MathDB

1

Part of 2012 Korea National Olympiad

Problems(2)

Korea Second Round 2012 Problem 1

Source: Korea Second Round 2012 Problem 1

8/19/2012
Let ABC ABC be an obtuse triangle with ∠A>90∘ \angle A > 90^{\circ} . Let circle O O be the circumcircle of ABC ABC . D D is a point lying on segment AB AB such that AD=AC AD = AC . Let AK AK be the diameter of circle O O . Two lines AK AK and CD CD meet at L L . A circle passing through D,K,L D, K, L meets with circle O O at P(≠K) P ( \ne K ) . Given that AK=2,∠BCD=∠BAP=10∘ AK = 2, \angle BCD = \angle BAP = 10^{\circ} , prove that DP=sin⁡(∠A2) DP = \sin ( \frac{ \angle A}{2} ).
geometrycircumcircletrigonometryrectangletrapezoidgeometry proposed
Consecutive terms of sequence

Source: Korea National 2012 Problem 5

8/19/2012
p>3 p >3 is a prime number such that p∣2p−1−1 p | 2^{p-1} -1 and p∤2x−1 p \not | 2^x - 1 for x=1,2,⋯ ,p−2 x = 1, 2, \cdots , p-2 . Let p=2k+3 p = 2k+3 . Now we define sequence {an} \{ a_n \} as ai=ai+k=2i(1≤i≤k), aj+2k=ajaj+k (j≥1) a_i = a_{i+k}= 2^i ( 1 \le i \le k ) , \ a_{j+2k} = a_j a_{j+k} \ ( j \ge 1 ) Prove that there exist 2k2k consecutive terms of sequence ax+1,ax+2,⋯ ,ax+2k a_{x+1} , a_{x+2} , \cdots , a_{x+2k} such that for all 1≤i<j≤2k 1 \le i < j \le 2k , ax+i≢ax+j (mod p) a_{x+i} \not \equiv a_{x+j} \ (mod \ p) .
inductionmodular arithmeticnumber theory proposednumber theory