10th Mexican Mathematical Olympiad Problems 1996

10th Mexican Mathematical Olympiad Problems 1996

A1.  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.

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Posted in: Mexican Mathematical Olympiad

9th Mexican Mathematical Olympiad Problems 1995

9th Mexican Mathematical Olympiad Problems 1995

A1.  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 three of the points form an equilateral triangle with side 1.

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Posted in: Mexican Mathematical Olympiad

8th Mexican Mathematical Olympiad Problems 1994

8th Mexican Mathematical Olympiad Problems 1994

A1.  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.

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Posted in: Mexican Mathematical Olympiad

7th Mexican Mathematical Olympiad Problems 1993

7th Mexican Mathematical Olympiad Problems 1993

A1.  ABC is a triangle with ∠A = 90^o. Take E such that the triangle AEC is outside ABC and AE = CE and ∠AEC = 90^o. 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 of ABC.

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Posted in: Mexican Mathematical Olympiad

6th Mexican Mathematical Olympiad Problems 1992

6th Mexican Mathematical Olympiad Problems 1992

A1.  The tetrahedron OPQR has the ∠POQ = ∠POR = ∠QOR = 90^o. 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?

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Posted in: Mexican Mathematical Olympiad

5th Mexican Mathematical Olympiad Problems 1991

5th Mexican Mathematical Olympiad Problems 1991

A1.  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 integers.

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Posted in: Mexican Mathematical Olympiad

4th Mexican Mathematical Olympiad Problems 1990

4th Mexican Mathematical Olympiad Problems 1990

A1.  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?

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Posted in: Mexican Mathematical Olympiad

3rd Mexican Mathematical Olympiad Problems 1989

3rd Mexican Mathematical Olympiad Problems 1989

A1.  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.

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Posted in: Mexican Mathematical Olympiad

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.

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Posted in: Long Division

Division When the Divisor Is a Decimal

Division When the Divisor Is a Decimal - Division of Decimals by Whole Numbers

The 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).

• Place the divisor (17) before the division bracket and place the dividend (0.4131) under it.

          
 17)0.4131

• Proceed with the division as you normally would except put the decimal point in the answer or quotient exactly above where it occurs in the dividend. For example:

   0.0243
17)0.4131

Division of Decimals by Decimals

The procedure for the division of decimals is very similar to the division of whole numbers. Make the divisor into a whole number by multiplying both it and the dividend by the same number (such as 10, 100, 1000 etc.). An easy way to do this is to move the decimal point to the right end of the divisor and move the decimal point of the dividend the same number of places.
How to divide a four digit decimal number by a two digit decimal number (e.g 0.4131 ÷ 0.17).

• Place the divisor before the division bracket and place the dividend (0.4131) under it.

           
0.17)0.4131

• Multiply both the divisor and dividend by 100 so that the divisor is not a decimal but a whole number. In other words move the decimal point two places to the right in both the divisor and dividend

        
17)41.31

• Proceed with the division as you normally would except put the decimal point in the answer or quotient exactly above where it occurs in the dividend. For example:

    2.43
17)41.31

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2nd Mexican Mathematical Olympiad Problems 1988

2nd Mexican Mathematical Olympiad Problems 1988

A1.  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?

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Posted in: Mexican Mathematical Olympiad

1st Mexican Mathematical Olympiad Problems 1987

1st Mexican Mathematical Olympiad Problems 1987

A1.  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! ?

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Posted in: Mexican Mathematical Olympiad

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.

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4th Brazilian Mathematical Olympiad Problems 1982

4th Brazilian Mathematical Olympiad Problems 1982

1.  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.

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Posted in: Brazilian Mathematical Olympiad

3rd Brazilian Mathematical Olympiad Problems 1981

3rd Brazilian Mathematical Olympiad Problems 1981

1.  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?

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Posted in: Brazilian Mathematical Olympiad

2nd Brazilian Mathematical Olympiad Problems 1980

2nd Brazilian Mathematical Olympiad Problems 1980

1.  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?

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Posted in: Brazilian Mathematical Olympiad

1st Brazilian Mathematical Olympiad Problems 1979

1st Brazilian Mathematical Olympiad Problems 1979

1.  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]?

