B3. Do there exist polynomials p(x), q(x), r(x) whose coefficients are positive integers such that p(x) = (x2 - 3x + 3) q(x) and q(x) = (x2/20 - x/15 + 1/12) r(x)?
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31st Vietnamese Mathematical Olympiad 1993 Problems
A1. f : [√1995, ∞) → R is defined by f(x) = x(1993 + √(1995 - x2) ). Find its maximum and minimum values.
A2. ABCD is a quadrilateral such that AB is not parallel to CD, and BC is not parallel to AD. Variable points P, Q, R, S are taken on AB, BC, CD, DA respectively so that PQRS is a parallelogram. Find the locus of its center.
A3. Find a function f(n) on the positive integers with positive integer values such that f( f(n) ) = 1993 n1945 for all n.
B1. The tetrahedron ABCD has its vertices on the fixed sphere S. Find all configurations which minimise AB2 + AC2 + AD2 - BC2 - BD2 - CD2.
B2. 1993 points are arranged in a circle. At time 0 each point is arbitrarily labeled +1 or -1. At times n = 1, 2, 3, ... the vertices are relabeled. At time n a vertex is given the label +1 if its two neighbours had the same label at time n-1, and it is given the label -1 if its two neighbours had different labels at time n-1. Show that for some time n > 1 the labeling will be the same as at time 1.
B3. Define the sequences a0, a1, a2, ... and b0, b1, b2, ... by a0 = 2, b0 = 1, an+1 = 2anbn/(an + bn), bn+1 = √(an+1bn). Show that the two sequences converge to the same limit, and find the limit.
30th Vietnamese Mathematical Olympiad 1992 Problems
A1. ABCD is a tetrahedron. The three face angles at A sum to 180o, and the three face angles at B sum to 180o. Two of the face angles at C, ∠ACD and ∠BCD, sum to 180o. Find the sum of the areas of the four faces in terms of AC + CB = k and ∠ACB = x.
A2. For any positive integer n, let f(n) be the number of positive divisors of n which equal ±1 mod 10, and let g(n) be the number of positive divisors of n which equal ±3 mod 10. Show that f(n) ≥ g(n).
A3. Given a > 0, b > 0, c > 0, define the sequences a, bn, cn by a0 = a, b0 = b, c0 = c, an+1 = an + 2/(bn + cn), bn+1 = 2/(cn + an), cn+1 = cn + 2/(an + bn). Show that an tends to infinity.
B1. Label the squares of a 1991 x 1992 rectangle (m, n) with 1 ≤ m ≤ 1991 and 1 ≤ n ≤ 1992. We wish to color all the squares red. The first move is to color red the squares (m, n), (m+1, n+1), (m+2, n+1) for some m < 1990, n < 1992. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way?
B2. ABCD is a rectangle with center O and angle AOB ≤ 45o. Rotate the rectangle about O through an angle 0 < x < 360o. Find x such that the intersection of the old and new rectangles has the smallest possible area.
B3. Let p(x) be a polynomial with constant term 1 and every coefficient 0 or 1. Show that p(x) does not have any real roots > (1 - √5)/2.
29th Vietnamese Mathematical Olympiad 1991 Problems
A1. Find all real-valued functions f(x) on the reals such that f(xy)/2 + f(xz)/2 - f(x) f(yz) ≥ 1/4 for all x, y, z.
A2. For each positive integer n and odd k > 1, find the largest number N such that 2 N divides k n - 1.
A3. The lines L, M, N in space are mutually perpendicular. A variable sphere passes through three fixed points A on L, B on M, C on N and meets the lines again at A', B', C'. Find the locus of the midpoint of the line joining the centroids of ABC and A'B'C'. B1. 1991 students sit in a circle. Starting from student A and counting clockwise round the remaining students, every second and third student is asked to leave the circle until only one remains. (So if the students clockwise from A are A, B, C, D, E, F, ... , then B, C, E, F are the first students to leave.) Where was the surviving student originally sitting relative to A? B2. The triangle ABC has centroid G. The lines GA, GB, GC meet the circumcircle again at D, E, F. Show that 3/R ≤ 1/GD + 1/GE + 1/GF ≤ √3 (1/AB + 1/BC + 1/CA), where R is the circumradius. B3. Show that x 2y/z + y 2z/x + z 2x/y ≥ x 2 + y 2 + z 2 for any non-negative reals x, y, z. [This is false, (1,2,3), (1,1,1), (1,2,8) give >, =, < . Does anyone know the correct question?]Solution 29th VMO 1991Problem A1 Find all real-valued functions f(x) on the reals such that f(xy)/2 + f(xz)/2 - f(x) f(yz) ≥ 1/4 for all x, y, z. Answer f(x) = 1/2 for all x Solution Put x = y = z = 0, then (1/2 - f(0)) 2 ≤ 0, so f(0) = 1/2. Put z = 0, then f(xy) ≥ f(x) for all x,y. Taking x = 1 we get f(x) ≥ f(1) for all x. Taking y = 1/x we get f(1) ≥ f(x) for all x except possibly x = 0, so f(x) = f(1) for all x except possibly x = 0. But putting x = y = z = 1 we get (1/2 - f(1)) 2 ≤ 0, so f(1) = 1/2. Hence f(x) = 1/2 for all x. Thanks to Suat Namli
28th Vietnamese Mathematical Olympiad 1990 Problems
A1. -1 < a < 1. The sequence x1, x2, x3, ... is defined by x1 = a, xn+1 = ( √(3 - 3xn2) - xn)/2. Find necessary and sufficient conditions on a for all members of the sequence to be positive. Show that the sequence is always periodic.
