[Đề thi] Các đề thi học sinh giỏi quốc gia môn toán của VN qua từng năm

[Bùi Thanh Tùng (tbhnams2002)]

41st Vietnam 2003 problems

A1. Let R be the reals and f: R ® R a function such that f( cot x ) = cos 2x + sin 2x for all 0 < x < p. Define g(x) = f(x) f(1-x) for -1 <= x <= 1. Find the maximum and minimum values of f on the closed interval [-1, 1].

A2. The circles C1 and C2 touch externally at M and the radius of C2 is larger than that of C1. A is any point on C2 which does not lie on the line joining the centers of the circles. B and C are points on C1 such that AB and AC are tangent to C1. The lines BM, CM intersect C2 again at E, F respectively. D is the intersection of the tangent at A and the line EF. Show that the locus of D as A varies is a straight line.

A3. Let Sn be the number of permutations (a1, a2, ... , an) of (1, 2, ... , n) such that 1 <= |ak - k | <= 2 for all k. Show that ¾ Sn-1 < Sn < 2 Sn-1 for n > 6.
B1. Find the largest positive integer n such that the following equations have integer solutions in x, y1, y2, ... , yn:
(x + 1)2 + y12 = (x + 2)2 + y22 = ... = (x + n)2 + yn2.
B2. Define p(x) = 4x3 - 2x2 - 15x + 9, q(x) = 12x3 + 6x2 - 7x + 1. Show that each polynomial has just three distinct real roots. Let A be the largest root of p(x) and B the largest root of q(x). Show that A2 + 3 B2 = 4.
B3. Let R+ be the set of positive reals and let F be the set of all functions f : R+ ® R+ such that f(3x) >= f( f(2x) ) + x for all x. Find the largest A such that f(x) >= A x for all f in F and all x in R+.

 

[Bùi Thanh Tùng (tbhnams2002)]

40th Vietnam 2001 problems

A1. Solve the following equation: Ö(4 - 3Ö(10 - 3x)) = x - 2.

A2. ABC is an isosceles triangle with AB = AC. O is a variable point on the line BC such that the circle center O radius OA does not have the lines AB or AC as tangents. The lines AB, AC meet the circle again at M, N respectively. Find the locus of the orthocenter of the triangle AMN.

A3. m < 2001 and n < 2002 are fixed positive integers. A set of distinct real numbers are arranged in an array with 2001 rows and 2002 columns. A number in the array is bad if it is smaller than at least m numbers in the same column and at least n numbers in the same row. What is the smallest possible number of bad numbers in the array?

B1. If all the roots of the polynomial x3 + a x2 + bx + c are real, show that 12ab + 27c <= 6a3 + 10(a2 - 2b)3/2. When does equality hold?

B2. Find all positive integers n for which the equation a + b + c + d = nÖ(abcd) has a solution in positive integers.

B3. n is a positive integer. Show that the equation 1/(x - 1) + 1/(22x - 1) + ... + 1/(n2x - 1) = 1/2 has a unique solution xn > 1. Show that as n tends to infinity, xn tends to 4.

 

[Bùi Thanh Tùng (tbhnams2002)]

39th Vietnam 2001 problems

A1. A circle center O meets a circle center O' at A and B. The line TT' touches the first circle at T and the second at T'. The perpendiculars from T and T' meet the line OO' at S and S'. The ray AS meets the first circle again at R, and the ray AS' meets the second circle again at R'. Show that R, B and R' are collinear.

A2. Let N = 6n, where n is a positive integer, and let M = aN + bN, where a and b are relatively prime integers greater than 1. M has at least two odd divisors greater than 1. Find the residue of M mod 6 12n.

A3. For real a, b define the sequence x0, x1, x2, ... by x0 = a, xn+1 = xn + b sin xn. If b = 1, show that the sequence converges to a finite limit for all a. If b > 2, show that the sequence diverges for some a.

B1. Find the maximum value of 1/Öx + 2/Öy + 3/Öz, where x, y, z are positive reals satisfying 1/Ö2 <= z <= min(xÖ2, yÖ3), x + zÖ3 >= Ö6, yÖ3 + zÖ10 >= 2Ö5.

