1st Vietnamese Secondary Mathematics Contest

Math Contests: High School, Vol. 4- School Years: 1996-97 through 2000-2001

Math Olympiad Contest Problems for Elementary and Middle Schools, Vol. 1

Math Contests, Grades 7 & 8, Vol. 1- School Years: 1977-78 Through 1981-82

Competition Math for Middle School

A1. Find all pair of natural numbers x, y such that: (2^x + 1)(2^x + 2)(2^x + 3)(2^x + 4) - 5^y = 11879

(For 6^th grade)

A2. Let n be a positive integer and let U(n) = {d₁; d₂;...d_m} be the set of all positive divisors of n. Prove that
d₁² + d₂² + ...+ d_m² <

(For 7^th grade)

A3. Prove that

where a, b, c are there positive numbers satisfying abc = 1.

A4. Solve the equation

A5. Let ABCD be a square, M is a point lying on CD (M≠C, M≠D). Through the point C draw

a line perpendicular to AM at H, BH meets AC at K. Prove that:

1) MK is always parallel to a fixed line when M moves on the side CD.

2) The circumcenter of the quadrilateral ADMK lies on a fixed line.

A6. Let a, b, c be positive real numbers such that abc = 1. Prove that

(For upper secondary schools)

A7. Conside all triangles ABC where A < B < C≤

Find the least value of the expression: M = cot²A + cot²B + cot²C + 2(cotA - cotB)(cotB - cotC)(cotC - cotA).

A8. Suppose that the tetrahedron ABCD statisfies the following conditions: All faces are acute triangles and BC is perpendicular to AD. Let h_a, h_d be respectively the lengths of the altitudes from A, D onto the opposite faces, and let 2α

be the measure of the dihedral angle at edge BC, d is the distance between BC and AD. Prove the inequalityl: