World Explorer Books - Children's Books About Exploration

Follow the Dream: The Story of Christopher Columbuswritten and illustrated by Peter Sis; ages 5 and upOver 500 years ago a little boy was born in the city of Genoa, Italy. His father was a weaver, but Christopher Columbus dreamed of faraway places, adventure, and discovery. He observed the ships that sailed into the harbor and listened to the sailors and merchants as they told tales of their

Exploration Kids Books - American Exploer Books

Who Was First? Discovering the AmericasRussell Freedman; ages 11 and upHistorian Russell Freeman explores the various claims to the "discovery" of the American continents. Every U.S. school child knows the story of Columbus, but what about the Chinese explorer, Zheng He? This lavishly illustrated volume traces explorers' journeys with archival maps, charts, and timelines. Freeman discusses the

Reading Readiness Books: ages 3-4

Get your 3 to 4 year old ready to read and learn with these age-appropriate story recommendations.How can we help our children as they are learning to read?  One of the building blocks of reading competency is phonetic awareness. What are the sounds that make up a written word? Phonemic awareness refers to the ability to hear and tell the difference between words, sounds, and syllables in

Best Music Books For Children

Teach your child all about instruments, melody, and more with these recommended music books for preschoolersKids Go!by They Might Be Giants; illustrated by Pascal CampionHipster rock group They Might Be Giants return after their Grammy-winning CD for kids, Here Come the 123s, with a book and song combination to get kids off the couch and get moving. The song was originally created in 2008 for a

Top 10 Kids Math Web Sites

FunbrainA math arcade and interactive math games are only two of the many features of this site, which offers games in a wide variety of topics. Classic games on this site include math car racing. Check out the teacher's resource page and the curriculum guide. Time4LearningComplete online curriculum for preschool through the eighth grade. 

Middle School Educational Software

QuickStudy English VocabularyTake the quick path to writing success! Quickstudy English Vocabulary provides a solid educational foundation that will raise grades and test scores and improve vocabulary and writing skills in the classroom and beyond. The curriculum-based lessons are designed by educators to help students expand their vocabulary in an engaging, interactive learning

Software For Kids

Millie's Math HouseDevelop a love for math with Millie! In seven fun-filled activities, kids explore fundamental math concepts as they learn about numbers, shapes, sizes, quantities, patterns, sequencing, addition, and subtraction. They count critters, build mouse houses, create crazy-looking bugs, make jellybean cookies for Harley the horse, and find just the right shoes for Little,

Boom!

Boom! by Mark Haddon

2010

for ages 8-12 years

Jim and Charlie are regular boys until they overhear two teachers in the teacher's lounge. Mr. Kidd and Mrs. Pearce appear to be normal teachers, but they talk to each other in a very strange language when no one else is around. Out of curiosity Charlie says, "spudvetch" to Mr. Kidd and that's when they know things will never be the same.

Charlie soon disappears and it's up to Jim to find him. With the help of his sister Becky, Jim heads to the Isle of Sky in Scotland with nothing but mysterious coordinates to guide him. What he finds there is beyond belief! Before he knows it he must save mankind.-sc-

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Gobble, Gobble

Thanksgiving is more than stuffing and mashed potatoes! Take some time to read about why we celebrate this holiday.


Thanksgiving Day by Rebecca Rissman
Big color photos and simple text explain why we celebrate this holiday.

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Pilgrims by Mary Pope Osborne
What it was really like to sail to the New World on the Mayflower.

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Pardon That Turkey by Susan Sloate
Explains how Thanksgiving became a national holiday and why the president always pardons a turkey.

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Happy Thanksgiving by Abbie Mercer
All the fun of the holiday plus how to make a pinecone turkey.

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Celebrate Thanksgiving Day by Elaine A. Kule
For older readers, this includes the history and traditions and ways to give back to your community on this holiday.

