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17th Mexican Mathematical Olympiad Problems 2003

17th Mexican Mathematical Olympiad Problems 2003A1.  Find all positive integers with two or more digits such that if we insert a 0 between the units and tens digits we get a multiple of the original number. A2.  A, B, C

16th Mexican Mathematical Olympiad Problems 2002

16th Mexican Mathematical Olympiad Problems 2002A1.  The numbers 1 to 1024 are written one per square on a 32 x 32 board, so that the first row is 1, 2, ... , 32, the second row is 33, 34, ... , 64 and so on. Then the board is divided into four 16 x 16 boards and the position of these boards is moved round clockwise, so that AB goes to DADC CBthen each of the 16 x 16

15th Mexican Mathematical Olympiad Problems 2001

15th Mexican Mathematical Olympiad Problems 2001A1.  Find all 7-digit numbers which are multiples of 21 and which have each digit 3 or 7. A2.  Given some colored balls (at least three different colors) and at least three boxes. The balls are put into the boxes so that no box is empty and we cannot find three balls of different colors which are in three

14th Mexican Mathematical Olympiad Problems 2000

14th Mexican Mathematical Olympiad Problems 2000A1.  A, B, C, D are circles such that A and B touch externally at P, B and C touch externally at Q, C and D touch externally at R, and D and A touch externally at S. A does not intersect C, and B does not intersect D. Show that PQRS is cyclic. If A and C have radius 2, B and D have radius 3, and the distance between the

13th Mexican Mathematical Olympiad Problems 1999

13th Mexican Mathematical Olympiad Problems 1999A1.  1999 cards are lying on a table. Each card has a red side and a black side and can be either side up. Two players play alternately. Each player can remove any number of cards showing the same color from the table or turn over any number of cards of the same color. The winner is the player who removes the last card.

12th Mexican Mathematical Olympiad Problems 1998

12th Mexican Mathematical Olympiad Problems 1998A1.  Given a positive integer we can take the sum of the squares of its digits. If repeating this operation a finite number of times gives 1 we call the number tame. Show that there are infinitely many pairs (n, n+1) of consecutive tame integers. A2.  ABC is a triangle with ∠B = 90o and

3rd Mexican Mathematical Olympiad Problems 1989

3rd Mexican Mathematical Olympiad Problems 1989A1.  The triangle ABC has AB = 5, the medians from A and B are perpendicular and the area is 18. Find the lengths of the other two sides. A2.  Find integers m and n

How do you do long division with decimals?

How do you do long division with decimals?
When we are given a long division to do it will not always work out to a whole number. Sometimes there will be numbers left over. We can use the long division process to work out the answer to a number of decimal places.
The secret to working out a long division to decimal places is the

Division When the Divisor Is a Decimal

Division When the Divisor Is a Decimal - Division of Decimals by Whole NumbersThe procedure for the division of decimals is very similar to the division of whole numbers. How to divide a four digit decimal number by a two digit number (e.g. 0.4131 ÷ 17). A2.  If m and n

1st Mexican Mathematical Olympiad Problems 1987

1st Mexican Mathematical Olympiad Problems 1987A1.  a/b and c/d are positive fractions in their lowest terms such that a/b + c/d = 1. Show that b = d. A2.  How many positive integers divide 20! ? A3.  L and L' are parallel lines and P is a point midway between them. The variable point A lies L, and A' lies on L' so that ∠APA' = 90o. X is the foot of the

How to Make Multiplication Homework Fun

For a grade-schooler, learning the basics of math can be hard especially if it is not taught properly. Multiplication Tool is an online math study tool that helps students master the “art” of multiplying several digits. This website should help children improve on this essential math skill which can benefit a lot in tackling more difficult number problems.2.  The remainder on dividing the

54th Polish Mathematical Olympiad Problems 2003

54th Polish Mathematical Olympiad Problems 2003A1.  ABC is acute-angled. M is the midpoint of AB. A line through M meets the lines CA, CB at K, L with CK = CL. O is the circumcenter of CKL and CD is an altitude of ABC. Show that OD = OM. A2.  P is a point inside a regular tetrahedron with

51st Polish Mathematical Olympiad Problems 2000

51st Polish Mathematical Olympiad Problems 2000A1.  How many solutions in non-negative reals are there to the equations: x1 + xn2 = 4xn x2 + x12 = 4x1 ... xn + xn-12 = 4xn-1? A2.  The triangle ABC has AC

50th Polish Mathematical Olympiad Problems 1999

50th Polish Mathematical Olympiad Problems 1999A1.  D is a point on the side BC of the triangle ABC such that AD > BC. E is a point on the side AC such that AE/EC = BD/(AD-BC). Show that AD > BE. A2.  Given 101