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Posted in: Brazilian Mathematical Olympiad

54th Polish Mathematical Olympiad Problems 2003

54th Polish Mathematical Olympiad Problems 2003

A1.  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.

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Posted in: Polish Mathematical Olympiad

53rd Polish Mathematical Olympiad Problems 2002

53rd Polish Mathematical Olympiad Problems 2002

A1.  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.

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Posted in: Polish Mathematical Olympiad

52nd Polish Mathematical Olympiad Problems 2001

52nd Polish Mathematical Olympiad Problems 2001

A1.  Show that x₁ + 2x₂ + 3x₃ + ... + nx_n ≤ ½n(n-1) + x₁ + x₂² + x₃³ + ... + x_nⁿ for all non-negative reals x_i.

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Posted in: Polish Mathematical Olympiad

51st Polish Mathematical Olympiad Problems 2000

51st Polish Mathematical Olympiad Problems 2000

A1.  How many solutions in non-negative reals are there to the equations:
x₁ + x_n² = 4x_n
x₂ + x₁² = 4x₁
...
x_n + x_n-1² = 4x_n-1?

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Posted in: Polish Mathematical Olympiad

50th Polish Mathematical Olympiad Problems 1999

50th Polish Mathematical Olympiad Problems 1999

A1.  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.

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Posted in: Polish Mathematical Olympiad

49th Polish Mathematical Olympiad Problems 1998

49th Polish Mathematical Olympiad Problems 1998

A1.  Find all solutions in positive integers to a + b + c = xyz, x + y + z = abc.

A2.  Fn is the Fibonacci sequence F₀ = F₁ = 1, F_n+2 = F_n+1 + F_n. Find all pairs m > k ≥ 0 such that the sequence x₀, x₁, x₂, ... defined by x₀ = F_k/F_m and x_n+1 = (2x_n - 1)/(1 - x_n) for x_n ≠ 1, or 1 if x_n = 1, contains the number 1.

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Posted in: Polish Mathematical Olympiad

48th Polish Mathematical Olympiad Problems 1997

48th Polish Mathematical Olympiad Problems 1997

A1.  The positive integers x₁, x₂, ... , x₇ satisfy x₆ = 144, x_n+3 = x_n+2(x_n+1+x_n) 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.

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Posted in: Polish Mathematical Olympiad

47th Polish Mathematical Olympiad Problems 1996

47th Polish Mathematical Olympiad Problems 1996

A1.  Find all pairs (n,r) with n a positive integer and r a real such that 2x²+2x+1 divides (x+1)ⁿ - r.

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Posted in: Polish Mathematical Olympiad

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?

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Posted in: Polish Mathematical Olympiad

45th Polish Mathematical Olympiad Problems 1994

45th Polish Mathematical Olympiad Problems 1994

A1.  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.

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Posted in: Polish Mathematical Olympiad

44th Polish Mathematical Olympiad Problems 1993

44th Polish Mathematical Olympiad Problems 1993

A1.  Find all rational solutions to:
t² - w² + z² = 2xy
t² - y² + w² = 2xz
t² - w² + x² = 2yz.

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Posted in: Polish Mathematical Olympiad

43rd Polish Mathematical Olympiad Problems 1992

43rd Polish Mathematical Olympiad Problems 1992

A1.  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.

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Posted in: Polish Mathematical Olympiad

42nd Polish Mathematical Olympiad Problems 1991

42nd Polish Mathematical Olympiad Problems 1991

A1.  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?

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Posted in: Polish Mathematical Olympiad

41st Polish Mathematical Olympiad Problems 1990

41st Polish Mathematical Olympiad Problems 1990

A1.  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) + ... + x_n²/(x_n²+x¹x₂) ≤ n-1.

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Posted in: Polish Mathematical Olympiad

40th Polish Mathematical Olympiad Problems 1989

40th Polish Mathematical Olympiad Problems 1989

A1.  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.

