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2007 iTest Problems

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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?

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.

Problem 28

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

Problem 29

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

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 $a+b+c$ .

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.

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]$

Problem 33

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

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 $a+b$ .

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.

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 $m+n$ .

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.

Problem 38

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

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 $a+b$ .

Problem 40

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

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 $f(2007)$ .

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 $a+b+c+d$ .

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.

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 $a+b$ .

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, $a\neq 0$ .

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 $m$ .

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 $2007$ .

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 $a+b$ .

Problem 49

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

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 $V$ ).

Ultimate Question

In the next 10 problems, the problem after will require the answer of the current problem. TNFTPP stands for the number from the previous problem.

For those who want to try these problems without having to find the T-values of the previous problem, a link will be here. Also, all solutions will have the T-values substituted.

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.

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.

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.

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.

See Also

iTest Problems and Solutions

2007 iTest (Problems, Answer Key)
Preceded by: 2006 iTest	Followed by: 2008 iTest
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4

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