MathDB

5

Part of 2023 Durer Math Competition (First Round)

Problems(2)

products of two different numbers from n different reals

Source: (2022 -) 2023 XVI Dürer Math Competition Regional E5

5/25/2024
Let n≥3n \ge 3 be an integer. Timi thought of nn different real numbers and then wrote down the numbers which she could produce as the product of two different numbers she had in mind. At most how many different positive prime numbers did she write down (depending on nn)?
number theory
H_DB^2 + H_EC^2 + H_FA^2 = H_DC^2 + H_EA^2 + H_FB^2

Source: (2022 -) 2023 XVI Dürer Math Competition Regional E+5

5/25/2024
Consider an acute triangle ABCABC. Let DD, EE and FF be the feet of the altitudes through vertices AA, BB and CC. Denote by A′A', B′B', C′C' the projections of AA, BB, CC onto lines EFEF, FDFD, DEDE, respectively. Further, let HDH_D, HEH_E, HFH_F be the orthocenters of triangles DB′C′DB'C', EC′A′EC'A', FA′B′FA'B'. Show that HDB2+HEC2+HFA2=HDC2+HEA2+HFB2.H_DB^2 + H_EC^2 + H_FA^2 = H_DC^2 + H_EA^2 + H_FB^2.
geometry