Fibonacci numbers, all greater than $T$, are in arithmetic progression. Let $N$ be the smallest possible value of their sum. Find the remainder when $N$ is divided by $2007$.

Problem 54

Let $T=\text{TNFTPP}$. Consider the sequence $(1, 2007)$. Inserting the difference between $1$ and $2007$ between them, we get the sequence $(1, 2006, 2007)$. Repeating the process of inserting differences between numbers, we get the sequence $(1, 2005, 2006, 1, 2007)$. A third iteration of this process results in $(1, 2004, 2005, 1, 2006, 2005, 1, 2006, 2007)$. A total of $2007$ iterations produces a sequence with $2^{2007}+1$ terms. If the integer $4T$ (that is, $4$ times the integer $T$) appears a total of $N$ times among these $2^{2007}+1$ terms, find the remainder when $N$ gets divided by $2007$.

Problem 55

Let $T=\text{TNFTPP}$, and let $R=T-914$. Let $x$ be the smallest real solution of $3x^2+Rx+R=90x\sqrt{x+1}$. Find the value of $\lfloor x\rfloor$.

Problem 56

Let $T=\text{TNFTPP}$. In the binary expansion of $\dfrac{2^{2007}-1}{2^T-1}$, how many of the first $10,000$ digits to the right of the radix point are $0$'s?

Problem 57

Let $T=\text{TNFTPP}$. How many positive integers are within $T$ of exactly $\lfloor \sqrt T\rfloor$ perfect squares? (Note: $0^2=0$ is considered a perfect square.)

Problem 58

Let $T=\text{TNFTPP}$. For natural numbers $k,n\geq 2$, we define \[S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfloor\frac{3^{n+1}+1}{3^{n-1}+1}\right\rfloor+\cdots+\left\lfloor\frac{k^{n+1}+1}{k^{n-1}+1}\right\rfloor\] Compute the value of $S(10,T+55)-S(10,55)+S(10,T-55)$.

Problem 59

Let $T=\text{TNFTPP}$. Fermi and Feynman play the game $\textit{Probabicloneme}$ in which Fermi wins with probability $a/b$, where $a$ and $b$ are relatively prime positive integers such that $a/b<1/2$. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play $\textit{Probabicloneme}$ so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is $(T-332)/(2T-601)$. Find the value of $a$.

Problem 60

Let $T=\text{TNFTPP}$. Triangle $ABC$ has $AB=6T-3$ and $AC=7T+1$. Point $D$ is on $BC$ so that $AD$ bisects angle $BAC$. The circle through $A, B$, and $D$ has center $O_1$ and intersects line $AC$ again at $B'$, and likewise the circle through $A, C$, and $D$ has center $O_2$ and intersects line $AB$ again at $C'$. If the four points $B', C', O_1$, and $O_2$ lie on a circle, find the length of $BC$.

Tiebreaker Questions

Problem TB1

The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.

Solution

Problem TB2

Factor completely over integer coefficients the polynomial $p(x)=x^8+x^5+x^4+x^3+x+1$. Demonstrate that your factorization is complete.

Solution

Problem TB3

4014 boys and 4014 girls stand in a line holding hands, such that only the two people at the ends are not holding hands with exactly two people (an ordinary line of people). One of the two people at the ends gets tired of the hand-holding fest and leaves. Then, from the remaining line, one of the two people at the ends leaves. Then another from an end, and then another, and another. This continues until exactly half of the people from the original line remain. Prove that no matter what order the original 8028 people were standing in, that it is possible that exactly 2007 of the remaining people are girls.

Solution

Problem TB4

Circle $O$ is the circumcircle of non-isosceles triangle $ABC$. The tangent lines to circle $O$ at points $B$ and $C$ intersect at $L_a$, and the tangents at $A$ and $C$ intersect at $L_b$. The external angle bisectors of triangle $ABC$ at $B$ and $C$ meet at $I_a$ and the external bisectors at $A$ and $C$ intersect at $I_b$. Prove that lines $L_aI_a$, $L_bI_b$, and $AB$ are concurrent.

Solution


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