MathDB

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Part of 2023 Czech-Polish-Slovak Junior Match

Problems(2)

n-S(n), n+S(n) are both perfect squares

Source: 2023 Czech-Polish-Slovak Match Junior, Team p1 CPSJ

5/5/2024
Let S(n)S(n) denote the sum of all digits of natural number nn. Determine all natural numbers nn for which both numbers n+S(n)n + S(n) and n−S(n)n - S(n) are square powers of non-zero integers.
number theoryPerfect SquaresPerfect Squaresum of digits
AM_|_MD if BC = 2 AC, AD = 2BD, BM=//MC

Source: 2023 Czech-Polish-Slovak Match Junior, individual p1 CPSJ

5/5/2024
Given a triangle ABCABC, BC=2⋅ACBC = 2 \cdot AC. Point MM is the midpoint of side BC BC and point DD lies on ABAB, with AD=2⋅BDAD = 2 \cdot BD. Prove that the lines AMAM and MDMD are perpendicular.
perpendiculargeometry