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Posted in: Polish Mathematical Olympiad

38th Polish Mathematical Olympiad Problems 1987

38th Polish Mathematical Olympiad Problems 1987

A1.  There are n ≥ 2 points in a square side 1. Show that one can label the points P₁, P₂, ... , P_n such that ∑_i=1ⁿ |P_i-1 - P_i|² ≤ 4, where we use cyclic subscripts, so that P₀ means P_n.

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39th Polish Mathematical Olympiad Problems 1988

39th Polish Mathematical Olympiad Problems 1988

A1.  The real numbers x₁, x₂, ... , x_n belong to the interval (0,1) and satisfy x₁ + x₂ + ... + x_n = m + r, where m is an integer and r ∈ [0,1). Show that x₁² + x₂² + ... + x_n² ≤ m + r².

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Posted in: Polish Mathematical Olympiad

37th Polish Mathematical Olympiad Problems 1986

37th Polish Mathematical Olympiad Problems 1986

A1.  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.

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Posted in: Polish Mathematical Olympiad

36th Polish Mathematical Olympiad Problems 1985

36th Polish Mathematical Olympiad Problems 1985

A1.  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).

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Posted in: Polish Mathematical Olympiad

35th Polish Mathematical Olympiad Problems 1984

35th Polish Mathematical Olympiad Problems 1984

A1.  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?

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Posted in: Polish Mathematical Olympiad

34th Polish Mathematical Olympiad Problems 1983

34th Polish Mathematical Olympiad Problems 1983

A1.  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|.

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Posted in: Polish Mathematical Olympiad

16th Balkan Mathematical Olympiad Problems 1999

16th Balkan Mathematical Olympiad Problems 1999

A1.  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 AY.

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Posted in: Balkan Mathematical Olympiad

15th Balkan Mathematical Olympiad Problems 1998

15th Balkan Mathematical Olympiad Problems 1998

A1.  How many different integers can be written as [n²/1998] for n = 1, 2, ... , 1997?
A2.  x_i are distinct positive reals satisfying x₁ < x₂ < ... < x_2n+1. Show that x₁ - x₂ + x₃ - x₄ + ... - x_2n + x_2n+1 < (x₁ⁿ - x₂ⁿ + ... - x_2nⁿ + x_2n+1ⁿ)^1/n.

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Posted in: Balkan Mathematical Olympiad

14th Balkan Mathematical Olympiad Problems 1997

14th Balkan Mathematical Olympiad Problems 1997

A1.  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.

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Posted in: Balkan Mathematical Olympiad

13th Balkan Mathematical Olympiad Problems 1996

13th Balkan Mathematical Olympiad Problems 1996

A1.  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).

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Posted in: Balkan Mathematical Olympiad

12th Balkan Mathematical Olympiad Problems 1995

12th Balkan Mathematical Olympiad Problems 1995

A1.  Define a_n by a₃ = (2 + 3)/(1 + 6), a_n = (a_n-1 + n)/(1 + n a_n-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 meets CA at M. BD meets the circle again at L and AF at N. Show that (KE/KM) (LN/LD) = (O'E/OD).

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Posted in: Balkan Mathematical Olympiad

11th Balkan Mathematical Olympiad Problems 1994

11th Balkan Mathematical Olympiad Problems 1994

A1.  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.

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Posted in: Balkan Mathematical Olympiad

10th Balkan Mathematical Olympiad Problems 1993

10th Balkan Mathematical Olympiad Problems 1993

A1.  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?

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Posted in: Balkan Mathematical Olympiad

9th Balkan Mathematical Olympiad Problems 1992

9th Balkan Mathematical Olympiad Problems 1992

A1.  Let a(n) = 3⁴ⁿ. For which n is (m^a(n)+6 - m^a(n)+4 - m⁵ + m³) always divisible by 1992?

A2.  Prove that (2n² + 3n + 1)ⁿ ≥ 6ⁿn! n! for all positive integers.

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Posted in: Balkan Mathematical Olympiad

8th Balkan Mathematical Olympiad Problems 1991

8th Balkan Mathematical Olympiad Problems 1991

A1.  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.

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Posted in: Balkan Mathematical Olympiad

7th Balkan Mathematical Olympiad Problems 1990

7th Balkan Mathematical Olympiad Problems 1990

A1.  The sequence u_n is defined by u₁ = 1, u₂ = 3, u_n = (n+1) u_n-1 - n u_n-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.

