MathDB

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Part of 2022 Durer Math Competition Finals

Problems(4)

circle (X,XA) tangent to FB, sqaure and equilateral related

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 1 E1

11/29/2022
To the exterior of side ABAB of square ABCDABCD, we have drawn the regular triangle ABEABE. Point AA reflected on line BEBE is FF, and point EE reflected on line BFBF is GG. Let the perpendicular bisector of segment FGFG meet segment ADAD at XX. Show that the circle centered at XX with radius XAXA touches lineFB FB.
geometrytangentEquilateralSqaure
a_n = c x \phi (a_{n-1}) , is bounded

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 1 E+1

11/29/2022
Let c≥2c \ge 2 be a fixed integer. Let a1=ca_1 = c and for all n≥2n \ge 2 let an=c⋅ϕ(an−1)a_n = c \cdot \phi (a_{n-1}). What are the numbers cc for which sequence (an)(a_n) will be bounded?
ϕ\phi denotes Euler’s Phi Function, meaning that ϕ(n)\phi (n) gives the number of integers within the set {1,2,...,n}\{1, 2, . . . , n\} that are relative primes to nn. We call a sequence (xn)(x_n) bounded if there exist a constant DD such that ∣xn∣<D|x_n| < D for all positive integers nn.
Euler s Phi Functionphi functionnumber theorySequence
3 letters only in duck language

Source: (2021-) 2022 XV D&amp;uuml;rer Math Competition Finals Day2 E1

12/1/2022
In duck language, only letters qq, aa, and kk are used. There is no word with two consonants after each other, because the ducks cannot pronounce them. However, all other four-letter words are meaningful in duck language. How many such words are there? In duck language, too, the letter aa is a vowel, while qq and kk are consonants.
combinatorics
no of 10-digit sequences

Source: (2021-) 2022 XV 15th D&uuml;rer Math Competition Finals Day 2 E+1

12/12/2022
How many 1010-digit sequences are there, made up of 11 four, 22 threes, 33 twos, and 44 ones, in which there is a two in between any two ones, a three in between any two twos, and a four in between any two threes?
Digitsnumber theory