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Grasshopper jumping from point (1,1) to (m,n) with area =1/2

Source: Vietnamese TST 2011 P1

April 27, 2011
geometrycalculusintegrationinductionmodular arithmeticcombinatorics unsolvedcombinatorics

Problem Statement

A grasshopper rests on the point (1,1)(1,1) on the plane. Denote by O,O, the origin of coordinates. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point AA to B,B, then the area of △AOB\triangle AOB is equal to 12.\frac 12. (a)(a) Find all the positive integral poijnts (m,n)(m,n) which can be covered by the grasshopper after a finite number of steps, starting from (1,1).(1,1). (b)(b) If a point (m,n)(m,n) satisfies the above condition, then show that there exists a certain path for the grasshopper to reach (m,n)(m,n) from (1,1)(1,1) such that the number of jumps does not exceed ∣m−n∣.|m-n|.
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