49th Polish Mathematical Olympiad Problems 1998

49th Polish Mathematical Olympiad Problems 1998A1.  Find all solutions in positive integers to a + b + c = xyz, x + y + z = abc.A2.  Fn is the Fibonacci sequence F0 = F1 = 1, Fn+2 = Fn+1 + Fn. Find all pairs m > k ≥ 0 such that the sequence x0, x1, x2, ... defined by x0 = Fk/Fm and xn+1 = (2xn - 1)/(1 - xn) for xn ≠ 1, or 1 if xn = 1, contains the

48th Polish Mathematical Olympiad Problems 1997

48th Polish Mathematical Olympiad Problems 1997A1.  The positive integers x1, x2, ... , x7 satisfy x6 = 144, xn+3 = xn+2(xn+1+xn) for n = 1, 2, 3, 4. Find x7. A2.  Find all real solutions to 3(x2 + y2 + z2) = 1, x2y2 + y2z2 + z2x2 = xyz(x + y + z)3. A2.  P is a point inside the triangle ABC such that ∠PBC = ∠PCA <

46th Polish Mathematical Olympiad Problems 1995

46th Polish Mathematical Olympiad Problems 1995 A1.  How many subsets of {1, 2, ... , 2n} do not contain two numbers with sum 2n+1?A2.  The diagonals of a convex pentagon divide it into a small pentagon and ten triangles. What is the largest number of the triangles that can have the same area? A2.  L, L' are parallel lines. C is a circle that does not

44th Polish Mathematical Olympiad Problems 1993

44th Polish Mathematical Olympiad Problems 1993A1.  Find all rational solutions to: t2 - w2 + z2 = 2xy t2 - y2 + w2 = 2xz t2 - w2 + x2 = 2yz. A2.  A circle center O is inscribed in the quadrilateral ABCD

43rd Polish Mathematical Olympiad Problems 1992

43rd Polish Mathematical Olympiad Problems 1992A1.  Segments AC and BD meet at P, and |PA| = |PD|, |PB| = |PC|. O is the circumcenter of the triangle PAB. Show that OP and CD are perpendicular.A2.  Find all functions f : Q

42nd Polish Mathematical Olympiad Problems 1991

42nd Polish Mathematical Olympiad Problems 1991A1.  Do there exist tetrahedra T1, T2 such that (1) vol T1 > vol T2, and (2) every face of T2 has larger area than any face of T1? A2.  Let F(n) be the number of paths P0, P1, ... , Pn of length n that go from P0 = (0,0) to a lattice point Pn on the line y = 0, such that each Pi is a lattice point and

41st Polish Mathematical Olympiad Problems 1990

41st Polish Mathematical Olympiad Problems 1990A1.  Find all real-valued functions f on the reals such that (x-y)f(x+y) - (x+y)f(x-y) = 4xy(x2-y2) for all x, y. A2.  For n > 1 and positive reals x1, x2, ... , xn, show that x12/(x12+x2x3) + x22/(x22+x3x4) + ... + xn2/(xn2+x1x2) ≤ n-1.A2.  For

16th Balkan Mathematical Olympiad Problems 1999

16th Balkan Mathematical Olympiad Problems 1999A1.  O is the circumcenter of the triangle ABC. XY is the diameter of the circumcircle perpendicular to BC. It meets BC at M. X is closer to M than Y. Z is the point on MY such that MZ = MX. W is the midpoint of AZ. Show that W lies on the circle through the midpoints of the sides of ABC. Show that MW is perpendicular to

15th Balkan Mathematical Olympiad Problems 1998

15th Balkan Mathematical Olympiad Problems 1998A1.  How many different integers can be written as [n2/1998] for n = 1, 2, ... , 1997? A2.  xi are distinct positive reals satisfying x1 < x2 < ... < x2n+1. Show that x1 - x2 + x3 - x4 + ... - x2n + x2n+1 < (x1n - x2n + ... - x2nn + x2n+1n)1/n. A2.  The reals w, x

1st Balkan Mathematical Olympiad Problems 1984

1st Balkan Mathematical Olympiad Problems 1984A1.  Let x1, x2, ... , xn be positive reals with sum 1. Prove that x1/(2 - x1) + x2/(2 - x2) + ... + xn/(2 - xn) ≥ n/(2n - 1). A2.  ABCD is a cyclic quadrilateral.

New Parent Event on Tuesday, September 21, 2010, 2:00 p.m. - 3:00 p.m.

