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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 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 squ...
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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?
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).
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...
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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.
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...
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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.
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.
9th Asian Pacific Mathematics Olympiad 1997 ProblemsA1. Let Tn = 1 + 2 + ... + n = n(n+1)/2. Let Sn= 1/T1 + 1/T2 + ... + 1/Tn. Prove that 1/S1 + 1/S2 + ... + 1/S1996 > 1001. A2. Find an n in the range 100,...
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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.
5th Asian Pacific Mathematics Olympiad 1993 ProblemsA1. A, B, C is a triangle. X, Y, Z lie on the sides BC, CA, AB respectively, so that AYZ and XYZ are equilateral. BY and CZ meet at K. Prove that YZ2 = YK.YB. ...
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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?
3rd Asian Pacific Mathematics Olympiad 1991 ProblemsA1. ABC is a triangle. G is the centroid. The line parallel to BC through G meets AB at B' and AC at C'. Let A'' be the midpoint of BC, C'' the intersection of B'C and BG, and B'' the intersection of C'B and CG. Prove that A''B''C'' is similar to ABC.
2nd Asian Pacific Mathematics Olympiad 1990 ProblemsA1. Given θ in the range (0, π) how many (incongruent) triangles ABC have angle A = θ, BC = 1, and the following four points concyclic: A, the centroid, the midpoint of AB and the midpoint of AC?
1st Asian Pacific Mathematics Olympiad 1989 ProblemsA1. ai are positive reals. s = a1 + ... + an. Prove that for any integer n > 1 we have (1 + a1) ... (1 + an) < 1 + s + s2/2! + ... + sn/n! .A2. Prove that 5n2 = 36a2 + 18b2 + 6c2 has no integer solutions except a = b = c = n = 0. A3. ABC is a triangle. X lies on the segment AB so that AX/AB =...
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Indian National Mathematics Olympiad 2004 Problems1. ABCD is a convex quadrilateral. K, L, M, N are the midpoints of the sides AB, BC, CD, DA. BD bisects KM at Q. QA = QB = QC = QD, and LK/LM = CD/CB. Prove that ABCD is a square.
Indian National Mathematics Olympiad 2003 Problems1. ABC is acute-angled. P is an interior point. The line BP meets AC at E, and the line CP meets AB at F. AP meets EF at D. K is the foot of the perpendicular from D to BC. Show that KD bisects ∠EKF.
Indian National Mathematics Olympiad 2002 Problems1. ABCDEF is a convex hexagon. Consider the following statements. (1) AB is parallel to DE, (2) BC is parallel to EF, (3) CD is parallel to FA, (4) AE = BD, (5) BF = CE, (6) CA = DF. Show that if any five of these statements are true then the hexagon is cyclic.
Indian National Mathematics Olympiad 2001 Problems1. ABC is a triangle which is not right-angled. P is a point in the plane. A', B', C' are the reflections of P in BC, CA, AB. Show that [incomplete]. 2. Show that...
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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.
Indian National Mathematics Olympiad 1999 Problems1. ABC is an acute-angled triangle. AD is an altitude, BE a median, and CF an angle bisector. CF meets AD at M, and DE at N. FM = 2, MN = 1, NC = 3. Find the perimeter of ABC.
Indian National Mathematics Olympiad 1998 Problems1. C is a circle with center O. AB is a chord not passing through O. M is the midpoint of AB. C' is the circle diameter OM. T is a point on C'. The tangent to C' at T meets C at P. Show that PA2 + PB2 = 4 PT2.
Indian National Mathematics Olympiad 1997 Problems1. ABCD is a parallelogram. A line through C does not pass through the interior of ABCD and meets the lines AB, AD at E, F respectively. Show that AC2 + CE·CF = AB·AE + AD·AF.
Indian National Mathematics Olympiad 1996 Problems1. Given any positive integer n, show that there are distinct positive integers a, b such that a + k divides b + k for k = 1, 2, ... , n. If a, b are positive integers such that a + k divides b + k for all positive integers k, show that a = b.
Indian National Mathematics Olympiad 1995 Problems1. ABC is an acute-angled triangle with ∠A = 30o. H is the orthocenter and M is the midpoint of BC. T is a point on HM such that HM = MT. Show that AT = 2 BC. 2. ...
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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.
Relationship of Subtraction to DivisionThe result of division is to separate a group of objects into several equal smaller groups. The starting group is called the dividend. The number of groups that are separated out is called the divisor. The number of objects in each smaller group is called the quotient.The results of division can be obtained by repeated subtraction. If we are separating...
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Dividing Numbers in Scientific NotationDivide the base numbersSubtract the exponents of the tensAdjust the base number to have one digit before the decimal point by raising or lowering the resulting exponent of the tenTo divide numbers in scientific notation:
The main library building on SW Fifth and Hall is turning ten years old this September. Help us celebrate an amazing decade of library service by attending one of the many events we have planned. We will celebrate 10 years in 10 days of fun activities and programs. And thank you to the citizens of Beaverton and Washington County for a decade of support. We couldn't have done it without each and everyone of you. For more information about all the events planned please visit the 10th Birthday page on the library's website. -...
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One World, One Dayby Barbara KerleyAges 4–upSixty beautiful photographs follow children all around the world as they eat breakfast, go to school, and live a normal day. The global perspectives of the numerous photographers echo the theme of one global family.
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...
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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.
Book Summary of Trigonometry : Solving Trigonometric Equations & InequalitiesThe authors have made this book a thorough review of basic courses in Trigonometry. In addition to traditional studies of common trigonometric functions and basic trigonometric identities, the book covers various methods for solving trigonometric equations and inequalities. It also introduces an innovative grap...
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