Problem 1. Consider the following operation on positive real numbers written on a blackboard: choose a number r written on the blackboard, erase that number, and then write a pair of positive real numbers a and b satisfying the condition 2*r^2=ab on the board. Assume that you start out with just one positive real number r on the blackboard, and apply this operation (k^2 -1) times to end up with k^2 positive real numbers, not necessarily distinct. Show that there exists a number on the board which does not exceed k*r. Problem 2. Let a_1, a_2, a_3, a_4, a_5 be real numbers satisfying the following equations: a_1 a_2 a_3 a_4 a_5 1 ─── + ─── + ─── + ─── + ─── = ─── for k=1,2,3,4,5. k^2 +1 k^2 +2 k^2 +3 k^2 +4 k^2 +5 k^2 a_1 a_2 a_3 a_4 a_5 Find the value of ── + ── + ── + ── + ──. 37 38 39 40 41 (Express the value in a single fraction.) Problem 3. Let three circles Γ_1,Γ_2,Γ_3, which are non-overlapping and mutually external, be given in the plane. For each point P in the plane, outside the three circles, construct six points A_1, B_1, A_2, B_2, A_3, B_3 as follows: For each i=1,2,3, A_i, B_i are distinct points on the circle Γ_i such that the lines PA_i and PB_i are both tangents to Γ_i. Call the point P exceptional if, from the construction, three lines A_1B_1, A_2B_2, A_3B_3 are concurrent. Show that every exceptional point of the plane, if exists, lies on the same circle. Problem 4. prove that for any positive integer k, there exists an arithmetic sequence a_1 a_2 a_k ──, ──, ……, ── of rational numbers, where a_i, b_i are relatively b_1 b_2 b_k prime positive integers for each i=1,2,...,k, such that the positive integers a_1, b_1, a_2, b_2,...,a_k, b_k are all distinct. Problem 5. Larry and Rob are two robots travelling in one car from Argovia to Zillis. Both robots have control over the steering and steer according to the following algorithm: Larry makes a 90°left turn after every l kilometer driving from start' Rob makes a 90°right turn after every r kilometer driving from start, where l and r are relatively prime positive integers. In the event of both turns occurring simultaneously, the carwill keep going without changing direction. Assume that the ground is flat and the car can move in any direction. Let the car start from Argovia facing towards Zillis. For which choices of the pair (l,r) is the car guaranteed to reach Zillis, regardless of how far it is from Argovia? -- 錦瑟無端五十絃...一絃一柱思華年... 莊生曉夢迷蝴蝶...望帝春心託杜鵑... 滄海月明珠有淚...藍田日暖玉生煙... 此情可待成追憶...只是當時已惘然...多情者...情場殺手... --

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