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-gw-

Best Books for 10 Year Old Boys

There are many overlaps between books ‘for’ boys and books ‘for’ girls (and the gender divide was really driven by the twitter enquiry that prompted the list of best books for girls), but there are differences too. However much of an old-style Doc-Marten-wearing feminist Kate was (is…), and however much she swore that she would not encourage her own children into gender stereotypes, she’s come to

5th Grade Math Worksheets - 5th Grade Math Test (3)

Question 1: 6.2% written as a decimal is: 0.620.0620.00626.2Question 2: 13/25 may be written as a percent as:13%26%44%52%Question 3: 74 - 73 = 4924013432058Question 4: There are 12 milk containers in a box. Each container weighs 5/6 pounds. How many pounds does the box weigh?12 pounds5 pounds10 pounds8 poundsQuestion 5: m = 15n + 2What is m if n = 3?m = 2m = 15m = 45m = 47Question 6: 2(6x - 1) =

Math 5th Grade Test - 5th Grade Math Test (2)

Question 1: What is the perimeter of the rectangular in the figure below?6.2m6.6m8m8.2mQuestion 2: What is the area of triangle ABC if AD = BC = 6 inches?12 square inches16 square inches18 square inches22 square inchesQuestion 3: Which of the following is not a composite number?13141516Question 4: A group of middle school student has made $46.5 by selling lemonade. They charged $1.5 for a cup of

Math 5th Grade Test - 5th Grade Math Test (1)

Question 1: 4521 × 613 = 1,771,3731,871,3732,871,3732,771,373Question 2: 4/5 - 1/2 =3/102/51/103/5Question 3: How many integers between 1 and 50 contain the digit 3?12141518Question 4: A middle school has 116 students enrolled in four classes. If there is an equal number of students in each class, what is a way to determine the number of students in 5th grade?subtract 4 from 116add 116 to

15th Swedish Mathematical Society Problems 1975

1.  A is the point (1, 0), L is the line y = kx (where k > 0). For which points P (t, 0) can we find a point Q on L such that AQ and QP are perpendicular?2.  Is there a positive integer n such that the fractional part of (3 + √5)n > 0.99? 3.  Show that an + bn + cn ≥ abn-1 + bcn-1 + can-1 for real a, b, c ≥ 0 and n a positive integer. 4.  P1,

14th Swedish Mathematical Society Problems 1974

1.  Let an = 2n-1 for n > 0. Let bn = ∑r+s≤n aras. Find bn - bn-1, bn - 2bn-1 and bn.2.  Show that 1 - 1/k ≤ n(k1/n - 1) ≤ k - 1 for all positive integers n and positive reals k. 3.  Let a1 = 1, a2 = 2a1, a3 = 3a2, a4 = 4a3, ... , a9 = 9a8. Find the last two digits of a9. 4.  Find all polynomials p(x) such that p(x2) = p(x)

13th Swedish Mathematical Society Problems 1973

1.  log82 = 0.2525 in base 8 (to 4 places of decimals). Find log84 in base 8 (to 4 places of decimals). 2.  The Fibonacci sequence f1, f2, f3, ... is defined by f1 = f2 = 1, fn+2 = fn+1 + fn. Find all n such that fn = n2. 3.  ABC is a triangle with ∠A = 90o, ∠B = 60o. The points A1, B1, C1 on BC, CA, AB respectively are such that

12th Swedish Mathematical Society Problems 1972

1.  Find the largest real number a such that x - 4y = 1, ax + 3y = 1 has an integer solution. 2.  A rectangular grid of streets has m north-south streets and n east-west streets. For which m, n > 1 is it possible to start at an intersection and drive through each of the other intersections just once before returning to the start? 3.  A steak

11th Swedish Mathematical Society Problems 1971

1.  Show that (1 + a + a2)2 < 3(1 + a2 + a4) for real a ≠ 1. 2.  An arbitrary number of lines divide the plane into regions. Show that the regions can be colored red and blue so that neighboring regions have different colors. 3.  A table is covered by 15 pieces of paper. Show that we can remove 7 pieces so that the remaining 8 cover at least 8/15 of

10th Swedish Mathematical Society Problems 1970

1.  Show that infinitely many positive integers cannot be written as a sum of three fourth powers of integers. 2.  6 open disks in the plane are such that the center of no disk lies inside another. Show that no point lies inside all 6 disks.3.  A polynomial with integer coefficients takes the value 5 at five distinct integers. Show that it does not take the value 9

Chapter Book Chips - "The Year Money Grew On Trees" by Aaron Hawkins

Jackson Jones faces the prospect of a summer vacation spent working in a scrap yard. But fate intervenes in the form of his neighbor, Mrs. Nelson. She wants to find someone to take care of her late husband's apple orchard. She and Jackson sign a contract that says if he can sell $8,000.00 worth of apples she will sign over the orchard to him. Jackson thinks this sounds like a much better way to spend his summer. He enlists the help of his siblings and his cousins and together they set to work to raise and harvest the apples from 300 trees. But they have no idea what hard work they have let themselves in for. First the trees need pruning, then there is the danger of a freeze. After that, they must protect the trees from worms. And they need manure spread to fertilize them and an irrigation system set up to water them. They accomplish all of this with incredible dedication, but just when the apples are ready to harvest they face some bad luck. Will they get the apples picked in time? And will Mrs. Nelson live up to her agreement to give Jackson the orchard if he is successful? "The Year Money Grew on Trees" is a story of the satisfaction of seeing a project through to the finish and the joy of growing and harvesting a crop of your own.