B2. Find all real-valued continuous functions defined on the interval (-1, 1) such that (1 - x2) f(2x/(1 + x2) ) = (1 + x2)2 f(x) for all x.

B3. a1, a2, ... , a2n is a permutation of 1, 2, ... , 2n such that |ai - ai+1| ¹ |aj - aj+1| for i ¹ j. Show that a1 = a2n + n iff 1 <= a2i <= n for i = 1, 2, ... n.

 

[Bùi Thanh Tùng (tbhnams2002)]

38th Vietnam 2000 problems

A1. Define a sequence of positive reals x0, x1, x2, ... by x0 = b, xn+1 = Ö(c - Ö(c + xn)). Find all values of c such that for all b in the interval (0, c), such a sequence exists and converges to a finite limit as n tends to infinity.

A2. C and C' are circles centers O and O' respectively. X and X' are points on C and C' respectively such that the lines OX and O'X' intersect. M and M' are variable points on C and C' respectively, such that angle XOM = angle X'O'M' (both measured clockwise). Find the locus of the midpoint of MM'. Let OM and O'M' meet at Q. Show that the circumcircle of QMM' passes through a fixed point.

A3. Let p(x) = x3 + 153x2 - 111x + 38. Show that p(n) is divisible by 32000 for at least nine positive integers n less than 32000. For how many such n is it divisible?

B1. Given an angle a such that 0 < a < p, show that there is a unique real monic quadratic x2 + ax + b which is a factor of pn(x) = sin a xn - sin(na) x + sin(na-a) for all n > 2. Show that there is no linear polynomial x + c which divides pn(x) for all n > 2.

B2. Find all n > 3 such that we can find n points in space, no three collinear and no four on the same circle, such that the circles through any three points all have the same radius.

B3. p(x) is a polynomial with real coefficients such that p(x2 - 1) = p(x) p(-x). What is the largest number of real roots that p(x) can have?

 

[Bùi Thanh Tùng (tbhnams2002)]

37th Vietnam 1999 problems

A1. Find all real solutions to (1 + 42x-y)(5y-2x+1) = 22x-y+1 + 1, y3 + 4x + ln(y2 + 2x) + 1 = 0.

A2. ABC is a triangle. A' is the midpoint of the arc BC of the circumcircle not containing A. B' and C' are defined similarly. The segments A'B', B'C', C'A' intersect the sides of the triangle in six points, two on each side. These points divide each side of the triangle into three parts. Show that the three middle parts are equal iff ABC is equilateral.

A3. The sequence a1, a2, a3, ... is defined by a1 = 1, a2 = 2, an+2 = 3an+1 - an. The sequence b1, b2, b3, ... is defined by b1 = 1, b2 = 4, bn+2 = 3bn+1 - bn. Show that the positive integers a, b satisfy 5a2 - b2 = 4 iff a = an, b = bn for some n.

B1. Find the maximum value of 2/(x2 + 1) - 2/(y2 + 1) + 3/(z2 + 1) for positive reals x, y, z which satisfy xyz + x + z = y.

B2. OA, OB, OC, OD are 4 rays in space such that the angle between any two is the same. Show that for a variable ray OX, the sum of the cosines of the angles XOA, XOB, XOC, XOD is constant and the sum of the squares of the cosines is also constant.

B3. Find all functions f(n) defined on the non-negative integers with values in the set {0, 1, 2, ... , 2000} such that: (1) f(n) = n for 0 <= n <= 2000; and (2) f( f(m) + f(n) ) = f(m + n) for all m, n.

 

[Võ Mạnh Cường (cuong8I)]

Hix,có đại huynh nào tốt bụng dịch giùm em với được không ạ,em cần gấp lắm.Cảm ơn lắm lắm.

 

[Mai Trung Anh (tieumai03)]

Bạn đúng là không chịu khó! Tôi dốt Tiếng Anh, nhưng không lơ mơ đến mứa không biết tìm đề bằng tiếng việt, he he he