MathDB

P2

Part of 2023 Bangladesh Mathematical Olympiad

Problems(2)

Touching semicircles

Source: BdMO 2023 Secondary National P2

2/12/2023
Let the points A,B,CA,B,C lie on a line in this order. ABAB is the diameter of semicircle ω1\omega_1, ACAC is the diameter of semicircle ω2\omega_2. Assume both ω1\omega_1 and ω2\omega_2 lie on the same side of ACAC. DD is a point on ω2\omega_2 such that BD⊥ACBD\perp AC. A circle centered at BB with radius BDBD intersects ω1\omega_1 at EE. FF is on ACAC such that EF⊥ACEF\perp AC. Prove that BC=BFBC=BF.
geometrysemicirclepower of a point
Three elements of a set

Source: BdMO 2023 Higher Secondary National P2

2/12/2023
Let {a1,a2,⋯ ,ana_1, a_2,\cdots,a_n} be a set of nn real numbers whos sym equals S. It is known that each number in the set is less than Sn−1\frac{S}{n-1}. Prove that for any three numbers aia_i, aja_j and aka_k in the set, ai+aj>aka_i+a_j>a_k.
algebra