A2. n-1 or more numbers are removed from {1, 2, ... , 2n-1} so that if a is removed, so is 2a and if a and b are removed, so is a + b. What is the largest possible sum for the remaining numbers? A3. ABCD is a tetrahedron with volume V. We wish to make three plane cuts to give a parallelepiped three of whose faces and all of whose vertices belong to the surface of the tetrahedron. Find the intersection of the three planes if the volume of the parallelepiped is 11V/50. Can it be done so that the volume of the parallelepiped is 9V/40? B1. ABC is a triangle. P is a variable point. The feet of the perpendiculars from P to the lines BC, CA, AB are A', B', C' respectively. Find the locus of P such that PA PA' = PB PB' = PC PC'.
B2. The polynomial p(x) with degree at least 1 satisfies p(x) p(2x2) = p(3x3 + x). Show that it does not have any real roots.
B3. Some children are sitting in a circle. Each has an even number of tokens (possibly zero). A child gives half his tokens to the child on his right. Then the child on his right does the same and so on. If a child about to give tokens has an odd number, then the teacher gives him an extra token. Show that after several steps, all the children will have the same number of tokens, except one who has twice the number. Solution
28th VMO 1990Problem A1
-1 < a < 1. The sequence x1, x2, x3, ... is defined by x1 = a, xn+1 = ( √(3 - 3xn2) - xn)/2. Find necessary and sufficient conditions on a for all members of the sequence to be positive. Show that the sequence is always periodic. Answer 0 < a < (√3)/2
period 2 Solution Put x1 = sin k. Then x2 = (√3)/2 cos k - (1/2) sin k = sin 60o cos k - cos 60o sin k = sin(60o-k). Hence x3 = sin(60o - 60o + k) = x1. For sin k and sin(60o-k) to be positive we need 0 < k < 60o and hence 0 < a < (√3)/2. Thanks to Suat Namli
28th VMO 1990Problem B2
The polynomial p(x) with degree at least 1 satisfies p(x) p(2x2) = p(3x3 + x). Show that it does not have any real roots. Solution Suppose it has a positive root α, then 3α3+α is another root, which is larger than α. Proceeding, we get infinitely many roots. Contradiction. So there are no positive roots. Similarly, there are no negative roots. So the only possibility is 0. Suppose bxm is the lowest power of x in p(x). Then the lowest power of x in p(x)p(2x2) is x3m, but the lowest power of x in p(3x3 + x) is xm, so m = 0 and hence p(0) = b ≠ 0, so 0 is not a root. Thanks to Suat Namli
27th Vietnamese Mathematical Olympiad 1989 Problems
A1. Show that the absolute value of sin(kx) /N + sin(kx + x) /(N+1) + sin(kx + 2x) /(N+2) + ... + sin(kx + nx) /(N+n) does not exceed the smaller of (n+1) |x| and 1/(N sin(x/2) ), where N is a positive integer and k is real and satisfies 0 ≤ k ≤ N.
A2. Let a1 = 1, a2 = 1, an+2 = an+1 + an be the Fibonacci sequence. Show that there are infinitely many terms of the sequence such that 1985an2 + 1956an + 1960 is divisible by 1989. Does there exist a term such that 1985an2 + 1956an + 1960 + 2 is divisible by 1989?
A3. ABCD is a square side 2. The segment AB is moved continuously until it coincides with CD (note that A is brought into coincidence with the opposite corner). Show that this can be done in such a way that the region passed over by AB during the motion has area < 5π/6.
B1. Do there exist integers m, n not both divisible by 5 such that m2 + 19n2 = 198·101989?
