# Difference between revisions of "2006 iTest Problems"

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Acute triangle <math>ABC</math> satisfies <math>AB=2AC</math> and <math>AB^4+BC^4+CA^4 = 2006\cdot 10^{10}</math>. Tetrahedron <math>DEFP</math> is formed by choosing points <math>D</math>, <math>E</math>, and <math>F</math> on the segments <math>BC</math>, <math>CA</math>, and <math>AB</math> (respectively) and folding <math>A</math>, <math>B</math>, <math>C</math>, over <math>EF</math>, <math>FD</math>, and <math>DE</math> (respectively) to the common point <math>P</math>. Let <math>R</math> denote the circumradius of <math>DEFP</math>. Compute the smallest positive integer <math>N</math> for which we can be certain that <math>n\geq R</math>. It may be helpful to use <math>\sqrt[4]{1239} = 5.9329109\ldots</math>. | Acute triangle <math>ABC</math> satisfies <math>AB=2AC</math> and <math>AB^4+BC^4+CA^4 = 2006\cdot 10^{10}</math>. Tetrahedron <math>DEFP</math> is formed by choosing points <math>D</math>, <math>E</math>, and <math>F</math> on the segments <math>BC</math>, <math>CA</math>, and <math>AB</math> (respectively) and folding <math>A</math>, <math>B</math>, <math>C</math>, over <math>EF</math>, <math>FD</math>, and <math>DE</math> (respectively) to the common point <math>P</math>. Let <math>R</math> denote the circumradius of <math>DEFP</math>. Compute the smallest positive integer <math>N</math> for which we can be certain that <math>n\geq R</math>. It may be helpful to use <math>\sqrt[4]{1239} = 5.9329109\ldots</math>. | ||

− | ==Problems 41 to 43 are | + | ==Ultimate Question== |

+ | In the next 2 problems, the problem after will require the answer of the current problem. TNFTPP stands for the number from the previous problem. Problem 41 requires the answer to the third problem. Problem 42 requires the answer to the seventh problem. '''Problem 43, however, requires the sum of the answers to all ten questions.''' | ||

+ | |||

+ | For those who want to try these problems without having to find the T-values of the previous problem, a link will be [[2006 iTest Problems/Ultimate Question|here]]. Also, all solutions will have the T-values substituted. | ||

+ | |||

+ | ===Problem 41=== | ||

+ | |||

+ | ====Problem U1==== | ||

+ | |||

+ | Find the real number <math>x</math> such that | ||

+ | <cmath>\sqrt{x-9} + \sqrt{x-6} = \sqrt{x-1}.</cmath> | ||

+ | |||

+ | ====Problem U2==== | ||

+ | |||

+ | Let <math>T=TNFTPP</math>. Points <math>A</math> and <math>B</math> lie on a circle centered at <math>O</math> such that <math>\angle AOB</math> is right. Points <math>C</math> and <math>D</math> lie on radii <math>OA</math> and <math>OB</math> respectively such that <math>AC = T-3</math>, <math>CD = 5</math>, and <math>BD = 6</math>. Determine the area of quadrilateral <math>ACDB</math>. | ||

+ | |||

+ | <asy> | ||

+ | draw(circle((0,0),10)); | ||

+ | draw((0,10)--(0,0)--(10,0)--(0,10)); | ||

+ | draw((0,3)--(4,0)); | ||

+ | label("O",(0,0),SW); | ||

+ | label("C",(0,3),W); | ||

+ | label("A",(0,10),N); | ||

+ | label("D",(4,0),S); | ||

+ | label("B",(10,0),E); | ||

+ | </asy> | ||

+ | |||

+ | ====Problem U3==== | ||

+ | |||

+ | Let <math>T = TNFTPP</math>. When properly sorted, <math>T - 35</math> math books on a shelf are arranged in alphabetical order from left to right. An eager student checked out and read all of them. Unfortunately, the student did not realize how the books were sorted, and so after finishing the student put the books back on the shelf in a random order. If all arrangements are equally likely, the probability that exactly <math>6</math> of the books were returned to their correct (original) position can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>. | ||

