# In the figure below we have that ae is parallel to cd

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13.In the figure below we have thatAEis parallel toCD,BCis parallel toDE,AEmeasures 2,ABCmeasures 45 degrees andBCDmeasures 75 degrees. A B C D E The distance from the point E to the segment AB is: (a)3(b)3/2(c)2(d) 1(e) none of these 14.Leta= 26·34·75. Then, the number of divisors ofawhich are relatively primeto 152012and for which the least common multiple with 14 is divisible by 28 is: 15.Three fair dice are thrown. Whats is the probability that the three numbers onthe top sides are (possibly after rearrangement) consecutive? 16.Let4ABCbe a triangle with sides measuring 1/2, 1 and3/2. Find the radiusof the circle inscribed in the triangle, i.e., of the circle inside the triangle and tangentto its three sides. 3 + 1 2 3 + 1 3 3 4 4
17.Ten couples must sit around a round table in such a way that every couple sitstogether. How many possible seating arrangements are there if we only regard whosits next to whom? (In other words, rotating everyone a fixed number of seats doesnotyield a different configuration.) (a)2010(b) 9!·210(c)202(d) 10!·219(e) none of these 10 18.The rhombusABCDshown below has sides measuring 1, andMandNare themidpoints of the sidesABandBCrespectively. A B C D M N If the length of the segment MN is 14 / 4, then the length of the segment DM is: 19.The sum of the first one hundredoddpositive integers which are not divisible by3 is: 5
20.A circular garden of diameter 12 feet is cut by a path 3 feet wide, one side ofwhich goes through the center of the garden. 12 3 PATH The garden area which is not part of the path (in square feet) is: 21.We have two urns, urn 1 and urn 2. Urn 1 contains two black balls and one whiteball. Urn 2 contains two black balls and two white balls. An urn is chosen at randomand one ball is removed (and not replaced). Then, another ball is removed followingthe same process. What is the probability that both balls removed are black? (a)12(b)23(c)924(d)724(e) none of these 6
22.Suppose that the equationx3-px2=qm, wherepandqare positive real numberswithq6= 1 andmis a positive integer, has three positive real solutionsa,bandc.Then,logq ( abc ( a 2 + b 2 + c 2 ) a + b + c ) is equal to:
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