Maths Olympiad Exam

Consider an n × n array of numbers:
0BBBB@
a11 a12 a13 · · · a1n
a21 a22 a23 · · · a2n
...
...
an1 an2 an3 · · · ann
1CCCCA
Suppose each row consists of the n numbers 1, 2, 3, . . . , n in some order and aij = aji for
i = 1, 2, . . . , n and j = 1, 2, . . . , n. If n is odd, prove that the numbers a11, a22, a33, . . . , ann
are 1, 2, 3, . . . , n in some order.

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Maths Olympiad Other Question
Find the number of positive integers x which satisfy the condition
h x
99 i = h x
101 i.
(Here [z] denotes, for any real z, the largest integer not exceeding z; e.g. [7/4] = 1.) In a triangle ABC, D is a point on BC such that AD is the internal bisector of A. Suppose
B = 2C and CD = AB. Prove that A = 72◦