MathDB

1

Part of 2016 Korea Winter Program Practice Test

Problems(4)

tangential circles

Source: 2016 Korea Winter Program Test1 Day1 #1

1/27/2016
There is circle ω\omega and A,BA, B on it. Circle γ1\gamma_1 tangent to ω\omega on TT and ABAB on DD. Circle γ2\gamma_2 tangent to ω\omega on SS and ABAB on EE. (like the figure below) Let AB∩TS=CAB\cap TS=C.
Prove that CA=CBCA=CB iff CD=CECD=CE
geometrygeometry proposedtangent circles
Find all {a_n}

Source: 2016 Korea Winter Program Test1 Day2 #5

1/25/2016
Find all {an}n≥0\{a_n\}_{n\ge 0} that satisfies the following conditions.
(1) an∈Za_n\in \mathbb{Z} (2) a0=0,a1=1a_0=0, a_1=1 (3) For infinitly many mm, am=ma_m=m (4) For every n≥2n\ge2, {2ai−ai−1∣i=1,2,3,⋯ ,n}≡{0,1,2,⋯ ,n−1}\{2a_i-a_{i-1} | i=1, 2, 3, \cdots , n\}\equiv \{0, 1, 2, \cdots , n-1\} mod  n\mod n
algebraalgebra proposedInteger sequencenumber theory
a^2+b^2=m^2-n^2, ab=2mn

Source: 2016 Korea Winter Program Test2 Day1 #1

1/25/2016
Solve: a,b,m,n∈Na, b, m, n\in \mathbb{N} a2+b2=m2−n2,ab=2mna^2+b^2=m^2-n^2, ab=2mn
number theory proposedDiophantine equationnumber theory
Perpendicular bisector

Source: 2016 Korea Winter Camp 2nd Test #5

1/25/2016
Let there be an acute triangle ABCABC with orthocenter HH. Let BH,CHBH, CH hit the circumcircle of △ABC\triangle ABC at D,ED, E. Let PP be a point on ABAB, between BB and the foot of the perpendicular from CC to ABAB. Let PH∩AC=QPH \cap AC = Q. Now △AEP\triangle AEP's circumcircle hits CHCH at SS, △ADQ\triangle ADQ's circumcircle hits BHBH at RR, and △AEP\triangle AEP's circumcircle hits △ADQ\triangle ADQ's circumcircle at J(≠A)J (\not=A). Prove that RSRS is the perpendicular bisector of HJHJ.
geometryperpendicular bisector