(b) Prove that there is no natural number n for which 7 divides 2n + 1.
a2(b + c - a) + b2(c + a - b) + c2(a + b - c) ≤ 3abc.
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A1. (a) Find all natural numbers n for which 7 divides 2n - 1. (b) Prove that there is no natural number n for which 7 divides 2n + 1. |
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A2. Suppose that a, b, c are the sides of a triangle. Prove that:
a2(b + c - a) + b2(c + a - b) + c2(a + b - c) ≤ 3abc. |
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| A3. Triangle ABC has sides a, b, c. Tangents to the inscribed circle are constructed parallel to the sides. Each tangent forms a triangle with the other two sides of the triangle and a circle is inscribed in each of these three triangles. Find the total area of all four inscribed circles. |
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| B1. Each pair from 17 people exchange letters on one of three topics. Prove that there are at least 3 people who write to each other on the same topic. [In other words, if we color the edges of the complete graph K17 with three colors, then we can find a triangle all the same color.] |
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| B2. 5 points in a plane are situated so that no two of the lines joining a pair of points are coincident, parallel or perpendicular. Through each point lines are drawn perpendicular to each of the lines through two of the other 4 points. Determine the maximum number of intersections these perpendiculars can have. |
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| B3. ABCD is a tetrahedron and D0 is the centroid of ABC. Lines parallel to DD0 are drawn through A, B and C and meet the planes BCD, CAD and ABD in A0, B0, and C0 respectively. Prove that the volume of ABCD is one-third of the volume of A0B0C0D0. Is the result true if D0 is an arbitrary point inside ABC? |
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