MathDB

3

Part of 2008 Indonesia TST

Problems(4)

10 people attended a party

Source: 2008 Indonesia TST stage 2 test 1 p3

12/14/2020
1010 people attended a party. For every 33 people, there exist at least 22 people who don’t know each other. Prove that there exist 44 people who don’t know each other.
combinatorics
cyclic quadrilaterals and angle bisector wanted, tangent circles

Source: 2008 Indonesia TST stage 2 test 2 p3

12/14/2020
Let Γ1\Gamma_1 and Γ2\Gamma_2 be two circles that tangents each other at point NN, with Γ2\Gamma_2 located inside Γ1\Gamma_1. Let A,B,CA, B, C be distinct points on Γ1\Gamma_1 such that ABAB and ACAC tangents Γ2\Gamma_2 at DD and EE, respectively. Line NDND cuts Γ1\Gamma_1 again at KK, and line CKCK intersects line DEDE at II. (i) Prove that CKCK is the angle bisector of ∠ACB\angle ACB. (ii) Prove that IECNIECN and IBDNIBDN are cyclic quadrilaterals.
geometrycyclic quadrilateralangle bisectortangent circles
Parallel and not parallel lines.

Source:

4/30/2017
Let ABCDABCD be a convex quadrilateral with ABAB is not parallel to CDCD Circle Γ1\Gamma_{1} with center O1O_1 passes through AA and BB, and touches segment CDCD at PP. Circle Γ2\Gamma_{2} with center O2O_2 passes through CC and DD, and touches segment ABAB at QQ. Let EE and FF be the intersection of circles Γ1\Gamma_{1} and Γ2\Gamma_{2}. Prove that EFEF bisects segment PQPQ if and only if BCBC is parallel to ADAD.
geometry
gcd(n^a + 1, n^b + 1) <= n^{gcd(a,b)} + 1

Source: 2008 Indonesia TST stage 2 test 3 p3

12/15/2020
Let nn be an arbitrary positive integer. (a) For every positive integers aa and bb, show that gcd(na+1,nb+1)≤ngcd(a,b)+1gcd(n^a + 1, n^b + 1) \le n^{gcd(a,b)} + 1. (b) Show that there exist infinitely many composite pairs (a,b)a, b), such that each of them is not a multiply of the other number and equality holds in (a).
GCDgreatest common divisornumber theory