Are you a brand new parent of an infant under the age of one? Are you interested in finding out what the Beaverton City Library has to offer for families? Please join us on Tuesday, September 21, 2010 from 2:00 p.m. to 3:00 p.m for our Fall New Parent Event. We will have a storytime for infants, a play area, craft area and snacks for babies and grown-ups. We will be joined by a nutrition expert from the OSU Extension Service who can answer questions about your baby's nutritional needs. No registration is required. We hope to see you and your new baby at this fun program. -gw-

35th Canadian Mathematical Olympiad Problems 2003

35th Canadian Mathematical Olympiad Problems 20031.  The angle between the hour and minute hands of a standard 12-hour clock is exactly 1o. The time is an integral number n of minutes after noon (where 0 < n < 720). Find the possible values of n. 2.  What are the last three digits of 2003N, where N = 20022001. 3.  Show that if n > 1 is an integer then (1 + 1/3 + 1/5

29th Canadian Mathematical Olympiad Problems 1997

29th Canadian Mathematical Olympiad Problems 19971.  How many pairs of positive integers have greatest common divisor 5! and least common multiple 50! ? 2.  A finite number of closed intervals of length 1 cover the interval [0, 50]. Show that we can find a subset of at least 25 intervals with every pair disjoint. 3. 

26th Canadian Mathematical Olympiad Problems 1994

26th Canadian Mathematical Olympiad Problems 19941.  Find -3/1! + 7/2! - 13/3! + 21/4! - 31/5! + ... + (19942 + 1994 + 1)/1994! 2.  Show that every power of (√2 - 1) can be written in the form √(k+1) - √k. 3.

25th Canadian Mathematical Olympiad Problems 1993

25th Canadian Mathematical Olympiad Problems 19931.  Show that there is a unique triangle such that (1) the sides and an altitude have lengths with are 4 consecutive integers, and (2) the foot of the altitude is an integral distance from each vertex. 2.  Show that the real number k is rational iff the sequence k, k + 1, k + 2, k + 3, ... contains three (distinct)

24th Canadian Mathematical Olympiad Problems 1992

24th Canadian Mathematical Olympiad Problems 19921.  Show that n! is divisible by (1 + 2 + ... + n) iff n+1 is not an odd prime. 2.  Show that x(x - z)2 + y(y - z)2 ≥ (x - z)(y - z)(x + y - z) for all non-negative reals x, y, z. When does equality hold? 2.  Three

17th Canadian Mathematical Olympiad Problems 1985

17th Canadian Mathematical Olympiad Problems 19851.  A triangle has sides 6, 8, 10. Show that there is a unique line which bisects the area and the perimeter. 2.  Is there an integer which is doubled by moving its first digit to the end? [For example, 241 does not work because 412 is not 2 x 241.] 2.  Show that two

10th Canadian Mathematical Olympiad Problems 1978

10th Canadian Mathematical Olympiad Problems 19781.  A square has tens digit 7. What is the units digit? 2.  Find all positive integers m, n such that 2m2 = 3n3. 3.  Find the real solution x, y, z to x + y + z = 5,

Truck Day!

Come join us Monday September 20th for Truck Day at the library!

Big trucks will be parked in the west farmer's market parking lot of the library.

They will be here 10:30 -11:30 a.m. Children aged 0-6 years and their families are welcome to come and see some big trucks up close!-SC-

9th Canadian Mathematical Olympiad Problems 1977

9th Canadian Mathematical Olympiad Problems 19771.  Show that there are no positive integers m, n such that 4m(m+1) = n(n+1). 2.  X is a point inside a circle center O other than O. Which points P on the circle maximise ∠OPX? 2.  Given the

5th Canadian Mathematical Olympiad Problems 1973

5th Canadian Mathematical Olympiad Problems 19731.  (1) For what x do we have x < 0 and x < 1/(4x) ? (2) What is the greatest integer n such that 4n + 13 < 0 and n(n+3) > 16? (3) Give an example of a rational number between 11/24 and 6/13. (4) Express 100000 as a product of two integers which are not divisible by 10. (5) Find 1/log236 + 1/log336. 2.  D is a point on the side AB of the triangle ABC such that AB = 4·AD. P is a point on the

38th British Mathematical Olympiad 2002 Problems

38th British Mathematical Olympiad 2002 Problems1.  From the foot of an altitude in an acute-angled triangle perpendiculars are drawn to the other two sides. Show that the distance between their feet is independent of the choice of altitude. 2.

