Ημέρα 1η
1. Find, with proof, all positive integers
for which 
is a perfect square.
2. Let
be positive real numbers such that 
. Prove that
3. For a point 
in the coordinate plane, let 
denote the line passing through 
with slope 
. Consider the set of triangles with vertices of the form 
, such that the intersection of the lines 
form an equilateral triangle 
. Find the locus of the center of as 
ranges over all such triangles.
1. Find, with proof, all positive integers
for which is a perfect square. 2. Let
be positive real numbers such that . Prove that in the coordinate plane, let denote the line passing through with slope . Consider the set of triangles with vertices of the form , such that the intersection of the lines form an equilateral triangle . Find the locus of the center of as ranges over all such triangles. Ημέρα 2η
1. A word is defined as any finite string of letters. A word is a palindrome if it reads the same backwards and forwards. Let a sequence of words
be defined as follows: 
, and for 
, 
is the word formed by writing 
followed by 
. Prove that for any 
, the word formed by writing 
in succession is a palindrome.
2. Points
lie on a circle 
and point lies outside the circle. The given points are such that (i) lines 
and 
are tangent to , (ii) 
are collinear, and (iii) 
. Prove that 
bisects 
.
3. Consider the assertion that for each positive integer
, the remainder upon dividing 
by 
is a power of 4. Either prove the assertion or find (with proof) a counterexample.
1. A word is defined as any finite string of letters. A word is a palindrome if it reads the same backwards and forwards. Let a sequence of words
be defined as follows: , and for , is the word formed by writing followed by . Prove that for any , the word formed by writing in succession is a palindrome. 2. Points
lie on a circle and point lies outside the circle. The given points are such that (i) lines and are tangent to , (ii) are collinear, and (iii) . Prove that bisects . 3. Consider the assertion that for each positive integer
, the remainder upon dividing by is a power of 4. Either prove the assertion or find (with proof) a counterexample. 
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