-gw-


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9th Swedish Mathematical Society Problems 1969

1.  Find all integers m, n such that m3 = n3 + n. 2.  Show that tan π/3n is irrational for all positive integers n. 3.  a1 ≥ a2 ≥ ... ≥ an is a sequence of reals. b1, b2, b3, ... bn is any rearrangement of the sequence B1 ≥ B2 ≥ ... ≥ Bn. Show that ∑ aibi ≤ &sum aiBi. 4.  Define g(x) as the largest value of |y2 - xy| for

8th Swedish Mathematical Society Problems 1968

1.  Find the maximum and minimum values of x2 + 2y2 + 3z2 for real x, y, z satisfying x2 + y2 + z2 = 1. 2.  How many different ways (up to rotation) are there of labeling the faces of a cube with the numbers 1, 2, ... , 6? 3.  Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When

7th Swedish Mathematical Society Problems 1967

1.  p parallel lines are drawn in the plane and q lines perpendicular to them are also drawn. How many rectangles are bounded by the lines? 2.  You are given a ruler with two parallel straight edges a distance d apart. It may be used (1) to draw the line through two points, (2) given two points a distance ≥ d apart, to draw two parallel lines, one

6th Swedish Mathematical Society Problems 1966

1.  Let {x} denote the fractional part of x = x - [x]. The sequences x1, x2, x3, ... and y1, y2, y3, ... are such that lim {xn} = lim {yn} = 0. Is it true that lim {xn + yn} = 0? lim {xn - yn} = 0? 2.  a1 + a2 + ... + an = 0, for some k we have aj ≤ 0 for j ≤ k and aj ≥ 0 for j > k. If ai are not all 0, show that a1 + 2a2 + 3a3 + ..

5th Swedish Mathematical Society Problems 1965

1.  The feet of the altitudes in the triangle ABC are A', B', C'. Find the angles of A'B'C' in terms of the angles A, B, C. Show that the largest angle in A'B'C' is at least as big as the largest angle in ABC. When is it equal? 2.  Find all positive integers m, n such that m3 - n3 = 999. 3.  Show that for every real x ≥ ½ there is an integer n such

4th Swedish Mathematical Society Problems 1964

1.  Find the side lengths of the triangle ABC with area S and ∠BAC = x such that the side BC is as short as possible. 2.  Find all positive integers m, n such that n + (n+1) + (n+2) + ... + (n+m) = 1000. 3.  Find a polynomial with integer coefficients which has √2 + √3 and √2 + 31/3 as roots. 4.  Points H1, H2, ... , Hn are arranged in the

3rd Swedish Mathematical Society Problems 1963

1.  How many positive integers have square less than 107? 2.  The squares of a chessboard have side 4. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board? 3.  What is the remainder on dividing 1234567 + 891011 by 12?

Top 10 Maths Books

Lecture Notes on Mathematical Olympiad Courses: For Junior Section (Mathematical Olympiad Series) http://amzn.to/OlympiadCoursesAmazon.com: Lecture Notes on Mathematical Olympiad Courses: For Junior Section (Mathematical Olympiaamzn.toAmazon.com: Lecture Notes on Mathematical Olympiad Courses: For Junior Section (Mathematical Olympiad Series) (9789814293532): Xu Jiagu: Books 15 000 problems

Ahhh, nostalgia

We love our books here at the library and I am going to venture to guess that those of us working in the Children's department have special memories of books and libraries from our own childhoods. I asked my colleagues to tell me about one of their favorite books when they were a child and why it was special to them.

The Stinky Cheese Man and Other Fairly Stupid Tales
By Jon Scieszka

"It's silly and a little bit gross and has the word 'stupid' right there in the title"
-Jennifer's Pick

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Burt Dow Deep-Water Man
By Robert McCloskey

"The thing that most fascinated me was the pink color inside the whale's tummy. I wanted to get swallowed by a whale just so I could check it out for myself."

--Ginny's Pick

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Edith & Mr. Bear A Lonely Doll Story
By Dare Wright

"The doll and the bear were kind of frightening, but also fascinating. I had never seen a book like this before. It was so real!"