B2. Define the sequence of polynomials p0(x), p1(x), p2(x), ... by p0(x) = 0, pn+1(x) = pn(x) + (x - pn(x)2)/2. Show that for any 0 ≤ x ≤ 1, 0 < √x - pn(x) ≤ 2/(n+1).
B3. ABCDA'B'C'D' is a parallelepiped (with edges AB, BC, CD, DA, AA', BB', CC', DD', A'B', B'C', C'D', D'A'). Show that if a line intersects three of the lines AB', BC', CA', AD', then it also intersects the fourth.
26th Vietnamese Mathematical Olympiad 1988 Problems
A1. 994 cages each contain 2 chickens. Each day we rearrange the chickens so that the same pair of chickens are never together twice. What is the maximum number of days we can do this?
A2. The real polynomial p(x) = xn - nxn-1 + (n2-n)/2 xn-2 + an-3xn-3 + ... + a1x + a0 (where n > 2) has n real roots. Find the values of a0, a1, ... , an-3.
A3. The plane is dissected into equilateral triangles of side 1 by three sets of equally spaced parallel lines. Does there exist a circle such that just 1988 vertices lie inside it?
B1. The sequence of reals x1, x2, x3, ... satisfies xn+2 <= (xn + xn+1)/2. Show that it converges to a finite limit.
B2. ABC is an acute-angled triangle. Tan A, tan B, tan C are the roots of the equation x3 + px2 + qx + p = 0, where q is not 1. Show that p ≤ √27 and q > 1.
B3. For a line L in space let R(L) be the operation of rotation through 180 deg about L. Show that three skew lines L, M, N have a common perpendicular iff R(L) R(M) R(N) has the form R(K) for some line K.
25th Vietnamese Mathematical Olympiad 1987 problems
A1. Let xn = (n+1)π/3974. Find the sum of all cos(± x1 ± x2 ± ... ± x1987).
A2. The sequences a0, a1, a2, ... and b0, b1, b2, ... are defined as follows. a0 = 365, an+1 = an(an1986 + 1) + 1622, b0 = 16, bn+1 = bn(bn3 + 1) - 1952. Show that there is no number in both sequences.
A3. There are n > 2 lines in the plane, no two parallel. The lines are not all concurrent. Show that there is a point on just two lines.
B1. x1, x2, ... , xn are positive reals with sum X and n > 1. h ≤ k are two positive integers. H = 2h and K = 2k. Show that x1K/(X - x1)H-1 + x2K/(X - x2)H-1 + x3K/(X - x3)H-1 + ... + xnK/(X - xn)H-1 ≥ XK-H+1/( (n-1)2H-1nK-H). When does equality hold?
B2. The function f(x) is defined and differentiable on the non-negative reals. It satisfies | f(x) | ≤ 5, f(x) f '(x) ≥ sin x for all x. Show that it tends to a limit as x tends to infinity.
B3. Given 5 rays in space from the same point, show that we can always find two with an angle between them of at most 90o.
24th Vietnamese Mathematical Olympiad 1986 Problems
A1. a1, a2, ... , an are real numbers such that 1/2 ≤ ai ≤ 5 for each i. The real numbers x1, x2, ... , xn satisfy 4xi2 - 4aixi + (ai - 1)2 = 0. Let S = (x1 + x2 + ... + xn)/n, S2 = (x12 + x22 + ... + xn2)/n. Show that √S2 ≤ S + 1.
A2. P is a pyramid whose base is a regular 1986-gon, and whose sloping sides are all equal. Its inradius is r and its circumradius is R. Show that R/r ≥ 1 + 1/cos(π/1986). Find the total area of the pyramid's faces when equality occurs.
A3. The polynomial p(x) has degree n and p(1) = 2, p(2) = 4, p(3) = 8, ... , p(n+1) = 2n+1. Find p(n+2).
B1. ABCD is a square. ABM is an equilateral triangle in the plane perpendicular to ABCD. E is the midpoint of AB. O is the midpoint of CM. The variable point X on the side AB is a distance x from B. P is the foot of the perpendicular from M to the line CX. Find the locus of P. Find the maximum and minimum values of XO.
B2. Find all n > 1 such that (x12 + x22 + ... + xn2) ≥ xn(x1 + x2 + ... + xn-1) for all real xi.
B3. A sequence of positive integers is constructed as follows. The first term is 1. Then we take the next two even numbers: 2, 4. Then we take the next three odd numbers: 5, 7, 9. Then we take the next four even numbers: 10, 12, 14, 16. And so on. Find the nth term of the sequence.
23rd Vietnamese Mathematical Olympiad 1985 Problems
A1. Find all integer solutions to m3 - n3 = 2mn + 8.
A2. Find all real-valued functions f(n) on the integers such that f(1) = 5/2, f(0) is not 0, and f(m) f(n) = f(m+n) + f(m-n) for all m, n.
