4th Austrian-Polish Mathematics Competition 1981 Problems

4th Austrian-Polish Mathematics Competition 1981 Problems1.  Find the smallest n for which we can find 15 distinct elements a1, a2, ... , a15 of {16, 17, ... , n} such that ak is a multiple of k. 2.  The rational

3rd Austrian-Polish Mathematics Competition 1980 Problems

3rd Austrian-Polish Mathematics Competition 1980 Problems1.  A, B, C are infinite arithmetic progressions of integers. {1, 2, 3, 4, 5, 6, 7, 8} is a subset of their union. Show that 1980 also belongs to their union.

2nd Austrian-Polish Mathematics Competition 1979 Problems

2nd Austrian-Polish Mathematics Competition 1979 Problems1.  ABCD is a square. E is any point on AB. F is the point on BC such that BF = BE. The perpendicular from B meets EF at G. Show that ∠DGF = 90o.2.  Find all

1st Austrian-Polish Mathematics Competition 1978 Problems

1st Austrian-Polish Mathematics Competition 1978 Problems1.  Find all real-valued functions f on the positive reals which satisfy f(x + y) = f(x2 + y2) for all positive x, y. 2.  A parallelogram has its vertices on the boundary of a regular hexagon and its center at the center of the hexagon. Show that its area is at most 2/3 the area of the hexagon.

15th Asian Pacific Mathematics Olympiad 2003 Problems

15th Asian Pacific Mathematics Olympiad 2003 Problems 1.  The polynomial a8x8 +a7x7 + ... + a0 has a8 = 1, a7 = -4, a6 = 7 and all its roots positive and real. Find the possible values for a0. 2.  A unit square

14th Asian Pacific Mathematics Olympiad 2002 Problems

14th Asian Pacific Mathematics Olympiad 2002 ProblemsA1.  xi are non-negative integers. Prove that x1! x2! ... xn! ≥ ( [(x1 + ... + xn)/n] ! )n (where [y] denotes the largest integer not exceeding y). When do you have equality? Getting

3rd Grade Arithmetic Facts Practice Sheets

All about these third grade arithmetic facts practice sheetsFree download! You can download our third grade arithmetic facts practice sheets for free from our website: www.meaningfulmathbooks.com.  On this website, there are also sheets designed for fourth and fifth grade, as well as a variety of other resources related to Making Math Meaningful books.A2.  Find an n in the range 100,

8th Asian Pacific Mathematics Olympiad 1996 Problems

8th Asian Pacific Mathematics Olympiad 1996 ProblemsA1.  ABCD is a fixed rhombus. P lies on AB and Q on BC, so that PQ is perpendicular to BD. Similarly P' lies on AD and Q' on CD, so that P'Q' is perpendicular to BD. The distance between PQ and P'Q' is more than BD/2. Show that the perimeter of the hexagon APQCQ'P' depends only on the distance between PQ and P'Q'.

Notes for Fractions and Decimals

Notes for Fractions and DecimalsDecimals  ·        Addition & Subtraction:  Line up the decimal points, then do the calculation.Example:   57.4 - 4.23           Solution:     57.40     (don't forget to add the extra zero!)                                                                   -  4.23                                                                      53.17·        Multiplication: 

7th Asian Pacific Mathematics Olympiad 1995 Problems

7th Asian Pacific Mathematics Olympiad 1995 ProblemsA1.  Find all real sequences x1, x2, ... , x1995 which satisfy 2√(xn - n + 1) ≥ xn+1 - n + 1 for n = 1, 2, ... , 1994, and 2√(x1995 - 1994) ≥ x1 + 1. A2. 

6th Asian Pacific Mathematics Olympiad 1994 Problems

6th Asian Pacific Mathematics Olympiad 1994 ProblemsA1.  Find all real-valued functions f on the reals such that (1) f(1) = 1, (2) f(-1) = -1, (3) f(x) ≤ f(0) for 0 < x < 1, (4) f(x + y) ≥ f(x) + f(y) for all x, y, (5) f(x + y) ≤ f(x) + f(y) + 1 for all x, y. A2.

4th Asian Pacific Mathematics Olympiad 1992 Problems

4th Asian Pacific Mathematics Olympiad 1992 Problems A1.  A triangle has sides a, b, c. Construct another triangle sides (-a + b + c)/2, (a - b + c)/2, (a + b - c)/2. For which triangles can this process be repeated arbitrarily many times? 2.  Show that a2

Ready, Set, Go! Georgia CRCT Grade 8 - Reading and English Language Arts (REA) [Paperback]

Indian National Mathematics Olympiad 2000 Problems

Indian National Mathematics Olympiad 2000 Problems1.  The incircle of ABC touches BC, CA, AB at K, L, M respectively. The line through A parallel to LK meets MK at P, and the line through A parallel to MK meets LK at Q. Show that the line PQ bisects AB and bisects AC. 2. 

Fun Math Books for Kids (read aloud to ages 9 and up)

FUN math books for kids (read aloud to ages 9 and up)

Estimating and Measuring's Books

Preparatory Mathematics: Answer Book

Multiplication Equations

Multiplication Sentences - Multiplication Equations An equation is a mathematical statement such that the expression on the left side of the equals sign (=) has the same value as the expression on the right side. An example of an equation is 9 * 8 = 72. One of the terms in an equation may not be known and needs to be determined. The unknown term may be represented by a letter such as x. (e.g.

Multiply Pounds and Ounces

Multiplying Pounds and OuncesHow to multiply pounds and ounces Convert pounds and ounces to ounces by multiplying pounds by 16 and adding the number of ouncesPerform the required multiplication to determine the number of ounces.Convert the ounces to pounds and ounces by dividing by 16.The quotient is the number of pounds and the remainder is the number of ounces.Example: Multiply 5 pounds 6

Multiply Gallons, Quarts and Pints

Multiplying Gallons, Quarts and PintsHow to multiply gallons, quarts and pints Convert gallons to pints by multiplying the number of gallons by 8.Convert quarts to pints by multiplying the number of quarts by 2.Add the above quantities and the number of original pints together.Perform the required multiplication to determine the number of pints.

Science & Nature books for Kids

City Dog, Country Frogby Mo Willems, Jon J MuthAges 4–8City Dog, enthusiastic about his new life without a leash, meets Country Frog in the spring and the two frolic throughout the summer and fall. But winter finds Country Frog’s favorite rock empty. In the spring, City Dog

Math Books for Kids

These math books are recommended for parents to use at home as well as for classroom use by teachers. All make good read-alouds and children like re-reading independently while enjoying the illustrations. Many can be used in the classroom to introduce concepts or units.