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Kamis, 30 September 2010
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,...
Posted in: Mexican Mathematical Olympiad16th 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...
Posted in: Mexican Mathematical Olympiad15th 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...
Posted in: Mexican Mathematical Olympiad14th 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...
Posted in: Mexican Mathematical Olympiad13th 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....
Posted in: Mexican Mathematical Olympiad12th 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.
11th Mexican Mathematical Olympiad Problems 1997A1. Find all primes p such that 8p4 - 3003 is a (positive) prime. A2. ABC is a triangle with centroid G. P, P' are points on the side BC, Q is a point on the side AC, R is a point on the side AB, such that AR/RB = BP/PC = CQ/QA = CP'/P'B. The lines AP' and QR meet at K. Show that P, G and K are collinear. ...
Posted in: Mexican Mathematical OlympiadRabu, 29 September 2010
10th Mexican Mathematical Olympiad Problems 1996
10th Mexican Mathematical Olympiad Problems 1996A1. ABCD is a quadrilateral. P and Q are points on the diagonal BD such that the points are in the order B, P, Q, D and BP = PQ = QD. The line AP meets BC at E, and the line Q meets CD at F. Show that ABCD is a parallelogram iff E and F are the midpoints of their sides. A2. 64 tokens are numbered 1, 2, ... , 64. The...
Posted in: Mexican Mathematical OlympiadSelasa, 28 September 2010
y3 racing games
Play y3 racing games Super Moto Bike Tags: HT83 Games, 19 games, Unrated games, New games, Flash, 1 Player, Racing, Motorcycle, Driving, Infomation: Race around 3 different tracks on your motorcycle. Reach checkpoints to gain more time. How to play: Use Arrow Keys to interact. Free Play at http://www.19games.com Quad Racer Tags: HT83 Games, 19 games, New games, Unrated games, Shockwave, 1 Player, Racing, Quad, Driving, Infomation: Drive your quad and race against other quad. How to play: Use Arrow Keys to interact. Free Play at http://www.19games.com 3D Jetski Racing Tags: HT83 Games, 19 games, Unrated games,...
9th Mexican Mathematical Olympiad Problems 1995
9th Mexican Mathematical Olympiad Problems 1995A1. N students are seated at desks in an m x n array, where m, n ≥ 3. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are 1020 handshakes, what is N? A2. 6 points in the plane have the property that 8 of the distances between them are 1. Show that...
Posted in: Mexican Mathematical Olympiad8th Mexican Mathematical Olympiad Problems 1994
8th Mexican Mathematical Olympiad Problems 1994A1. The sequence 1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ... is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to 1994.
7th Mexican Mathematical Olympiad Problems 1993A1. ABC is a triangle with ∠A = 90o. Take E such that the triangle AEC is outside ABC and AE = CE and ∠AEC = 90o. Similarly, take D so that ADB is outside ABC and similar to AEC. O is the midpoint of BC. Let the lines OD and EC meet at D', and the lines OE and BD meet at E'. Find area DED'E' in terms of the sides...
Posted in: Mexican Mathematical Olympiad6th Mexican Mathematical Olympiad Problems 1992
6th Mexican Mathematical Olympiad Problems 1992A1. The tetrahedron OPQR has the ∠POQ = ∠POR = ∠QOR = 90o. X, Y, Z are the midpoints of PQ, QR and RP. Show that the four faces of the tetrahedron OXYZ have equal area. A2. Given a prime number p, how many 4-tuples (a, b, c, d) of positive integers with 0 < a, b, c, d < p-1 satisfy ad = bc mod p?
5th Mexican Mathematical Olympiad Problems 1991A1. Find the sum of all positive irreducible fractions less than 1 whose denominator is 1991. A2. n is palindromic (so it reads the same backwards as forwards, eg 15651) and n = 2 mod 3, n = 3 mod 4, n = 0 mod 5. Find the smallest such positive integer. Show that there are infinitely many such positive intege...
Posted in: Mexican Mathematical OlympiadMinggu, 26 September 2010
Chapter Book Chips - "Emily's Fortune" by Phyllis Reynolds Naylor
Emily's Fortune by Phyllis Reynolds NaylorTumblin' Tarnation! What is poor Emily Wiggins going to do? She's been orphaned and only has her pet turtle, Rufus, to ease her sorrows. She wants to go and live with her Aunt Hilda, but some mean grown-ups have other ideas. Miss Catchum of Catchum Child-Catching Services and Emily's despicable Uncle Victor are both after her and the money she inherited. But Emily finds some confidence and inspiration from three ladies with funny names. And she meets a boy named Jackson, who helps her make her way to the town of Redbud and her Aunt Hilda. Along the way, Emily and Jackson take the ride of their lives on...
