MathDB

1

Part of Taiwan TST 2015 Round 1

Problems(4)

2015 Taiwan TST Round 1 Quiz 1 Problem 1

Source: 2015 Taiwan TST Round 1 Quiz 1 Problem 1

7/12/2015
Find all primes p,q,rp,q,r such that qr−1qr-1 is divisible by pp, pr−1pr-1 is divisible by qq, pq−1pq-1 is divisible by rr.
number theoryTaiwanTaiwan TST 2015BritishMathematicalOlympiad
2015 Taiwan TST Round 1 Quiz 2 Problem 1

Source: 2015 Taiwan TST Round 1 Quiz 2 Problem 1

7/12/2015
Prove that for any set containing 20472047 positive integers, there exists 10241024 positive integers in the set such that the sum of these positive integers is divisible by 10241024.
TaiwancombinatoricsTaiwan TST 2015
2015 Taiwan TST Round 1 Quiz 3 Problem 1

Source: 2015 Taiwan TST Round 1 Quiz 3 Problem 1

7/12/2015
Let a,b,c,da,b,c,d be any real numbers such that a+b+c+d=0a+b+c+d=0, prove that 1296(a7+b7+c7+d7)2≤637(a2+b2+c2+d2)71296(a^7+b^7+c^7+d^7)^2\le637(a^2+b^2+c^2+d^2)^7
inequalitiesTaiwanalgebraTaiwan TST 2015
2015 Taiwan TST Round 1 Mock IMO Day 2 Problem 1

Source: 2015 Taiwan TST Round 1 Mock IMO Day 2 Problem 1

7/12/2015
Let ABCABC be a triangle and MM be the midpoint of BCBC, and let AMAM meet the circumcircle of ABCABC again at RR. A line passing through RR and parallel to BCBC meet the circumcircle of ABCABC again at SS. Let UU be the foot from RR to BCBC, and TT be the reflection of UU in RR. DD lies in BCBC such that ADAD is an altitude. NN is the midpoint of ADAD. Finally let ASAS and MNMN meets at KK. Prove that ATAT bisector MKMK.
Taiwangeometrycircumcirclegeometric transformationreflectionTaiwan TST 2015