1

Part of 1991 Irish Math Olympiad

Problems(2)

Construct Triangle Given 3 Points

Source: 1991 IrMO Paper 1 Problem 1

X,Y

Z

are given that are, respectively, the circumcenter of a triangle

ABC

, the mid-point of

BC

, and the foot of the altitude from

B

AC

. Show how to reconstruct the triangle

ABC

Sum of consecutive squares

Source: Irish MO

Problem. The sum of two consecutive squares can be a square; for instance

3^2 + 4^2 = 5^2

.
(a) Prove that the sum of

m

consecutive squares cannot be a square for

m \in \{3, 4, 5, 6\}

. (b) Find an example of eleven consecutive squares whose sum is a square.
Can anyone help me with this? Thanks.

modular arithmeticDiophantine equationnumber theory unsolvednumber theory