Posted in: Chapter Book ChipsJumat, 24 September 2010
4th Mexican Mathematical Olympiad Problems 1990
4th Mexican Mathematical Olympiad Problems 1990A1. How many paths are there from A to the line BC if the path does not go through any vertex twice and always moves to the left? A2. ABC is a triangle with ∠B = 90o and ...
Posted in: Mexican Mathematical Olympiad3rd 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...
Posted in: Mexican Mathematical OlympiadKamis, 23 September 2010
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...
Posted in: Long DivisionDivision 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).
Fall is the time of the harvest moon. Learn more about earth's nearest neighbor in space, the magical, mysterious moon.Destination the Moon by Giles SparrowAll the facts about the moon for the very youngest reader.Check our catalog for this title. Moon by Jacqueline MittonStunning photographs illustrate this title.Check our catalog for this title. Moon: Science, History, and Mystery by Stewart RossScientific facts and the moon's influence on our culture.Check our catalog for this title. The Lunar Cycle: Phases of the Moon by Genevieve O'MaraWhy the moon looks different every night.Check our catalog for this title. Thirteen Moons on Turtle's...
Posted in: Non-Fiction2nd Mexican Mathematical Olympiad Problems 1988
2nd Mexican Mathematical Olympiad Problems 1988A1. In how many ways can we arrange 7 white balls and 5 black balls in a line so that there is at least one white ball between any two black balls? A2. If m and...
Posted in: Mexican Mathematical Olympiad1st 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...
Posted in: Mexican Mathematical OlympiadHow 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.
4th Brazilian Mathematical Olympiad Problems 19821. The angles of the triangle ABC satisfy ∠A/∠C = ∠B/∠A = 2. The incenter is O. K, L are the excenters of the excircles opposite B and A respectively. Show that triangles ABC and OKL are similar.
3rd Brazilian Mathematical Olympiad Problems 19811. For which k does the system x2 - y2 = 0, (x-k)2 + y2 = 1 have exactly (1) two, (2) three real solutions? 2. Show that there are at least 3 and at most 4 powers of 2 with m digits. For which m are there 4?
2nd Brazilian Mathematical Olympiad Problems 19801. Box A contains black balls and box B contains white balls. Take a certain number of balls from A and place them in B. Then take the same number of balls from B and place them in A. Is the number of white balls in A then greater, equal to, or less than the number of black balls in B?
1st Brazilian Mathematical Olympiad Problems 19791. Show that if a < b are in the interval [0, π/2] then a - sin a < b - sin b. Is this true for a < b in the interval [π, 3π/2]? 2. The remainder on dividing the...
Posted in: Brazilian Mathematical OlympiadJumat, 17 September 2010
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.
53rd Polish Mathematical Olympiad Problems 2002A1. Find all triples of positive integers (a, b, c) such that a2 + 1 and b2 + 1 are prime and (a2 + 1)(b2 + 1) = c2 + 1. A2. ABC is an acute-angled triangle. BCKL, ACPQ are rectangles on the outside of two of the sides and have equal area. Show that the midpoint of PK lies on the line through C and the circumcent...
Posted in: Polish Mathematical Olympiad52nd Polish Mathematical Olympiad Problems 2001
52nd Polish Mathematical Olympiad Problems 2001A1. Show that x1 + 2x2 + 3x3 + ... + nxn ≤ ½n(n-1) + x1 + x22 + x33 + ... + xnn for all non-negative reals xi. A2. P is a point inside a regular tetrahedron w...
Posted in: Polish Mathematical Olympiad51st 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...
Posted in: Polish Mathematical Olympiad50th 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...
Posted in: Polish Mathematical Olympiad49th 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...
Posted in: Polish Mathematical Olympiad48th 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.
Celebrate the release of the movie Ramona and Beezus with a party. We'll have games, trivia, prizes, crafts and more. For children in Grades 1-4 and accompanying adult.Registration is required. You can register at the Children's Desk or by callling (503) 350-3600.Monday, September 272:00-3:30 PMMeeting Rooms A & BIF YOU WILL BE LATE, THAT IS FINE. WE MAY EXTEND TIME SO THAT PEOPLE HAVE A CHANCE TO COME AFTERSCHOOL AND HAVE FUN!!!...