A3. A parallelepiped has side lengths a, b, c. Its center is O. The plane p passes through O and is perpendicular to one of the diagonals. Find the area of its intersection with the parallelepiped.
B1. a, b, m are positive integers. Show that there is a positive integer n such that (an - 1)b is divisible by m iff the greatest common divisor of ab and m is also the greatest common divisor of b and m.
B2. Find all real values a such that the roots of 16x4 - ax3 + (2a + 17)x2 - ax + 16 are all real and form an arithmetic progression.
B3. ABCD is a tetrahedron. The base BCD has area S. The altitude from B is at least (AC + AD)/2, the altitude from C is at least (AD + AB)/2, and the altitude from D is at least (AB + AC)/2. Find the volume of the tetrahedron.
Solution
23rd VMO 1985Problem A1
Find all integer solutions to m3 - n3 = 2mn + 8. Answer (m,n) = (2,0), (0,-2) Solution Put m = n+k. Then 3n2k+3nk2+k3 = 2n2+2nk+8, so (3k-2)n2+(3k2-2k)n+k3-8 = 0. For real solutions we require (3k2-2k)2 ≥ 4(3k-2)(k3-8) or (3k-2)(32-2k2-k3) ≥ 0. The first bracket is +ve for k ≥ 1, -ve for k ≤ 0, the second is +ve for k ≤ 2, -ve for k ≥ 3. Hence k = 1 or 2. If k = 1, then n2+n-7 = 0, which has no integer solutions. If k = 2, then 4n2+8n = 0, so n = 0 or -2. Thanks to Suat Namli
23rd VMO 1985Problem A2
Find all real-valued functions f(n) on the integers such that f(1) = 5/2, f(0) is not 0, and f(m) f(n) = f(m+n) + f(m-n) for all m, n. Answer f(n) = 2n + 1/2n Solution Putting m = n = 0, we get f(0)2 = 2f(0), so f(0) = 2. Putting n = 1, we get f(m+1) = 5/2 f(m) + f(m-1). That is a standard linear recurrence relation. The associated quadratic has roots 2, 1/2, so the general solution is f(n) = A 2n + B/2n. f(0) = 1 gives A + B = 2, f(1) = 5/2 gives 2A + B/2 = 5/2, so A = B = 1. It is now easy to check that this solutions satisfies the conditions. Thanks to Suat Namli
22nd Vietnamese Mathematical Olympiad 1984 Problems
A1. (1) Find a polynomial with integral coefficients which has the real number 21/2 + 31/3 as a root and the smallest possible degree.
(2) Find all real solutions to 1 + √(1 + x2) (√(1 + x)3 - √(1 - x)3 ) = 2 + √(1 - x2).
A2. The sequence a1, a2, a3, ... is defined by a1 = 1, a2 = 2, an+2 = 3an+1 - an. Find cot-1a1 + cot-1a2 + cot-1a3 + ... .
A3. A cube side 2a has ABCD as one face. S is the other vertex (apart from B and D) adjacent to A. M, N are variable points on the lines BC, CD respectively. (1) Find the positions of M and N such that the planes SMA and SMN are perpendicular, BM + DN ≥ 3a/2, and BM·DN has the smallest value possible. (2) Find the positions of M and N such that angle NAM = 45 deg, and the volume of SAMN is (a) a maximum, (b) a minimum, and find the maximum and minimum. (3) Q is a variable point (in space) such that ∠AQB = ∠AQD = 90o. p is the plane ABS. Q' is the intersection of DQ and p. Find the locus K of Q'. Let CQ meet K again at R. Let R' be the intersection of DR and p. Show that sin2Q'DB + sin2R'DB is constant. B1. (1) m, n are integers not both zero. Find the minimum value of | 5m2 + 11mn - 5n2 |. (2) Find all positive reals x such that 9x/10 = [x]/( x - [x] ).
B2. a, b are unequal reals. Find all polynomials p(x) which satisfy x p(x - a) = (x - b) p(x) for all x.
B3. (1) ABCD is a tetrahedron. ∠CAD = z, ∠BAC = y, ∠BAD = x, the angle between the planes ACB and ACD is X, the angle between the planes ABC and ABD is Z, the angle between the planes ADB and ADC is Y. Show that sin x/sin X = sin y/sin Y = sin z/sin Z and that x + y = 180o iff X + Y = 180o. (2) ABCD is a tetrahedron with ∠BAC = ∠CAD =∠DAB = 90o. Points A and B are fixed. C and D are variable. Show that ∠CBD + ∠ABD + ∠ABC is constant. Find the locus of the center of the insphere of ABCD.
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