34th British Mathematical Olympiad 1998 Problems

34th British Mathematical Olympiad 1998 Problems1.  A station issues 3800 tickets covering 200 destinations. Show that there are at least 6 destinations for which the number of tickets sold is the same. Show that this is not necessarily true for 7. 2.  The triangle ABC has ∠A > ∠C. P lies inside the triangle so that ∠PAC = ∠C. Q is taken outside the triangle so

33rd British Mathematical Olympiad 1997 Problems

33rd British Mathematical Olympiad 1997 Problems1.  M and N are 9-digit numbers. If any digit of M is replaced by the corresponding digit of N (eg the 10s digit of M replaced by the 10s digit of N), then the resulting integer is a multiple of 7. Show that if any digit of N is replaced by the corresponding digit of M, then the resulting integer must be a multiple of 7.

32nd British Mathematical Olympiad 1996 Problems

32nd British Mathematical Olympiad 1996 Problems1.  Find all non-negative integer solutions to 2m + 3n = k2. 2.  The triangle ABC has sides a, b, c, and the triangle UVW has sides u, v, w such that a2 = u(v + w - u), b2 = v(w + u - v), c2 = w(u + v - w). Show that ABC must be acute angled and express the angles U, V, W in terms of the angles A, B, C.

31st British Mathematical Olympiad 1995 Problems

31st British Mathematical Olympiad 1995 Problems1.  Find all positive integers a ≥ b ≥ c such that (1 + 1/a)(1 + 1/b)(1 + 1/c) = 2. 2.  ABC is a triangle. D, E, F are the midpoints of BC, CA, AB. Show that ∠DAC = ∠ABE iff ∠AFC = ∠ADB. 2.  Show that 12/(w + x + y + z) ≤ 1/(w + x

Math Books Patterns for Kids

Melissa & Doug Beginner Pattern BlocksThis set features 10 brightly-painted wooden patterns and 30 colorful shape pieces to replicate the fun pictures. They're perfect for early development of colors, shapes and matching skills. These puzzles are a tremendous value and a great learning set! Contains one each: fish, dog, butterfly, flowers, bird, and fire engine.Toy:  Great for early

27th British Mathematical Olympiad 1991 Problems

27th British Mathematical Olympiad 1991 Problems 1.  ABC is a triangle with ∠B = 90o and M the midpoint of AB. Show that sin ACM ≤ 1/3. 2.  Twelve dwarfs live in a forest. Some pairs of dwarfs are friends. Each has a

26th British Mathematical Olympiad 1990 Problems

26th British Mathematical Olympiad 1990 Problems1.  Show that if a polynomial with integer coefficients takes the value 1990 at four different integers, then it cannot take the value 1997 at any integer. 2.  The fractional part { x } of a real number is defined as x - [x]. Find a positive real x such that { x } + { 1/x } = 1 (*). Is there a rational x satisfying

25th British Mathematical Olympiad 1989 Problems

25th British Mathematical Olympiad 1989 Problems1.  Find the smallest positive integer a such that ax2 - bx + c = 0 has two distinct roots in the interval 0 < x < 1 for some integers b, c. 2.  Find the number of different

24th British Mathematical Olympiad 1988 Problems

24th British Mathematical Olympiad 1988 Problems1.  ABC is an equilateral triangle. S is the circle diameter AB. P is a point on AC such that the circle center P radius PC touches S at T. Show that AP/AC = 4/5. Find AT/AC. 2.  The sequence p1,

17th British Mathematical Olympiad 1981 Problems

17th British Mathematical Olympiad 1981 Problems 1.  ABC is a triangle. Three lines divide the triangle into four triangles and three pentagons. One of the triangle has its three sides along the new lines, the others each have just two sides along the new lines. If all four triangles are congruent, find the area of each in terms of the area of ABC. 2.  Find a set of seven consecutive positive integers and a polynomial

15th British Mathematical Olympiad 1979 Problems

15th British Mathematical Olympiad 1979 Problems1.  Find all triangles ABC such that AB + AC = 2 and AD + BD = √5, where AD is the altitude. 2.  Three rays in space have endpoints at O. The angles between the pairs are α, β, γ, where 0 < α < β < γ. Show that there are unique points A, B, C, one on each ray, so that the triangles OAB, OBC, OCA all have perimeter 2s.

14th British Mathematical Olympiad 1978 Problems

14th British Mathematical Olympiad 1978 Problems1.  Find the point inside a triangle which has the largest product of the distances to the three sides. 2.  Show that there is no rational number m/n with 0 < m < n < 101 whose decimal expansion has the consecutive digits 1, 6, 7 (in that order). 2. 

8th British Mathematical Olympiad 1972 Problems

8th British Mathematical Olympiad 1972 Problems 1.  The relation R is defined on the set X. It has the following two properties: if aRb and bRc then cRa for distinct elements a, b, c; for distinct elements a, b either aRb or bRa but not both. What is the largest possible number of elements in X? 2.  Graph x8 + xy + y8 = 0,