47th Polish Mathematical Olympiad Problems 1996
47th Polish Mathematical Olympiad Problems 1996A1. Find all pairs (n,r) with n a positive integer and r a real such that 2x2+2x+1 divides (x+1)n - r. A2. P is a point inside the triangle ABC such that ∠PBC = ∠PCA <...
Posted in: Polish Mathematical Olympiad46th 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?
45th Polish Mathematical Olympiad Problems 1994A1. Find all triples (x,y,z) of positive rationals such that x + y + z, 1/x + 1/y + 1/z and xyz are all integers. A2. L, L' are parallel lines. C is a circle that does ...
Posted in: Polish Mathematical Olympiad44th 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 A...
Posted in: Polish Mathematical Olympiad43rd 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 :...
Posted in: Polish Mathematical Olympiad42nd 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...
Posted in: Polish Mathematical Olympiad41st 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.
40th Polish Mathematical Olympiad Problems 1989A1. An even number of politicians are sitting at a round table. After a break, they come back and sit down again in arbitrary places. Show that there must be two people with the same number of people sitting between them as before the break.
39th Polish Mathematical Olympiad Problems 1988A1. The real numbers x1, x2, ... , xn belong to the interval (0,1) and satisfy x1 + x2 + ... + xn = m + r, where m is an integer and r ∈ [0,1). Show that x12 + x22 + ... + xn2 ≤ m + r2.
38th Polish Mathematical Olympiad Problems 1987A1. There are n ≥ 2 points in a square side 1. Show that one can label the points P1, P2, ... , Pn such that ∑i=1n |Pi-1 - Pi|2 ≤ 4, where we use cyclic subscripts, so that P0 means Pn.
37th Polish Mathematical Olympiad Problems 1986A1. A square side 1 is covered with m2 rectangles. Show that there is a rectangle with perimeter at least 4/m. A2. Find the maximum possible volume of a tetrahedron which has three faces with area 1.
36th Polish Mathematical Olympiad Problems 1985A1. Find the largest k such that for every positive integer n we can find at least k numbers in the set {n+1, n+2, ... , n+16} which are coprime with n(n+17).A2. Given a square side 1 and 2n positive reals a1, b1, ... , an, bn each ≤ 1 and satisfying ∑ aibi ≥ 100. Show that the square can be covered with...
Posted in: Polish Mathematical Olympiad35th Polish Mathematical Olympiad Problems 1984
35th Polish Mathematical Olympiad Problems 1984A1. X is a set with n > 2 elements. Is there a function f : X → X such that the composition f n-1 is constant, but f n-2 is not constant? A2. Given n we define ai,j as follows. For i, j = 1, 2, ... , n, ai,j = 1 for j = i, and 0 for j ≠ i. For i = 1, 2, ... , n, j = n+1, ... , 2n, ai,j = -1/n. Show that for ...
Posted in: Polish Mathematical Olympiad34th Polish Mathematical Olympiad Problems 1983
34th Polish Mathematical Olympiad Problems 1983A1. The angle bisectors of the angles A, B, C in the triangle ABC meet the circumcircle again at K, L, M. Show that |AK| + |BL| + |CM| > |AB| + |BC| + |CA|. A2. For...
Posted in: Polish Mathematical Olympiad16th 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...
Posted in: Balkan Mathematical Olympiad15th 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.
14th Balkan Mathematical Olympiad Problems 1997A1. ABCD is a convex quadrilateral. X is a point inside it. XA2 + XB2 + XC2 + XD2 is twice the area of the quadrilateral. Show that it is a square and that X is its center.
13th Balkan Mathematical Olympiad Problems 1996A1. Let d be the distance between the circumcenter and the centroid of a triangle. Let R be its circumradius and r the radius of its inscribed circle. Show that d2 ≤ R(R - 2r).
12th Balkan Mathematical Olympiad Problems 1995A1. Define an by a3 = (2 + 3)/(1 + 6), an = (an-1 + n)/(1 + n an-1). Find a1995. A2. Two circles centers O and O' meet at A and B, so that OA is perpendicular to O'A. OO' meets the circles at C, E, D, F, so that the points C, O, E, D, O', F lie on the line in that order. BE meets the circle again at K and...
Posted in: Balkan Mathematical Olympiad11th Balkan Mathematical Olympiad Problems 1994
11th Balkan Mathematical Olympiad Problems 1994A1. Given a point P inside an acute angle XAY, show how to construct a line through P meeting the line AX at B and the line AY at C such that the area of the triangle ABC is AP2.