+ | |||

+ | ===Problem 42=== | ||

+ | |||

+ | ====Problem U4==== | ||

+ | |||

+ | Let <math>T = TNFTPP</math>. As <math>n</math> ranges over the integers, the expression <math>n^4 - 898n^2 + T - 2160</math> evaluates to just one prime number. Find this prime. | ||

+ | |||

+ | ====Problem U5==== | ||

+ | |||

+ | Let <math>T = TNFTPP</math>, and let <math>S</math> be the sum of the digits of <math>T</math>. In triangle <math>ABC</math>, points <math>D</math>, <math>E</math>, and <math>F</math> are the feet of the angle bisectors of <math>\angle A</math>, <math>\angle B</math>, <math>\angle C</math> respectively. Let point <math>P</math> be the intersection of segments <math>AD</math> and <math>BE</math>, and let <math>p</math> denote the perimeter of <math>ABC</math>. If <math>AP = 3PD</math>, <math>BE = S - 1</math>, and <math>CF = 9</math>, then the value of <math>\frac{AD}{p}</math> can be expressed uniquely as <math>\frac{\sqrt{m}}{n}</math> where <math>m</math> and <math>n</math> are positive integers such that <math>m</math> is not divisible by the square of any prime. Find <math>m + n</math>. | ||

+ | |||

+ | ====Problem U6==== | ||

+ | |||

+ | Let <math>T = TNFTPP</math>. <math>x</math> and <math>y</math> are nonzero real numbers such that | ||

+ | |||

+ | <cmath>18x - 4x^2 + 2x^3 - 9y - 10xy - x^2y + Ty^2 + 2xy^2 - y^3 = 0</cmath> | ||

+ | |||

+ | The smallest possible value of <math>\frac{y}{x}</math> is equal to <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||

+ | |||

+ | ====Problem U7==== | ||

+ | |||

+ | Let <math>T = TNFTPP</math>. Triangle <math>ABC</math> has integer side lengths, including <math>BC = 100T - 4</math>, and a right angle, <math>\angle ABC</math>. Let <math>r</math> and <math>s</math> denote the inradius and semiperimeter of <math>ABC</math> respectively. Find the ''perimeter'' of the triangle ABC which minimizes <math>\frac{s}{r}</math>. | ||

+ | |||

+ | ===Problem 43=== | ||

+ | |||

+ | ====Problem U8==== | ||

+ | |||

+ | Let <math>T = TNFTPP</math>, and let <math>S</math> be the sum of the digits of <math>T</math>. Cyclic quadrilateral <math>ABCD</math> has side lengths <math>AB = S - 11</math>, <math>BC = 2</math>, <math>CD = 3</math>, and <math>DA = 10</math>. Let <math>M</math> and <math>N</math> be the midpoints of sides <math>AD</math> and <math>BC</math>. The diagonals <math>AC</math> and <math>BD</math> intersect <math>MN</math> at <math>P</math> and <math>Q</math> respectively. <math>\frac{PQ}{MN}</math> can be expressed as <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine <math>m + n</math>. | ||

+ | |||

+ | ====Problem U9==== | ||

+ | |||

+ | Let <math>T = TNFTPP</math>. Determine the number of 5 element subsets <math>S</math> of <math>{1,2,3, \cdots ,T + 100}</math> such that the sum of the elements of <math>S</math> is divisible by 5. | ||

+ | |||

+ | ====Problem U10==== | ||

+ | |||

+ | Let <math>T = TNFTPP</math> and let <math>S</math> be the sum of the digits of <math>T</math>. Point <math>P</math> in the interior of triangle <math>ABC</math> satisfies <math>AP = S + 51</math>, <math>BP = 156</math>, and <math>CP = 169</math>. If the sides of ABC satisfy | ||

+ | |||

+ | <cmath>\frac{BC}{13} = \frac{CA}{14} = \frac{AB}{15}</cmath> | ||

+ | |||

+ | then the area of triangle <math>ABC</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute the remainder obtained when <math>m + n</math> is divided by <math>2006</math>. | ||

