1) triangles T's vertices are at (0,0), (5,0) and (1,2). what is the slope of the
line through (0,0) that divides T into two triangles of equal area ?
2) In the sequence below, each angle is in radians. What is the largest number of consecutive terms of this sequence that can be postive ?
cos x , cos (x+1) , cos (x+2) , cos (x+3) , cos (x+4) ,cos (x+5) , cos (x+6)
3) If a,b,and c are positive integers, what is the largest value less than 1
representable by 1/a + 1/b + 1/c ?
4) Bricklayers Pat and Lee alternate turns in a game in which each player
removes from 1 to 100 bricks from a common pile that initially has 2008 bricks. The player taking the last brick wins. If Pat and Lee both play perfectly, and if Pat goes first, then how many bricks must Pat take on his first turn to
guarantee a win ?
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they are relly relly so hard
can anyone help me ^^*
Mathematics ! QUICK ! So hard
2008-03-22 1:15 pm
回答 (1)
2008-03-23 6:21 pm
✔ 最佳答案
1) by the geometry, the midpoint of line (5,0) (1,2) will split the triangle half.the mid-point = ( 3,1)
the slope = 1/3
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2) we know that cos has period of 2 pi ~ 6.28.
the positive value happen in half period ~ 3.1415
so, cos x , cos (x+1) , cos (x+2) , cos (x+3) , cos (x+4) ,cos (x+5) , cos (x+6)
included in 1 cycle.
the maximum number of positive terms is 4.
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3) 1/a + 1/b + 1/c <1
(2,2,c) X
(2,3,3) OK
any other possible value will lower the value.
the answer is (2,3,3)
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4)think about the last round.
in pat turn, the total brick left in the game should be less or equal to 100.
how about 101 bricks? if so, lee win. so, pat should avoid 101 appear.
how about 102 bricks? if so, pat will only pick 1 bricks, and lee lose.
103 bricks? if so, pat will only pick 2 bricks, and lee lose.
so, on,
at 200, pat will pick, 99 bricks, and lee lose.
201, pat will pick 100 bricks and lee lose
202? no matter how many pat picks, pat will lose. since pat pick 1 to 100, the position of win hand to lee.
between, 201 to 102, pat must win. if 101 or 202, pat lose. or, who get 101 or 202 in hand, he must lose.
203? pat should picks 1 brick.
204? 2 bricks.
302? 100 bricks.
we can see, who give 101, 202, 303 to other, he can win,
101X19 = 1919,
1919+89 = 2008.
pat should pick up 89 bricks, give lee 1919 bricks, no matter lee play, pat win. nice game! :D
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