MathDB

2016 Auckland Mathematical Olympiad

Part of Auckland Mathematical Olympiad

Subcontests

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2 player game with divisors of 328 (2016 Auckland MO J2)

The number 328328 is written on the board. Two players alternate writing positive divisors of 328328 on the board, subject to the following rules: ∙\bullet No divisor of a previously written number may be written. ∙\bullet The player who writes 328 loses. Who has a winning strategy, the first player or the second player?

computational, equal angles and a square (2016 Auckland MO S2)

In square ABCDABCD, AC‾\overline{AC} and BD‾\overline{BD} meet at point EE. Point FF is on CD‾\overline{CD} and ∠CAF=∠FAD\angle CAF = \angle FAD. If AF‾\overline{AF} meets ED‾\overline{ED} at point GG, and if EG‾=24\overline{EG} = 24 cm, then find the length of CF‾\overline{CF}.
3
2

area wanted, triangle inscribed in square (2016 Auckland MO J3)

Triangle XYZXYZ is inside square KLMNKLMN shown below so that its vertices each lie on three different sides of the square. It is known that: ∙\bullet The area of square KLMNKLMN is 11. ∙\bullet The vertices of the triangle divide three sides of the square up into these ratios: KX:XL=3:2KX : XL = 3 : 2 KY:YN=4:1KY : YN = 4 : 1 NZ:ZM=2:3NZ : ZM = 2 : 3 What is the area of the triangle XYZXYZ? (Note that the sketch is not drawn to scale). https://cdn.artofproblemsolving.com/attachments/8/0/38e76709373ba02346515f9949ce4507ed4f8f.png

in 2 weeks 3 cows eat all the grass on 2 hectares (2016 Auckland MO S3)

In two weeks three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks. In four weeks, two cows eat all the grass on two hectares of land, together with all the grass that regrows there during the four weeks. How many cows will eat all the grass on six hectares of land in six weeks, together with all the grass that regrows there over the six weeks?
(Assume: ∙\bullet the quantity of grass on each hectare is the same when the cows begin to graze, ∙\bullet the rate of growth of the grass is uniform during the time of grazing, ∙\bullet the cows eat the same amount of grass each week.)