10th Balkan Mathematical Olympiad Problems 1993A1. Given reals a1 ≤ a2 ≤ a3 ≤ a4 ≤ a5 ≤ a6 satisfying a1 + a2 + a3 + a4 + a5 + a6 = 10 and (a1 - 1)2 + (a2 - 1)2 + (a3 - 1)2 + (a4 - 1)2 + (a5 - 1)2 + (a6 - 1)2 = 6, what is the largest possible a6?
9th Balkan Mathematical Olympiad Problems 1992A1. Let a(n) = 34n. For which n is (ma(n)+6 - ma(n)+4 - m5 + m3) always divisible by 1992? A2. Prove that (2n2 + 3n + 1)n ≥ 6nn! n! for all positive integers.
8th Balkan Mathematical Olympiad Problems 1991A1. The circumcircle of the acute-angled triangle ABC has center O. M lies on the minor arc AB. The line through M perpendicular to OA cuts AB at K and AC at L. The line through M perpendicular to OB cuts AB at N and BC at P. MN = KL. Find angle MLP in terms of angles A, B and C.
7th Balkan Mathematical Olympiad Problems 1990A1. The sequence un is defined by u1 = 1, u2 = 3, un = (n+1) un-1 - n un-2. Which members of the sequence which are divisible by 11? A2. Expand (x + 2x2 + 3x3 + ... + nxn)2 and add the coefficients of xn+1 through x2n. Show that the result is n(n+1)(5n2 + 5n + 2)/24.
6th Balkan Mathematical Olympiad Problems 1989A1. Find all integers which are the sum of the squares of their four smallest positive divisors. A2. A prime p has decimal digits pnpn-1...p0 with pn > 1. Show that the polynomial pnxn + pn-1xn-1 + ... + p1x + p0 has no factors which are polynomials with integer coefficients and degree strictly betw...
Posted in: Balkan Mathematical Olympiad5th Balkan Mathematical Olympiad Problems 1988
5th Balkan Mathematical Olympiad Problems 1988A1. ABC is a triangle area 1. AH is an altitude, M is the midpoint of BC and K is the point where the angle bisector at A meets the segment BC. The area of the triangle AHM is 1/4 and the area of AKM is 1 - (√3)/2. Find the angles of the triangle.
4th Balkan Mathematical Olympiad Problems 1987A1. f is a real valued function on the reals satisfying (1) f(0) = 1/2, (2) for some real a we have f(x+y) = f(x) f(a-y) + f(y) f(a-x) for all x, y. Prove that f is constant.
3rd Balkan Mathematical Olympiad Problems 1986A1. A line through the incenter of a triangle meets the circumcircle and incircle in the points A, B, C, D (in that order). Show that AB·CD ≥ BC2/4. When do you have equality?
2nd Balkan Mathematical Olympiad Problems 1985A1. ABC is a triangle. O is the circumcenter, D is the midpoint of AB, and E is the centroid of ACD. Prove that OE is perpendicular to CD iff AB = AC. A2. The reals w,...
Posted in: Balkan Mathematical Olympiad1st 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....
Posted in: Balkan Mathematical OlympiadRabu, 15 September 2010
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. -...
Posted in: Special Programs35th 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.
34th Canadian Mathematical Olympiad Problems 20021. What is the largest possible number of elements in a subset of {1, 2, 3, ... , 9} such that the sum of every pair (of distinct elements) in the subset is different?
33rd Canadian Mathematical Olympiad Problems 20011. A quadratic with integral coefficients has two distinct positive integers as roots, the sum of its coefficients is prime and it takes the value -55 for some integer. Show that one root is 2 and find the other root.
32nd Canadian Mathematical Olympiad Problems 20001. Three runners start together and run around a track length 3L at different constant speeds, not necessarily in the same direction (so, for example, they may all run clockwise, or one may run clockwise). Show that there is a moment when any given runner is a distance L or more from both the other runners (where...
Posted in: CMO31st Canadian Mathematical Olympiad Problems 1999
31st Canadian Mathematical Olympiad Problems 19991. Find all real solutions to the equation 4x2 - 40[x] + 51 = 0. 2. ABC is equilateral. A circle with center on the line through A parallel to BC touches the segment BC. Show that the length of arc of the circle inside ABC is independent of the position of the circle.
30th Canadian Mathematical Olympiad Problems 19981. How many real x satisfy x = [x/2] + [x/3] + [x/5]? 2. Find all real x equal to √(x - 1/x) + √(1 - 1/x). 3. Show that if n > 1 is an integer then (1 + 1/3 + 1/5...