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Posted in: Balkan Mathematical Olympiad

6th Balkan Mathematical Olympiad Problems 1989

6th Balkan Mathematical Olympiad Problems 1989

A1.  Find all integers which are the sum of the squares of their four smallest positive divisors.

A2.  A prime p has decimal digits p_np_n-1...p₀ with p_n > 1. Show that the polynomial p_nxⁿ + p_n-1x^n-1 + ... + p₁x + p₀ has no factors which are polynomials with integer coefficients and degree strictly between 0 and n.

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Posted in: Balkan Mathematical Olympiad

5th Balkan Mathematical Olympiad Problems 1988

5th Balkan Mathematical Olympiad Problems 1988

A1.  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.

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Posted in: Balkan Mathematical Olympiad

4th Balkan Mathematical Olympiad Problems 1987

4th Balkan Mathematical Olympiad Problems 1987

A1.  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.

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Posted in: Balkan Mathematical Olympiad

3rd Balkan Mathematical Olympiad Problems 1986

3rd Balkan Mathematical Olympiad Problems 1986

A1.  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?

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Posted in: Balkan Mathematical Olympiad

2nd Balkan Mathematical Olympiad Problems 1985

2nd Balkan Mathematical Olympiad Problems 1985

A1.  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.

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Posted in: Balkan Mathematical Olympiad

1st Balkan Mathematical Olympiad Problems 1984

1st Balkan Mathematical Olympiad Problems 1984

A1.  Let x₁, x₂, ... , x_n be positive reals with sum 1. Prove that x₁/(2 - x₁) + x₂/(2 - x₂) + ... + x_n/(2 - x_n) ≥ n/(2n - 1).

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Posted in: Balkan Mathematical Olympiad

35th Canadian Mathematical Olympiad Problems 2003

35th Canadian Mathematical Olympiad Problems 2003

1.  The angle between the hour and minute hands of a standard 12-hour clock is exactly 1^o. 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 2003^N, where N = 2002²⁰⁰¹.

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Posted in: CMO

34th Canadian Mathematical Olympiad Problems 2002

34th Canadian Mathematical Olympiad Problems 2002

1.  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?

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Posted in: CMO

33rd Canadian Mathematical Olympiad Problems 2001

33rd Canadian Mathematical Olympiad Problems 2001

1.  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.

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Posted in: CMO

32nd Canadian Mathematical Olympiad Problems 2000

32nd Canadian Mathematical Olympiad Problems 2000

1.  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 distance is measured around the track in the shorter direction).

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Posted in: CMO

31st Canadian Mathematical Olympiad Problems 1999

31st Canadian Mathematical Olympiad Problems 1999

1.  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.

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Posted in: CMO

30th Canadian Mathematical Olympiad Problems 1998

30th Canadian Mathematical Olympiad Problems 1998

1.  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).

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Posted in: CMO

29th Canadian Mathematical Olympiad Problems 1997

29th Canadian Mathematical Olympiad Problems 1997

1.  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.

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Posted in: CMO

28th Canadian Mathematical Olympiad Problems 1996

28th Canadian Mathematical Olympiad Problems 1996

1.  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).

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Posted in: CMO

27th Canadian Mathematical Olympiad Problems 1995

27th Canadian Mathematical Olympiad Problems 1995

1.  Find g(1/1996) + g(2/1996) + g(3/1996) + ... + g(1995/1996) where g(x) = 9^x/(3 + 9^x).

2.  Show that x^xy^yz^z >= (xyz)^(x+y+z)/3 for positive reals x, y, z.

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26th Canadian Mathematical Olympiad Problems 1994

26th Canadian Mathematical Olympiad Problems 1994

1.  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.

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Posted in: CMO

25th Canadian Mathematical Olympiad Problems 1993

25th Canadian Mathematical Olympiad Problems 1993

1.  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) terms which form a geometric progression.

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Posted in: CMO

24th Canadian Mathematical Olympiad Problems 1992

24th Canadian Mathematical Olympiad Problems 1992

1.  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?