+ | |||

+ | '''Recall that you are turning in the sum of all ten answers, NOT the answer to this problem. |

## Revision as of 11:52, 18 October 2018

## Contents

- 1 Multiple Choice Section
- 1.1 Problem 1
- 1.2 Problem 2
- 1.3 Problem 3
- 1.4 Problem 4
- 1.5 Problem 5
- 1.6 Problem 6
- 1.7 Problem 7
- 1.8 Problem 8
- 1.9 Problem 9
- 1.10 Problem 10
- 1.11 Problem 11
- 1.12 Problem 12
- 1.13 Problem 13
- 1.14 Problem 14
- 1.15 Problem 15
- 1.16 Problem 16
- 1.17 Problem 17
- 1.18 Problem 18
- 1.19 Problem 19
- 1.20 Problem 20

- 2 Short Answer Section
- 2.1 Problem 21
- 2.2 Problem 22
- 2.3 Problem 23
- 2.4 Problem 24
- 2.5 Problem 25
- 2.6 Problem 26
- 2.7 Problem 27
- 2.8 Problem 28
- 2.9 Problem 29
- 2.10 Problem 30
- 2.11 Problem 31
- 2.12 Problem 32
- 2.13 Problem 33
- 2.14 Problem 34
- 2.15 Problem 35
- 2.16 Problem 36
- 2.17 Problem 37
- 2.18 Problem 38
- 2.19 Problem 39
- 2.20 Problem 40

- 3 Ultimate Question

## Multiple Choice Section

### Problem 1

Find the number of positive integral divisors of 2006.

### Problem 2

Find the harmonic mean of 10 and 20.

### Problem 3

Let be distinct positive integers such that the product . What is the largest possible value of the sum ?

### Problem 4

Four couples go ballroom dancing one evening. Their first names are Henry, Peter, Louis, Roger, Elizabeth, Jeanne, Mary, and Anne. If Henry's wife is not dancing with her husband (but with Elizabeth's husband), Roger and Anne are not dancing, Peter is playing the trumpet, and Mary is playing the piano, and Anne's husband is not Peter, who is Roger's wife?

### Problem 5

A line has y-intercept and forms a right angle to the line . Find the x-intercept of the line.

### Problem 6

What is the remainder when is divided by 7?

### Problem 7

The sum of consecutive integers is . Find the second largest integer.

### Problem 8

The point is a point on a circle with center . Perpendicular lines are drawn from to perpendicular diameters, and , meeting them at points and , respectively. If the diameter of the circle is , what is the length of ?

### Problem 9

If and is in the third quadrant, what is the absolute value of ?

### Problem 10

Find the number of elements in the first rows of Pascal's Triangle that are divisible by .

### Problem 11

Find the radius of the inscribed circle of a triangle with sides of length , , and .

### Problem 12

What is the highest possible probability of getting of these multiple choice questions correct, given that you don't know how to work any of them and are forced to blindly guess on each one?

### Problem 13

Suppose that are three distinct prime numbers such that . Find the maximum possible value for the product .

### Problem 14

Find , where is the smallest positive integer such that leaves a remainder of when divided by , , and .

### Problem 15

How many integers between and , inclusive, are perfect squares?

### Problem 16

The Minnesota Twins face the New York Mets in the 2006 World Series. Assuming the two teams are evenly matched (each has a probability of winning any game) what is the probability that the World Series (a best of 7 series of games which lasts until one team wins four games) will require the full seven games to determine a winner?

### Problem 17

Let . Find the numerical value of .

### Problem 18

*Every even number greater than 2 can be expressed as the sum of two prime numbers.'*

Name the mathematician for which this theorem was named, and then name the mathematician to whom he transmitted this theorem via letter in 1742.

### Problem 19

Questions 19 and 20 are Sudoku-related questions. Sudoku is a puzzle game that has one and only one solution for each puzzle. Digits from 1 to 9 must go into each space on the grid such that every row, column, and square contains one and only one of each digit.