Posted in: CMO29th 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.
28th Canadian Mathematical Olympiad Problems 19961. The roots of x3 - x - 1 = 0 are r, s, t. Find (1 + r)/(1 - r) + (1 + s)/(1 - s) + (1 + t)/(1 - t). 2. Find all real solutions to the equations x = 4z2/(1 + 4z2), y = 4x2/(1 + 4x2), z = 4y2/(1 + 4y2).
27th Canadian Mathematical Olympiad Problems 19951. Find g(1/1996) + g(2/1996) + g(3/1996) + ... + g(1995/1996) where g(x) = 9x/(3 + 9x). 2. Show that xxyyzz >= (xyz)(x+y+z)/3 for positive reals x, y, z. 3. ...
Posted in: CMO26th 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. ...
Posted in: CMO25th 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)...
Posted in: CMOSelasa, 14 September 2010
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?
23rd Canadian Mathematical Olympiad Problems 19911. Show that there are infinitely many solutions in positive integers to a2 + b5 = c3. 2. Find the sum of all positive integers which have n 1s and n 0s when written in base 2.
22nd Canadian Mathematical Olympiad Problems 19901. A competition is played amongst n > 1 players over d days. Each day one player gets a score of 1, another a score of 2, and so on up to n. At the end of the competition each player has a total score of 26. Find all possible values for (n, d). 2. n(n + 1)/2 distinct numbers are arranged at random into n rows....
Posted in: CMO21st Canadian Mathematical Olympiad Problems 1989
21St Canadian Mathematical Olympiad Problems 19891. How many permutations of 1, 2, 3, ... , n have each number larger than all the preceding numbers or smaller than all the preceding numbers? 2. Each vertex of a right angle triangle of area 1 is reflected in the opposite side. What is the area of the triangle formed by the three reflected points?
20th Canadian Mathematical Olympiad Problems 19881. For what real values of k do 1988x2 + kx + 8891 and 8891x2 + kx + 1988 have a common zero? 2. Given a triangle area A and perimeter p, let S be the set of all points a distance 5 or less from a point of the triangle. Find the area of S.
19th Canadian Mathematical Olympiad Problems 19871. Find all positive integer solutions to n! = a2 + b2 for n < 14. 2. Find all the ways in which the number 1987 can be written in another base as a three digit number with the digits having the same sum 25.
18th Canadian Mathematical Olympiad Problems 19861. The triangle ABC has angle B = 90o. The point D is taken on the ray AC, the other side of C from A, such that CD = AB. ∠CBD = 30o. Find AC/CD. 2. Three...
Posted in: CMO17th 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.]
16th Canadian Mathematical Olympiad Problems 19841. Show that the sum of 1984 consecutive positive integers cannot be a square. 2. You have keyring with n identical keys. You wish to color code the keys so that you can distinguish them. What is the smallest number of colors you need? [For example, you could use three colors for eight keys: R R R R G B R R....
Posted in: CMO15th Canadian Mathematical Olympiad Problems 1983
15th Canadian Mathematical Olympiad Problems 19831. Find all solutions to n! = a! + b! + c! . 2. Find all real-valued functions f on the reals whose graphs remain unchanged under all transformations (x, y) → (2kx, 2k(kx + y) ), where k is real.
14th Canadian Mathematical Olympiad Problems 19821. Given a quadrilateral ABCD and a point P, take A' so that PA' is parallel to AB and of equal length. Similarly take PB', PC', PD' equal and parallel to BC, CD, DA respectively. Show that the area of A'B'C'D' is twice that of ABCD.
13th Canadian Mathematical Olympiad Problems 19811. Show that there are no real solutions to [x] + [2x] + [4x] + [8x] + [16x] + [32x] = 12345. 2. The circle C has radius 1 and touches the line L at P. The point X lies on C and Y is the foot of the perpendicular from X to L. Find the maximum possible value of area PXY (as X varies).
12th Canadian Mathematical Olympiad Problems 19801. If the 5-digit decimal number a679b is a multiple of 72 find a and b. 2. The numbers 1 to 50 are arranged in an arbitrary manner into 5 rows of 10 numbers each. Then each row is rearranged so that it is in increasing order. Then each column is arranged so that it is in increasing order. Are the rows...
Posted in: CMOSenin, 13 September 2010
11th Canadian Mathematical Olympiad Problems 1979
11th Canadian Mathematical Olympiad Problems 19791. If a > b > c > d is an arithmetic progression of positive reals and a > h > k > d is a geometric progression of positive reals, show that bc ≥ hk. 2. Show that ...