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Posted in: CMO

23rd Canadian Mathematical Olympiad Problems 1991

23rd Canadian Mathematical Olympiad Problems 1991

1.  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.

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22nd Canadian Mathematical Olympiad Problems 1990

22nd Canadian Mathematical Olympiad Problems 1990

1.  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).

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21st Canadian Mathematical Olympiad Problems 1989

21St Canadian Mathematical Olympiad Problems 1989

1.  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?

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20th Canadian Mathematical Olympiad Problems 1988

20th Canadian Mathematical Olympiad Problems 1988

1.  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.

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19th Canadian Mathematical Olympiad Problems 1987

19th Canadian Mathematical Olympiad Problems 1987

1.  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.

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18th Canadian Mathematical Olympiad Problems 1986

18th Canadian Mathematical Olympiad Problems 1986

1.  The triangle ABC has angle B = 90^o. The point D is taken on the ray AC, the other side of C from A, such that CD = AB. ∠CBD = 30^o. Find AC/CD.

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17th Canadian Mathematical Olympiad Problems 1985

17th Canadian Mathematical Olympiad Problems 1985

1.  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.]

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16th Canadian Mathematical Olympiad Problems 1984

16th Canadian Mathematical Olympiad Problems 1984

1.  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. Starting with the blue key and moving away from the green key uniquely distinguishes each of the red keys.]

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15th Canadian Mathematical Olympiad Problems 1983

15th Canadian Mathematical Olympiad Problems 1983

1.  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) → (2^kx, 2^k(kx + y) ), where k is real.

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14th Canadian Mathematical Olympiad Problems 1982

14th Canadian Mathematical Olympiad Problems 1982

1.  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.

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13th Canadian Mathematical Olympiad Problems 1981

13th Canadian Mathematical Olympiad Problems 1981

1.  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).

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12th Canadian Mathematical Olympiad Problems 1980

12th Canadian Mathematical Olympiad Problems 1980

1.  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 necessarily still in increasing order?

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11th Canadian Mathematical Olympiad Problems 1979

11th Canadian Mathematical Olympiad Problems 1979

1.  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.

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10th Canadian Mathematical Olympiad Problems 1978

10th Canadian Mathematical Olympiad Problems 1978

1.  A square has tens digit 7. What is the units digit?
2.  Find all positive integers m, n such that 2m² = 3n³.

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9th Canadian Mathematical Olympiad Problems 1977

9th Canadian Mathematical Olympiad Problems 1977

1.  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?

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8th Canadian Mathematical Olympiad Problems 1976

8th Canadian Mathematical Olympiad Problems 1976

1.  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) x_n+1 = n(n-1) x_n - (n-2) x_n-1. Find x₀/x₁ + x₁x₂ + ... + x₅₀/x₅₁.

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7th Canadian Mathematical Olympiad Problems 1975

7th Canadian Mathematical Olympiad Problems 1975

1.  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, n²a_n = a₁ + a₂ + ... + a_n. Evaluate a_n.

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6th Canadian Mathematical Olympiad Problems 1974

6th Canadian Mathematical Olympiad Problems 1974

1.  (1) given x = (1 + 1/n)ⁿ, y = (1 + 1/n)ⁿ⁺¹, show that x^y = y^x. (2) Show that 1² - 2² + 3² - 4² + ... + (-1)ⁿ⁺¹n² = (-1)ⁿ⁺¹(1 + 2 + ... + n).

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5th Canadian Mathematical Olympiad Problems 1973

5th Canadian Mathematical Olympiad Problems 1973

1.  (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.

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4th Canadian Mathematical Olympiad Problems 1972

4th Canadian Mathematical Olympiad Problems 1972

1.  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, ... , x_n are non-negative reals. Let s = ∑_i<j x_ix_j. Show that at least one of the x_i has square not exceeding 2s/(n² - n).

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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?

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2nd Canadian Mathematical Olympiad Problems 1970

2nd Canadian Mathematical Olympiad Problems 1970

1.  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 > 90^o. The altitude from A is AD and the altitude from B is BE. Show that BC + AD ≥ AC + BE. When do we have equality?

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