1. Boddhi places some counters onto the squares of an 8 by 8 chessboard so that there is at most one counter in each of the 64 squares. Determine, with justification, the maximum number that he can place without having five or more counters in the same row, or in the same column, or on either of the two long diagonals.
2. Two circles S and T touch at X. They have a common tangent which meets S at A and T at B. The points A and B are different. Let AP be a diameter of S. Prove that B, X and P lie on a straight line.
3. Find all real numbers x, y and z which satisfy the simultaneous equations x^2 − 4y + 7 = 0, y^2 − 6z + 14 = 0 and z^2 − 2x − 7 = 0.
4. Find all positive integers n such that 12 n −119 and 75 n −539 are both perfect squares.
5. A triangle has sides of length at most 2, 3 and 4 respectively. Determine, with proof, the maximum possible area of the triangle.
6. Let ABC be a triangle. Let S be the circle through B tangent to CA at A and let T be the circle through C tangent to AB at A. The circles S and T intersect at A and D. Let E be the point where the line AD meets the circle ABC. Prove that D is the midpoint of AE.
<<< bmoc 201213 >>>
2. Two circles S and T touch at X. They have a common tangent which meets S at A and T at B. The points A and B are different. Let AP be a diameter of S. Prove that B, X and P lie on a straight line.
3. Find all real numbers x, y and z which satisfy the simultaneous equations x^2 − 4y + 7 = 0, y^2 − 6z + 14 = 0 and z^2 − 2x − 7 = 0.
4. Find all positive integers n such that 12 n −119 and 75 n −539 are both perfect squares.
5. A triangle has sides of length at most 2, 3 and 4 respectively. Determine, with proof, the maximum possible area of the triangle.
6. Let ABC be a triangle. Let S be the circle through B tangent to CA at A and let T be the circle through C tangent to AB at A. The circles S and T intersect at A and D. Let E be the point where the line AD meets the circle ABC. Prove that D is the midpoint of AE.
<<< bmoc 201213 >>>
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