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 sequence a0, a1, a2, ... satisfies an+1 = 2an2 - 2an + 1. Find all a0 for which there...

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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. 2.  1 = a1 < a2 < a3 < ... is an infinite sequence of integers such that an < 2n-1. Show that every positive integer is the difference of two members of...

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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 polynomials of degree n with real roots x1 ≤ x2 ≤ ... ≤ xn such that xk belongs to the...

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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. 3.  Let x = 1o. Show that...

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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 lies across two parallel lines a unit distance apart, so that two triangular areas of...

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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? A2.  Find all pairs m, n of positive integers such that m2 - n divides...

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13th Asian Pacific Mathematics Olympiad 2001 Problems

13th Asian Pacific Mathematics Olympiad 2001 Problems A1.  If n is a positive integer, let d be the number of digits in n (in base 10) and s be the sum of the digits. Let n(k) be the number formed by deleting the last k digits of n. Prove that n = s + 9 n(1) + 9 n(2) + ... + 9 n(d). A2.  Find the largest n so that the number of integers less than or equal to n and divisible by 3 equals...

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Arithmetic Facts Flashcards

Arithmetic Facts Flashcards Guidelines:The Three Stacks.  Initially, some (if not all) of the new flashcards are divided into a weekly stack and a daily stack.  After several weeks of daily work, a finished stack is created.  Getting Started.  Individual flashcards are initially created by cutting the flashcard sheets along the lines.  Answers should be written lightly in pencil on the back.  On the...

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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.Facts of the Week!  (See next page.)  Central to the idea behind...

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12th Asian Pacific Mathematics Olympiad 2000 Problems

12th Asian Pacific Mathematics Olympiad 2000 ProblemsA1.  Find a13/(1 - 3a1 + 3a12) + a23/(1 - 3a2 + 3a22) + ... + a1013/(1 - 3a101 + 3a1012), where an = n/101. A2.  Find all permutations a1, a2, ... , a9 of 1, 2, ... , 9 such that a1 + a2 + a3 + a4 = a4 + a5 + a6 + a7 = a7 + a8 + a9 + a1 and a12 + a22 + a32 + a42 = a42 + a52 + ...

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11th Asian Pacific Mathematics Olympiad 1999 Problems

11th Asian Pacific Mathematics Olympiad 1999 ProblemsA1.  Find the smallest positive integer n such that no arithmetic progression of 1999 reals contains just n integers. A2.  The real numbers x1, x2, x3, ... satisfy xi+j ≤ xi + xj for all i, j. Prove that x1 + x2/2 + ... + xn/n ≥ xn. A3.  Two circles touch the line AB at A and B and intersect each other at X and Y with...

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10th Asian Pacific Mathematics Olympiad 1998 Problems

10th Asian Pacific Mathematics Olympiad 1998 ProblemsA1.  S is the set of all possible n-tuples (X1, X2, ... , Xn) where each Xi is a subset of {1, 2, ... , 1998}. For each member k of S let f(k) be the number of elements in the union of its n elements. Find the sum of f(k) over all k in S. A2.  Show that (36m + n)(m + 36n) is not a power of 2 for any positive integers m, n....

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