Find the sum of by solving the Sudoku puzzle below.

1 _ _|3 5 8|_ _ 6 4 _ _|_ _ _|_ x 8 _ _ 9|_ 1 _|7 _ _---------------------_ z _|1 _ _|_ 5 _ _ _ 3|2 _ 4|8 _ _ _ 2 _|w _ 9|_ _ _---------------------_ _ 6|_ 2 _|9 _ _ 3 _ _|_ y _|_ _ 1 2 _ _|8 4 3|_ _ 7

### Problem 20

Sudoku is a puzzle game that has one and only one solution for each puzzle. Digits from 1 to 9 must go into each space on the grid such that every row, column, and square contains one and only one of each digit.

Find the sum of by solving the Sudoku puzzle below.

_ _ _|_ 4 _|_ z _ 1 _ 6|_ _ _|7 _ 3 5 _ _|9 _ _|_ _ 2---------------------_ 8 3|w 2 _|5 _ _ 2 _ _|5 _ 9|_ _ 7 _ _ 7|_ 8 _|9 2 _---------------------3 _ _|_ _ 1|_ _ 6 8 _ 9|x _ _|3 _ 5 _ y _|_ 3 _|_ _ _

## Short Answer Section

### Problem 21

What is the last (rightmost) digit of ?

### Problem 22

Triangle has sidelengths , , and . Point is the foot of the altitude from , and lies on segment such that . Find the area of the triangle .

### Problem 23

Jack and Jill are playing a chance game. They take turns alternately rolling a fair six sided die labeled with the integers 1 through 6 as usual (fair meaning the numbers appear with equal probability.) Jack wins if a prime number appears when he rolls, while Jill wins if when she rolls a number greater than 1 appears. The game terminates as soon as one of them has won. If Jack rolls first in a game, then the probability of that Jill wins the game can be expressed as where and are relatively prime positive integers. Compute .

### Problem 24

Points and are chosen on side of triangle such that is between and and , . If , the area of can be expressed as , where and are relatively prime positive integers and is a positive integer not divisible by the square of any prime. Compute .

### Problem 25

The expression reduces to , where and are relatively prime positive integers. Find .

### Problem 26

A rectangle has area and perimeter . The largest possible value of can be expressed as , where and are relatively prime positive integers. Compute .

### Problem 27

Line passes through and into the interior of the equilateral triangle . and are the orthogonal projections of and onto respectively. If and , then the area of can be expressed as , where and are positive integers and is not divisible by the square of any prime. Determine .

### Problem 28

The largest prime factor of is greater than . Determine the remainder obtained when this prime factor is divided by .

### Problem 29

The altitudes in triangle have lengths 10, 12, and 15. The area of can be expressed as , where and are relatively prime positive integers and is a positive integer not divisible by the square of any prime. Find .

### Problem 30

Triangle is equilateral. Points and are the midpoints of segments and respectively. is the point on segment such that . Let denote the intersection of and , The value of can be expressed as where and are relatively prime positive integers. Find .

### Problem 31

The value of the infinite series can be expressed as where and are relatively prime positive numbers. Compute .

### Problem 32

Triangle is scalene. Points and are on segment with between and such that , , and . If and trisect , then can be written uniquely as , where and are relatively prime positive integers and is a positive integer not divisible by the square of any prime. Determine .

### Problem 33

Six students sit in a group and chat during a complicated mathematical lecture. The professor, annoyed by the chatter, splits the group into two or more smaller groups. However, the smaller groups with at least two members continue to produce chatter, so the professor again chooses one noisy group and splits it into smaller groups. This process continues until the professor achieves the silence he needs to teach Algebraic Combinatorics. Suppose the procedure can be carried out in ways, where the order of group breaking matters (if A and B are disjoint groups, then breaking up group A and then B is considered different form breaking up group B and then A even if the resulting partitions are identical) and where a group of students is treated as an unordered set of people. Compute the remainder obtained when is divided by .

