Đề thi học sinh giỏi Toán toàn quốc năm 1978


Dịch sang tiếng Anh

A1.  Find all three digit numbers abc such that 2(abc) = bca + cab.
A2.  Find all values of the parameter m such that the equations x2 = 2|x| + |x| - y - m = 1 - y2 have only one root.
A3.  The triangle ABC has angle A = 30o and AB = 3/4 AC. Find the point P inside the triangle which minimises 5 PA + 4 PB + 3 PC.
B1.  Find three rational numbers a/d, b/d, c/d in their lowest terms such that they form an arithmetic progression and b/a = (a + 1)/(d + 1), c/b = (b + 1)/(d + 1).
B2.  A river has a right-angle bend. Except at the bend, its banks are parallel lines a distance a apart. At the bend the river forms a square with the river flowing in across one side and out across an adjacent side. What is the longest boat of length c and negligible width which can pass through the bend?
B3.  ABCDA'B'C'D' is a rectangular parallelepiped (so that ABCD and A'B'C'D' are faces and AA', BB', CC', DD' are edges). We have AB = a, AD = b, AA' = c. The perpendicular distances of A, A', D from the line BD' are p, q, r. Show that there is a triangle with sides p, q, r. Find a relation between a, b, c and p, q, r.