IMO 2018 [1] Let Γ be the circumcircle of acute triangle ABC. Points D and E are on gegments AB and AC respectively such that AD = AE. The perpendicular bisectors of BD and CE intersect minor arcs AB and AC of Γ at points F and G respectively. Prove that lines DE and FG are either parallel or they are the same line. suseum.jp/gq/question/2890
IMO 2018 [2] Find all integers n ≧ 3 for which there exist real numbers a_1，a_2，…，a_{n+2} satisfying a_{n+1} = a_1，a_{n+2} = a_2 and a_i a_{i+1} + 1 = a_{i+2} for i = 1，2，…，n. suseum.jp/gq/question/2891
IMO 2018 [5] Let a_1，a_2,… be an infinite sequence of positive integers. Suppose that there is an integer N > 1 such that，for each n ≧ N，the number a_1/a_2 + a_2/a_3 ＋ … + a_{n-1}/a_n + a_n/a_1 is an integer. Prove that there is a positive integer M such that a_m = a_{m+1} for all m ≧ M. suseum.jp/gq/question/2894
IMO 2018 [6] A convex quadrilateral ABCD satisfies AB･CD = BC･DA. Point X lies inside ABCD so that ∠XAB = ∠XCD and ∠XBC = ∠XDA. Prove that ∠BXA + ∠DXC = 180ﾟ. suseum.jp/gq/question/2895