Posted in: CMO10th 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 =...
Posted in: CMOTruck 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!-...
Minggu, 12 September 2010
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?
8th Canadian Mathematical Olympiad Problems 19761. Given four unequal weights in geometric progression, show how to find the heaviest weight using a balance twice. 2. The real sequence x0, x1, x2, ... is defined by x0 = 1, x1 = 2, n(n+1) xn+1 = n(n-1) xn - (n-2) xn-1. Find x0/x1 + x1x2 + ... + x50/x51.
7th Canadian Mathematical Olympiad Problems 19751. Evaluate (1·2·4 + 2·4·8 + 3·6·12 + 4·8·16 + ... + n·2n·4n)1/3/(1·3·9 + 2·6·18 + 3·9·27 + 4·12·36 + ... + n·3n·9n)1/3. 2. Define the real sequence a1, a2, a3, ... by a1 = 1/2, n2an = a1 + a2 + ... + an. Evaluate an.
6th Canadian Mathematical Olympiad Problems 19741. (1) given x = (1 + 1/n)n, y = (1 + 1/n)n+1, show that xy = yx. (2) Show that 12 - 22 + 32 - 42 + ... + (-1)n+1n2 = (-1)n+1(1 + 2 + ... + n). 2. Given the...
Posted in: CMO5th 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.
4th Canadian Mathematical Olympiad Problems 19721. Three unit circles are arranged so that each touches the other two. Find the radii of the two circles which touch all three. 2. x1, x2, ... , xn are non-negative reals. Let s = ∑ij xixj.="" show="" that="" at="" least="" one="" of="" the="" xi="" has="" square="" not="" exceeding="" 2s="" (n2="" -="" n).="" !--google_ad_client="pub-3460669521768433" ;
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3rd Canadian Mathematical Olympiad Problems 1971
3rd Canadian Mathematical Olympiad Problems 1971 1. A diameter and a chord of a circle intersect at a point inside the circle. The two parts of the chord are length 3 and 5 and one part of the diameter is length 1. What is the radius of the circle?
2nd Canadian Mathematical Olympiad Problems 19701. Find all triples of real numbers such that the product of any two of the numbers plus the third is 2. 2. The triangle ABC has angle A > 90o. The altitude from A is AD and the altitude from B is BE. Show that BC + AD ≥ AC + BE. When do we have equality?
1st Canadian Mathematical Olympiad Problems 1969 1. a, b, c, d, e, f are reals such that a/b = c/d = e/f; p, q, r are reals, not all zero; and n is a positive integer. Show that (a/b)n = (p an + q cn + r en)/(p bn + q dn + r fn ). 2. If x is a real number not less than 1, which is larger: √(x+1) - √x or √x - √(x-1)?
Fall is all about leaves. Get to know all about them in the following books. Why Do Leaves Change Color? by Betsy MaestroEveryone wants to know the answer to this question!Check our catalog for this title.Look What I Did With A Leaf! by Morteza E. SohiAll sorts of ideas for things you can make with a leaf.Check our catalog for this title.Leaves in Fall by Martha E. H. Rustad Leaves put on their biggest show in autumn.Check our catalog for this title. Leaves by John Farndon The different kinds of leaves, how they grow and function.Check our catalog for this title.Leaf Jumpers by Carole GerberRhymes about leaves and the trees they fall from.Check...
Posted in: Non-FictionPicture Books For The Littlest Ones
A selection of new, and not quite new, picture books for toddlers and preschoolers. Each of these titles features beautiful, bold colors and are perfect for capturing the attention and interest of your little ones. Thank you to Seven Impossible Things Before Breakfast, the blog that inspired this post.All Things Bright And BeautifulBy: Ashley BryanAn illustrated version of the famous hymn beautifully realized through Bryan's paper-cut illustrations.Find this title in our catalogSleepy Oh So SleepyBy: Denise FlemingAs mama puts her baby to bed we see other sleepy animal babies on their way to bed.Find this title in our catalogAll of Baby Nose...
40th British Mathematical Olympiad 2004 Problems
40th British Mathematical Olympiad 2004 Problems1. ABC is an equilateral triangle. D is a point on the side BC (not at the endpoints). A circle touches BC at D and meets the side AB at M and N, and the side AC at P and Q. Show that BD + AM + AN = CD + AP + AQ.
