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| | ==Problem 1== | | ==Problem 1== |
| − | Let <math>\mathbb{N}</math> be the set of all positive integers and <math>S = {(a,b,c,d) \in \mathbb{N}^{4} : a^{2} + b^{2} + c^{2} = d^{2}}</math>. Find the largest positive integer <math>m</math> such that <math>m</math> divides <math>abcd</math> for all <math>(a,b,c,d) \in S</math>.
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| | ==Problem 2== | | ==Problem 2== |
| − | Let <math>\omega</math> be a semicircle with <math>AB</math> as the bounding diameter and let <math>CD</math> be a variable chord of the semicircle of constant length such that <math>C,D</math> lie in the interior of the arc <math>AB</math>. Let <math>E</math> be a point on the diameter <math>AB</math> such that <math>CE</math> and <math>DE</math> are equally inclined to the line <math>AB</math>. Prove that
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| − | (a) the measure of <math>\angle CED</math> is a constant;
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| − | (b) the circumcircle of triangle <math>CED</math> passes through a fixed point.
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| | ==Problem 3== | | ==Problem 3== |
| − | For any natural number <math>n</math>, expressed in base <math>10</math>, let <math>s(n)</math> denote the sum of all its digits. Find all natural numbers <math>m</math> and <math>n</math> such that <math>m < n</math> and
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| − | <math>(s(n))^{2} = m</math> and <math>(s(m))^{2} = n</math>.
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| | ==Problem 4== | | ==Problem 4== |
| − | Let <math>\Omega_1,\Omega_2</math> be two intersecting circles with centres <math>O_1,O_2</math> respectively. Let <math>l</math> be a line that intersects <math>\Omega_1</math> at points <math>A,C</math> and <math>\Omega_2</math> at points <math>B,D</math> such that <math>A, B, C, D</math> are collinear in that order. Let the perpendicular bisector of segment <math>AB</math> intersect <math>\Omega_1</math> at points <math>P,Q</math>; and the perpendicular bisector of segment <math>CD</math> intersect <math>\Omega_1</math> at points <math>R,S</math> such that <math>P,R</math> are on the same side of <math>l</math>. Prove that the midpoints of <math>PR, QS</math> and <math>\Omega_{1} \Omega_{2}</math> are collinear.
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| | ==Problem 5== | | ==Problem 5== |
| − | Let <math>n>k>1</math> be positive integers. Determine all positive real numbers <math>a_1, a_2, ..., a_n</math> which satisfy <math>\sum_{i=1}^{n}</math> <math>\sqrt {\frac {ka_{i}^{k}}{k-1a_{i}^{k}+1}}</math> <math>=\sum_{i=1}^{n}</math> <math>a_i</math> <math>=n</math>.
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| | ==Problem 6== | | ==Problem 6== |
| − | Consider a set of <math>16</math> points arranged in a <math>4\times4</math> square grid formation. Prove that if any <math>7</math> of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.
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