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Kyoto University 2013 Entrance Examination—Science Course Math

1.  (35 points)


In parallelogram ABCD, let E, F, and G be internal dividing points such that sides AB, BC, and CD are divided into ratios of 1:1, 2:1, and 3:1, respectively. Let P be the intersection of sides CE and FG, and let Q be the intersection of the extension of line segment AP and side BC. Calculate the ratio AP:PQ.



2.
  (35 points)


Let N be a natural number greater than 1, and let an (n = 1, 2, …) be a sequence that satisfies conditions (i) and (ii).
(i)    a1 = 2N – 3,
(ii)    For n = 1, 2, …, an +1 = an/2 for an even number of an, while an+1 = an − 1/2 for odd number of an.
Prove that

M
Σ
n=1
an   ≤  2N+1 – N – 5

holds for any natural number M.



3.  (35 points)


Let n be a natural number, and let ax + b be the remainder of integral expression Xn divided by integral expression X2 − 2X−1. Prove that both a and b are integers. Additionally, prove that there is not a prime number by which both a and b are divisible.



4.
  (35 points)


Find the maximum value of  cos

√3
 4
 x2   when 

π
 2
 ≤ 

≤ 
π
 2
  .

Herein both relation π > 3.1 and √3 > 1.7 can be used without a proof.



5. 
(30 points)


In the xy-plane, let circle C with center P, which is on the y-axis, touch two curves

            C1: y = √3 log (1 + x), C2: y = √3 log (1 − x)

at points A and B, respectively. In addition, let ∆PAB be an equilateral triangle on which points A and B are situated in symmetrical positions relative to the y-axis. Find the area that is surrounded by three curves C, C1, and C2. Herein the notion that two curves touch at a specific point means that they share a single point and have a common tangential line at the point.



6.  (35 points)


Consider a coin for which the probability of heads or tails is exactly same when tossing. Place a stone on a number line. When the coin is heads upon tossing, the stone is moved to the symmetrical position relative to the origin on the number line. When the coin is tails upon tossing, the stone is moved to the symmetrical position relative to coordinate 1.

(1)    Let the stone be at coordinate x. Find the probability that the stone will be at coordinate x after the coin is tossed twice.

(2)    Let the stone be at the origin and let n be a natural number. Find the probability that the stone will be at the coordinate 2n – 2 after the coin is tossed 2n times.


(End of the problems)

*The answers are available and please confirm if necessary.