MathDB

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Part of 2013 China Second Round Olympiad

Problems(3)

Trisecting a chord

Source: China Second Round (A) 2013 Q1

5/15/2016
ABAB is a chord of circle ω\omega, PP is a point on minor arc ABAB, E,FE,F are on segment ABAB such that AE=EF=FBAE=EF=FB. PE,PFPE,PF meets ω\omega at C,DC,D respectively. Prove that EF⋅CD=AC⋅BDEF\cdot CD=AC\cdot BD.
geometry
2013 China Second Round Olympiad (B) Test 2 Q1

Source: 13 Oct 2013

10/14/2013
For any positive integer nn , Prove that there is not exist three odd integer x,y,zx,y,z satisfing the equation (x+y)n+(y+z)n=(x+z)n(x+y)^n+(y+z)^n=(x+z)^n.
modular arithmeticnumber theory proposednumber theoryChinaBPSQ
2013 China Second Round Olympiad (C) Test 2 Q1

Source: 13 Oct 2013

10/16/2013
Let nn be a positive odd integer , a1,a2,⋯ ,ana_1,a_2,\cdots,a_n be any permutation of the positive integers 1,2,⋯ ,n1,2,\cdots,n . Prove that :(a1−1)(a22−2)(a33−3)⋯(ann−n)(a_1-1)(a^2_2-2)(a^3_3-3)\cdots (a^n_n-n) is an even number.
modular arithmeticnumber theory proposednumber theoryParity