AoPS Community
2017 Japan EGMO Preliminary
www.artofproblemsolving.com/community/c3281214
by parmenides51, Kouichi Nakagawa
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There is a trapezoid ABCD with parallel sides AB and DC, and the diagonals intersect perpendicularly at the point P . Let Q be the reflection of P wrt line AB. Prove that ∠AM B + ∠CQD =
90o , where M is the midpoint of side CD.
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A sequence of integers a1 , a2 , . . . satisfies a1 = 1, ak+1 = a2k + 1. (k = 1, 2, . . .)
Then, show that there are infinitely many primes p that satisfy the following conditions.
Condition: A multiple of p exists in a1 , a2 , . . ..
3
As shown in the figure, there is a board with 16 squares removed from a 9 × 9 grid.
Find the minimum possible number of rectangles that can be divided when dividing this board
into rectangles along the squares.
4
Let x, y be changeable to any value in all positive real numbers.
Find the smallest possible value of following
(x2 + y)(x + y 2 )
.
xy(x + y)
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