--Sandy's Pick


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Where the Wild Things Are
By Maurice Sendak

" I guess I liked the boy's costume and how his bedroom became a forest. I think I also liked that, although they were monsters, they were nice (they sort of reminded me of the giant muppets)."

--Sarah's Pick

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Bread and Jam For Frances
By Russell Hoban

This was definitely my favorite in the Frances series. I
have always loved the songs Frances invents, especially when she is singing to her poached egg. I also remember being envious of Frances' packed lunch that included a vase of violets."

--Carson's Pick

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--cm

Geography Books

Free world geography quiz & games at http://www.worldgeographyquiz.net

Scientific Method Limitations

ntroduction to Scientific Method Limitations:              This article is showing the limitations for scientific method. Scientific method is otherwise said to be as scientific notation which helps to show the big numbers into simplest form of a number. Scientific method is of positive and negative method. Example for scientific Method:Positive Method: 170000000 this can be expressed as

How to do long division with two digit quotients?

We've learned that we can figure out the single-digit quotient, or answer, for two-digit divisor problems using estimation. This time around, we'll use estimation to determine two-digit quotients for two-digit divisor problems.Let's look at this division problem, which has a two-digit quotient:To start, it's important to determine the first part of 741 that we can divide by 32. That

Two-digit divisor in long division - How to do long division with 2 digit divisor?

When the divisor has two digits, the procedure works the same way but instead of using facts from multiplication tables when dividing, we might have to use pencil and paper to perform helping multiplications on the side, so to speak. 14    7  4  3  4  14 goes into 7 zero times, so we look at 74. To find how many times does 14 go into 74, you probably have to do

How to do long division with 2 digit divisor

Introduction of division with 2 digit divisor:          In mathematics, especially in elementary arithmetic, division (÷) is the arithmetic operation that is the inverse of multiplication.Specifically, if c time’s b equals a, written:           c x b = aWhere b is not zero, then a divided by b equals c, written:             = c.In the above expression, a is called the dividend. Source: Wikipedia

Long Division of Polynomials Step by Step

Introduction to long division of polynomials:               In arithmetic, long division is the standard procedure suitable for dividing simple or complex multi digit numbers. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient.  (Source

No Storytimes October 26, 27 & 28. Join us for Spooky Babies!

Please note: There will be no library storytimes October 26th, 27th and 28th, 2010. Please join us for Spooky Babies instead.

Spooky Babies
Tuesday, October 26, 2010
10:30 a.m. - 11:30 a.m.
Meeting Rooms A & B

Get your wiggly shivers out and dance, dance, dance to some spooky fun music! Don't forget to come in your Halloween costume. For ages birth to six years and accompanying adult.

No registration required.

-sv-

Math Games and Activities for kids

Mathematics is said to be the study of the structure, change, quantity and space. Mathematicians, or those who work in this field, tend to study patterns and conjectures using the principles of deduction derived from definitions and axioms. Looking at this definition of mathematics, you will certainly remember those days when you were dreaded by the thought of learning math. Well, your kids

The Free Ride In Public Schools

To protect children's self-esteem or deflect complaints by parents, many public schools today automatically advance failing students to the next grade level. In other schools, some students are left back a maximum of one year, then promoted again regardless of their academic skills.The No Child Left Behind Act tries to solve this problem. The federal government is pressuring public schools to set

Math Homework Help

Are you struggling with mathematics in school or college, TutorsOnnet math tutoring can provide help in a convenient and effective way. Math can be a major struggle for children ' starting in Grade 1 and continuing throughout their high school years and beyond. Children who find the subject less intuitive than others can often face hours of frustrating homework, alongside equally frustrated

Information YOU need to know that you shouldn't share on the internet.

Your first assignment is to use a program you are familiar with to answer some questions you have heard before. You are to open the application and type in the answers to the following questions:

1. What is your NAME (first and Last)?
2. When were you born (date of birth-DOB)?
3. What is your address (where do you live)?
4. What is your mother of father's phone number?

You are to use at least two different type faces or font styles. You are to use two different point sizes and you are to use two colors. Once you complete these tasks you may decorate the page, BUT the answers to your questions must remain readable.
These questions are considered personal information. In the event of an emergency or if you were to get lost you should know this information. This information though should never be given to anyone you don't know, unless it is someone in a position of authority ( a police officer, fireman, school administrator). Do you know people who might ask you for this information over the internet? In many cases you don't, if you are asked for this information you should tell your parents or an adult who is monitoring your computer activity. This is one of the most important rules of Internet Safety, which we will be discussing this year.