1. Find all integers 0 < a < b < c such that b - a = c - b and none of a, b, c have a prime factor greater than 3. 2. D is a point on the side AB of the triangle ABC such that AB = 4·AD. P is a point on the...
Posted in: British Mathematical Olympiad38th 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.
37th British Mathematical Olympiad 2001 Problems1. A has a marbles and B has b < a marbles. Starting with A each gives the other enough marbles to double the number he has. After 2n such transfers A has b marbles. Find a/b in terms of n. 2. Find all integer solutions to m2n + 1 = m2 + 2mn + 2m + n.
36th British Mathematical Olympiad 2000 Problems1. Two circles meet at A and B and touch a common tangent at C and D. Show that triangles ABC and ABD have the same area. 2. Find the smallest value of x2 + 4xy + 4y2 + 2z2 for positive reals x, y, z with product 32.
35th British Mathematical Olympiad 1999 Problems1. Let Xn = {1, 2, 3, ... , n}. For which n can we partition Xn into two parts with the same sum? For which n can we partition Xn into three parts with the same sum? ...
Posted in: British Mathematical Olympiad34th 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...
Posted in: British Mathematical Olympiad33rd 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....
Posted in: British Mathematical Olympiad32nd 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. ...
Posted in: British Mathematical Olympiad31st 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.
30th British Mathematical Olympiad 1994 Problems1. Find the smallest integer n > 1 such that (12 + 22 + 32 + ... + n2)/n is a square. 2. How many incongruent triangles have integer sides and perimeter 1994?
29th British Mathematical Olympiad 1993 Problems1. The angles in the diagram below are measured in some unknown unit, so that a, b, ... , k, l are all distinct positive integers. Find the smallest possible value of a + b + c and give the corresponding values of a, b, ... , k, l. 2. p > 3 is prime. m = (4p - 1)/3. Show that 2m-1 = 1 mod m. 3. P...
Posted in: British Mathematical Olympiad28th British Mathematical Olympiad 1992 Problems
28th British Mathematical Olympiad 1992 Problems1. p is an odd prime. Show that there are unique positive integers m, n such that m2 = n(n + p). Find m and n in terms of p. 2. Show that 12/(w + x + y + z) ≤ 1/(w +...
Posted in: British Mathematical OlympiadRabu, 08 September 2010
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...
Posted in: Math Books for Kids27th 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...
Posted in: British Mathematical Olympiad26th 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...
Posted in: British Mathematical Olympiad25th 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...
Posted in: British Mathematical Olympiad24th 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.
22th British Mathematical Olympiad 1986 Problems - Further International Selection Test 1. A rational point is a point both of whose coordinates are rationals. Let A, B, C, D be rational points such that AB and CD are not both equal and parallel. Show that there is just one point P such that the triangle PCD can be obtained from the triangle PAB by enlargement and...
Posted in: British Mathematical OlympiadSelasa, 07 September 2010
21th British Mathematical Olympiad 1985 Problems
21th British Mathematical Olympiad 1985 Problems1. Prove that ∑1n ∑1n | xi - xj | ≤ n2 for all real xi such that 0 ≤ xi ≤ 2. When does equality hold? 2. (1) The incircle of the triangle ABC touches BC at L. LM is a diameter of the incircle. The ray AM meets BC at N. Show that | NL | = | AB - AC |.
1. In the triangle ABC, ∠C = 90o. Find all points D such that AD·BC = AC·BD = AB·CD/√2. 2. ABCD is a tetrahedron such that DA = DB = DC = d and AB = BC = CA = e. M and N are the midpoints of AB and CD. A variable plane through MN meets AD at P and BC at Q. Show that AP/AD = BQ/BC. Find the value of this ratio in terms of d and e which minimises the area of M...
Posted in: British Mathematical Olympiad19th British Mathematical Olympiad 1983 Problems
19th British Mathematical Olympiad 1983 Problems - Further International Selection Test1. Given points A and B and a line l, find the point P which minimises PA2 + PB2 + PN2, where N is the foot of the perpendicular from P to l. State without proof a generalisation to three points.
18th BMO 1982 - Further International Selection Test1. ABC is a triangle. The angle bisectors at A, B, C meet the circumcircle again at P, Q , R respectively. Show that AP + BQ + CR > AB + BC + CA. 2. The sequence p1,...
Posted in: British Mathematical Olympiad17th 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.
1. Show that there are no solutions to an + bn = cn, with n > 1 is an integer, and a, b, c are positive integers with a and b not exceeding n. 2. Find a set of seven consecutive positive integers and a polynomial...
