Solutions
Notes. A sequence x1 , x2 , , xk is in arithmetic progression iff xi+1 xi is constant for 1 i k 1.
A triangular number is a positive integer of the form
T (x)
1
x(x + 1) = 1 + 2 + + x ,
2
where x is a positive integer.
[ ] refers to the area of
Solutions for October Problems
Comment on problems 339 and 342. In both these problems, a condition was left out and made
each of them trivial. Accordingly, problem 339 is marked out of 4 and problem 342 out of 3, for the basic
solution. However, addition
On Heron Triangles, III
Jozsef Sandor
Babes-Bolyai University, 3400 Cluj-Napoca, Romania
1. Let ABC be a triangle with lengths of sides BC = a, AC = b, AB = c positive
integers. Then ABC is called a Heron triangle (or simply, H-triangle) if its area =
Are
283. (a) Determine all quadruples (a, b, c, d) of positive integers for which the greatest common divisor of its
elements is 1,
a
c
=
b
d
and a + b + c = d.
(b) Of those quadruples found in (a), which also satisfy
1 1 1
1
+ + = ?
b
c d
a
(c) For quadruple
325. Solve for positive real values of x, y, t:
(x2 + y 2 )2 + 2tx(x2 + y 2 ) = t2 y 2 .
Are there infinitely many solutions for which the values of x, y, t are all positive integers? What is the
smallest value of t for a positive integer solution?
Soluti
318. Solve for integers x, y, z the system
1 = x + y + z = x3 + y 3 + z 2 .
[Note that the exponent of z on the right is 2, not 3.]
Solution 1. Substituting the first equation into the second yields that
x3 + y 3 + [1 (x + y)]2 = 1
which holds if and only
297. The point P lies on the side BC of triangle ABC so that P C = 2BP , 6 ABC = 45 and 6 AP C = 60 .
Determine 6 ACB.
Solution 1. Let D be the image of C under a reflection with axis AP . Then 6 AP C = 6 AP D =
DP B = 60 , P D = P C = 2BP , so that 6 DBP
290. The School of Architecture in the Olymon University proposed two projects for the new Housing Campus
of the University. In each project, the campus is designed to have several identical dormitory buildings,
with the same number of one-bedroom apartme
Solutions for November Problems
Notes. A real-valued function f (x) of a real variable is increasing if and only if u < v implies that
f (u) f (v). The circumcircle of a triangle is that circle that passes through its three vertices; its centre
is the cir
Solutions
311. Given a square with a side length 1, let P be a point in the plane such that the sum of the distances
from P to the sides of the square (or their extensions) is equal to 4. Determine the set of all such points
P.
Solution. If the square is
The Erdos-Mordell inequality
George Tsintsifas, Thessaloniki, Greece.
Let ABC be a triangle and an interior point M. We denote by AM =
x1 , BM = x2 , CM = x3 . The distances of the point M from BC, CA, AB
are denoted by p1 , p2 , p3 .
Erdos-Mordell inequa