49th Eötvös Competition Problems 1945

1.  Knowing that 23 October 1948 was a Saturday, which is more frequent for New Year's Day, Sunday or Monday? 2.  A convex polyhedron has no diagonals (every pair of vertices are connected by an edge). Prove that it is a tetrahedron. 3.  ABC is

42nd Eötvös Competition Problems 1938

1.  Show that a positive integer n is the sum of two squares iff 2n is the sum of two squares. 2.  Show that 1/n + 1/(n+1) + 1/(n+2) + ... + 1/n2 > 1 for integers n > 1. 3.  Show that for

41st Eötvös Competition Problems 1937

1.  a1, a2, ... , an is any finite sequence of positive integers. Show that a1! a2! ... an! < (S + 1)! where S = a1 + a2 + ... + an. 2.  P, Q, R are three points in space. The circle CP passes through Q and R, the circle CQ passes through R and P, and the circle CR passes through P and Q. The tangents to CQ and CR at P coincide. Similarly, the

40th Eötvös Competition Problems 1936

1.  Show that 1/(1·2) + 1/(3·4) + 1/(5·6) + ... + 1/( (2n-1)·2n) = 1/(n+1) + 1/(n+2) + ... + 1/(2n). 2.  ABC is a triangle. Show that, if the point P inside the triangle is such that the triangles PAB, PBC, PCA have equal area, then P must be the centroid. 3.  Find the

34th Eötvös Competition Problems 1930

1.  How many integers (1) have 5 decimal digits, (2) have last digit 6, and (3) are divisible by 3? 2.  A straight line is drawn on an 8 x 8 chessboard. What is the largest possible number of the unit squares with interior points on the line? 3.  A circle or

29th Eötvös Competition Problems 1925

1.  Given four integers, show that the product of the six differences is divisible by 12. 2.  How many zeros does the the decimal representation of 1000! end with? 3.  Show that the inradius of a right-angled

27th Eötvös Competition Problems 1923

1.  The circles OAB, OBC, OCA have equal radius r. Show that the circle ABC also has radius r. 2.  Let x be a real number and put y = (x + 1)/2. Put an = 1 + x + x2 + ... + xn, and bn = 1 + y + y2 + ... + yn. Show that ∑0n am (n+1)C(m+1) = 2n bn, where aCb is the binomial coefficient a!/( b! (a-b)! ). 3.  Show that an infinite arithmetic progression

26th Eötvös Competition Problems 1922

1.  Show that given four non-coplanar points A, B, P, Q there is a plane with A, B on one side and P, Q on the other, and with all the points the same distance from the plane. 2.  Show that we cannot factorise x4 + 2x2 + 2x + 2 as the product of two quadratics with integer coefficients. 3.  A, B are two points inside a given

23rd Eötvös Competition Problems 1916

1.  a, b are positive reals. Show that 1/x + 1/(x-a) + 1/(x+b) = 0 has two real roots one in [a/3, 2a/3] and the other in [-2b/3, -b/3]. 2.  ABC is a triangle. The bisector of ∠C meets AB at D. Show that CD2 < CA·CB. 3.  If d is the

19th Eötvös Competition Problems 1912

1.  How many n-digit decimal integers have all digits 1, 2 or 3. How many also contain each of 1, 2, 3 at least once? 2.  Prove that 5n + 2 3n-1 + 1 = 0 (mod 8). 3.  ABCD is a quadrilateral with vertices in

18th Eötvös Competition Problems 1911

1.  Real numbers a, b, c, A, B, C satisfy b2 < ac and aC - 2bB + cA = 0. Show that B2 ≥ AC. 2.  L1, L2, L3, L4 are diameters of a circle C radius 1 and the angle between any two is either π/2 or π/4. If P is a point on the circle, show that the sum of the fourth powers of the distances from P to the four diameters is 3/2. 3.  ABC is a triangle with angle C = 120o. Find the

16th Eötvös Competition Problems 1909

1.  Prove that (n + 1)3 ≠ n3 + (n - 1)3 for any positive integer n. 2.  α is acute. Show that α < (sin α + tan α)/2. 3.  ABC is a triangle. The feet of the altitudes from A, B, C are P, Q, R respectively, and P,

15th Eötvös Competition Problems 1908

1.  m and n are odd. Show that 2k divides m3 - n3 iff it divides m - n. 2.  Let a right angled triangle have side lengths a > b > c. Show that for n > 2, an > bn + cn. 3.  Let the vertices of a regular 10