### Problem 34

For each positive integer let denote the set of positive integers such that is divisible by . Define the function by the rule Let be the least upper bound of and let be the number of integers such that and . Compute the value of .

### Problem 35

Compute the of ordered quadruples of complex numbers (not necessarily nonreal) such that the following system is satisfied:

### Problem 36

Let denote . The recursive sequence satisfies and, for all positive integers , Suppose that the series can be expressed uniquely as , where and are coprime positive integers and is not divisible by the square of any prime. Find the value of .

### Problem 37

The positive reals , , satisfy the relations The value of can be expressed uniquely as , where , , , are positive integers such that is not divisible by the square of any prime and no prime dividing divides both and . Compute .

### Problem 38

Segment is a diameter of circle . Point lies in the interior of segment such that , and is a point on such that . Segment is a diameter of the circle . A third circle, , is drawn internally tangent to , externally tangent to , and tangent to segment . If is centered on the opposite side of as , then the radius of can be expressed as , where and are relatively prime positive integers. Compute .

### Problem 39

is a regular dodecagon. The number 1 is written at the vertex A, and 0's are written at each of the other vertices. Suddenly and simultaneously, the number at each vertex is replaced by the arithmetic mean of the two numbers appearing at the adjacent vertices. If this procedure is repeated a total of times, then the resulting number at A can be expressed as , where and are relatively prime positive integers. Compute the remainder obtained when is divided by .

### Problem 40

Acute triangle satisfies and . Tetrahedron is formed by choosing points , , and on the segments , , and (respectively) and folding , , , over , , and (respectively) to the common point . Let denote the circumradius of . Compute the smallest positive integer for which we can be certain that . It may be helpful to use .

## Ultimate Question

In the next 2 problems, the problem after will require the answer of the current problem. TNFTPP stands for the number from the previous problem. Problem 41 requires the answer to the third problem. Problem 42 requires the answer to the seventh problem. **Problem 43, however, requires the sum of the answers to all ten questions.**

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 41

#### Problem U1

Find the real number such that

#### Problem U2

Let . Points and lie on a circle centered at such that is right. Points and lie on radii and respectively such that , , and . Determine the area of quadrilateral .

#### Problem U3

Let . When properly sorted, math books on a shelf are arranged in alphabetical order from left to right. An eager student checked out and read all of them. Unfortunately, the student did not realize how the books were sorted, and so after finishing the student put the books back on the shelf in a random order. If all arrangements are equally likely, the probability that exactly of the books were returned to their correct (original) position can be expressed as , where and are relatively prime positive integers. Compute .

### Problem 42

#### Problem U4

Let . As ranges over the integers, the expression evaluates to just one prime number. Find this prime.

#### Problem U5

Let , and let be the sum of the digits of . In triangle , points , , and are the feet of the angle bisectors of , , respectively. Let point be the intersection of segments and , and let denote the perimeter of . If , , and , then the value of can be expressed uniquely as where and are positive integers such that is not divisible by the square of any prime. Find .

#### Problem U6

Let . and are nonzero real numbers such that

The smallest possible value of is equal to where and are relatively prime positive integers. Find .

#### Problem U7

Let . Triangle has integer side lengths, including , and a right angle, . Let and denote the inradius and semiperimeter of respectively. Find the *perimeter* of the triangle ABC which minimizes .

### Problem 43

#### Problem U8

Let , and let be the sum of the digits of . Cyclic quadrilateral has side lengths , , , and . Let and be the midpoints of sides and . The diagonals and intersect at and respectively. can be expressed as where and are relatively prime positive integers. Determine .

#### Problem U9

Let . Determine the number of 5 element subsets of such that the sum of the elements of is divisible by 5.

#### Problem U10

Let and let be the sum of the digits of . Point in the interior of triangle satisfies , , and . If the sides of ABC satisfy

then the area of triangle can be expressed as , where and are relatively prime positive integers. Compute the remainder obtained when is divided by .

**Recall that you are turning in the sum of all ten answers, NOT the answer to this problem.**