MathDB

4

Part of 2014 Saudi Arabia GMO TST

Problems(3)

concyclic wanted, angle bisectors related

Source: 2014 Saudi Arabia GMO TST day I p4

7/31/2020
Let ABCABC be a triangle, DD the midpoint of side BCBC and EE the intersection point of the bisector of angle ∠BAC\angle BAC with side BCBC. The perpendicular bisector of AEAE intersects the bisectors of angles ∠CBA\angle CBA and ∠CDA\angle CDA at MM and NN, respectively. The bisectors of angles ∠CBA\angle CBA and ∠CDA\angle CDA intersect at PP . Prove that points A,M,N,PA, M, N, P are concyclic.
geometryangle bisectorConcyclic
a_1a_2(a_1 - a_2) + a_2a_3(a_2 - a_3) +...+ a_{n-1}a_n(a_{n-1} - a_n) \ge a_1a_n

Source: 2014 Saudi Arabia GMO TST II p4

7/26/2020
Let a1≥a2≥...≥an>0a_1 \ge a_2 \ge ... \ge a_n > 0 be real numbers. Prove that a1a2(a1−a2)+a2a3(a2−a3)+...+an−1an(an−1−an)≥a1an(a1−an)a_1a_2(a_1 - a_2) + a_2a_3(a_2 - a_3) +...+ a_{n-1}a_n(a_{n-1} - a_n) \ge a_1a_n(a_1 - a_n)
algebrainequalities
|x- z|/ |y - z |= 2 or |y - z|/ |x- z |= 2

Source: 2014 Saudi Arabia GMO TST III p4

7/26/2020
Let XX be a set of rational numbers satisfying the following two conditions: (a) The set XX contains at least two elements, (b) For any x,yx, y in XX, if x≠yx \ne y then there exists zz in XX such that either ∣x−zy−z∣=2\left| \frac{x- z}{y - z} \right|= 2 or ∣y−zx−z∣=2\left| \frac{y -z}{x - z} \right|= 2 . Prove that XX contains infinitely many elements.
rationalsetalgebra