14th Eötvös Competition Problems 1907

1.  Show that the quadratic x2 + 2mx + 2n has no rational roots for odd integers m, n. 2.  Let r be the radius of a circle through three points of a parallelogram. Show that any point inside the parallelogram is a distance ≤ r from at least one of its vertices. A2.  ABC is a triangle. The incircle has center I and

15th Iberoamerican Mathematical Olympiad Problems 2000

15th Iberoamerican Mathematical Olympiad Problems 2000A1.  Label the vertices of a regular n-gon from 1 to n > 3. Draw all the diagonals. Show that if n is odd then we can label each side and diagonal with a number from 1 to n different from the labels of its endpoints so that at each vertex the sides and diagonals all have different labels.

Amazing Number Facts No (part 2)

Here is a remarkable formula: f(n) = n2-n+41 f(1) = 12-1+41 = 41 a prime number f(2) = 22-2+41 = 43 a prime number f(3) = 32-3+41 = 47 a prime number f(4) = 42-4+41 = 53 a prime number   How many prime numbers does this formula produce? A formula that always produces prime numbers in this way has never been found. So just how

Amazing Number Facts No (part 1)

Since 32 + 42 = 52 does it follow that 33 + 43 + 53 = 63 ? Check it out ! Does this pattern continue to be true?The factors of 28 (not including itself) are 1, 2, 4, 7 and 14. Astonishingly these

Scumble

by Ingrid Law (Childrens New Books)(J Law)(2010)

for ages 8-12 years

In this follow up to Savvy, when you turn 13 in Legder Kale's family you find yourself with a savvy. In the first book Ledge's cousin Mibs turns 13 with a bang. Now it's Ledge's turn, but his savvy is a let down. He can disassemble watches, but what good is that?

His family sets out for Wyoming for a savvy-filled family wedding. Just about the time he gets there he realizes his savvy can disassemble a lot more than just watches! To make things worse, an outsider named Sarah Jane witnesses his skills. Not only that, but she's the town snoop, too!

Ledge must learn how to scumble his savvy, which just means he needs to learn to finesse it. He must also keep Sarah Jane's dad from foreclosing on his uncle's ranch, and keep Sarah Jane from revealing all his family's secrets!

This book has just the right amount of action and intrigue to keep any reader wanting to read more!-sc-

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7th Junior Balkan Mathematical Olympiad Problems 2003

1.  Let A = 44...4 (2n digits) and B = 88...8 (n digits). Show that A + 2B + 4 is a square. 2.  A1, A2, ... , An are points in the plane, so that if we take the points in any order B1, B2, ... , Bn, then the broken line B1B2...Bn does not intersect itself. What is the largest possible value of n? 3.  ABC is a triangle. E, F are

3rd Junior Balkan Mathematical Olympiad Problems 1999

1.  a, b, c are distinct reals and there are reals x, y such that a3 + ax + y = 0, b3 + bx + y = 0 and c3 + cx + y = 0. Show that a + b + c = 0. 2.  Let an = 23n + 36n+2 + 56n+2. Find gcd(a0, a1, a2, ... , a1999). 3.

1st Junior Balkan Mathematical Olympiad Problems 1997

1.  Show that given any 9 points inside a square side 1 we can always find three which form a triangle with area < 1/8. 2.  Given reals x, y with (x2 + y2)/(x2 - y2) + (x2 - y2)/(x2 + y2) = k, find (x8 + y8)/(x8 - y8) + (x8 - y8)/(x8 + y8) in terms of k. 2.  A square is divided into n parallel strips (parallel to

13th All Soviet Union Mathematical Olympiad Problems 1979

1.  T is an isosceles triangle. Another isosceles triangle T' has one vertex on each side of T. What is the smallest possible value of area T'/area T? 2.  A grasshopper hops about in the first quadrant (x, y >= 0).

12th All Soviet Union Mathematical Olympiad Problems 1978

1.  an is the nearest integer to √n. Find 1/a1 + 1/a2 + ... + 1/a1980. 2.  ABCD is a quadrilateral. M is a point inside it such that ABMD is a parallelogram. ∠CBM = ∠CDM. Show that ∠ACD = ∠BCM. 3.  Show that there is no

10th All Soviet Union Mathematical Olympiad Problems 1976

1.  50 watches, all keeping perfect time, lie on a table. Show that there is a moment when the sum of the distances from the center of the table to the center of each dial equals the sum of the distances from the center of the table to the tip of each minute hand.