Posted in: British Mathematical OlympiadKamis, 02 September 2010
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....
Posted in: British Mathematical Olympiad14th 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).
13th British Mathematical Olympiad 1977 Problems1. f(n) is a function on the positive integers with non-negative integer values such that: (1) f(mn) = f(m) + f(n) for all m, n; (2) f(n) = 0 if the last digit of n is 3; (3) f(10) = 0. Show that f(n) = 0 for all n.
12th British Mathematical Olympiad 1976 Problems1. ABC is a triangle area k. Let d be the length of the shortest line segment which bisects the area of the triangle. Find d. Give an example of a curve which bisects the area and has length < d.
11th British Mathematical Olympiad 1975 Problems1. Find all positive integer solutions to [11/3] + [21/3] + ... + [(n3 - 1)1/3] = 400. 2. The first k primes are divided into two groups. n is the product of the first group and n is the product of the second group. M is any positive integer divisible only by primes in the first group and N is any positive...
Posted in: British Mathematical Olympiad10th British Mathematical Olympiad 1974 Problems
10th British Mathematical Olympiad 1974 Problems1. C is the curve y = 4x2/3 for x ≥ 0 and C' is the curve y = 3x2/8 for x ≥ 0. Find curve C" which lies between them such that for each point P on C" the area bounded by C, C" and a horizontal line through P equals the area bounded by C", C and a vertical line through P.
9th British Mathematical Olympiad 1973 Problems1. A variable circle touches two fixed circles at P and Q. Show that the line PQ passes through one of two fixed points. State a generalisation to ellipses or conics. 2. ...
Posted in: British Mathematical Olympiad8th 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?
The Pirate Cruncherby Jonny Duddle (New Childrens Books)(2009)for ages 3 and upAaarg! Who is ready for another pirate story? Some greedy pirates led by Captain Purplebeard are lured by the promise of treasure, of course. A mysterious fiddler tells the tale in rhyme, but only after boarding ship does he caution them about a certain pirate crunching monster. The pirates are naturally undeterred, "And if there's a beast, it better beware. I can smell that gold! We're almost there!" They finally find the tiny island and a giant fold out page contains the surprise at the end. Throughout visual clues tip the reader off about what may lay ahead and...
Posted in: New Book ReviewNew Fiction Book Review: The Dancing Pancake
By Eileen SpinelliBest for grades 3-6Call number: J Spinelli, E.Copyright 2010Bindi's whole life is changing. First her father, after months of frusterating unemployment moves out, then her aunt has a great idea to open a pancake restaurant with her mother as a partner. What follows is a novel in verse with a colorful cast of characters, from her little cousin and Inky, his pet rubber spider, to Grace a regular at the restaurant who pushes a cart with all the stuff she owns inside. Bindi must come to terms with her feelings about her parents, having to move, and the new responsibilities and relationships she finds herself in.The Dancing Pancake...
Rabu, 01 September 2010
7th British Mathematical Olympiad 1971 Problems
7th British Mathematical Olympiad 1971 Problems 1. Factorise (a + b)7 - a7 - b7. Show that 2n3 + 2n2 + 2n + 1 is never a multiple of 3. 2. Let a = 99 , b = 9a, c = 9b. Show that the last two digits of b and c are equal. What are they?
6th British Mathematical Olympiad 1970 Problems 1. (1) Find 1/log2a + 1/log3a + ... + 1/logna as a quotient of two logs to base 2. (2) Find the sum of the coefficients of (1 + x - x2)3(1 - 3x + x2)2 and the sum of the coefficients of its derivative.
5th British Mathematical Olympiad 1969 Problems1. Find the condition on the distinct real numbers a, b, c such that (x - a)(x - b)/(x - c) takes all real values. Sketch a graph where the condition is satisfied and another where it is not. 2. Find all real solutions to cos x + cos5x + cos 7x = 3.
4th British Mathematical Olympiad 1968 Problems 1. C is the circle center the origin and radius 2. Another circle radius 1 touches C at (2, 0) and then rolls around C. Find equations for the locus of the point P of the second circle which is initially at (2, 0) and sketch the locus.
3rd British Mathematical Olympiad 1967 Problems1. a, b are the roots of x2 + Ax + 1 = 0, and c, d are the roots of x2 + Bx + 1 = 0. Prove that (a - c)(b - c)(a + d)(b + d) = B2 - A2. 2. Graph x8 + xy + y8 = 0,...
Posted in: British Mathematical Olympiad
