Combinatorics

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Problem: Construct a linear homogeneous difference equation (of the minimum possible order) with constant real coefficients, the partial solutions of which are the functions \[ (-3)^{s}, 5^{s}, s \cdot 5^{s}, 12^{s} \cos \frac{11 \pi(s+1)}{6} \text {. } \]

6.6.32 Combinatorics

2.03 $

Problem: Prove the statement using mathematical induction: \( \left(n^{3}+11 n\right) \vdots 6 \), for all natural numbers \( n \geq 1 \).

6.6.33 Combinatorics

0 $

Problem: Find a sequence \( \left\{a_{n}\right\} \), satisfying the recurrence relation \( a_{n+2}-3 a_{n+1}+2 a_{n}=0 \) and the initial conditions \( a_{1}=3, a_{2}=7 \).

6.6.34 Combinatorics

1.02 $

Problem: How many positive three-digit numbers are there that are divisible by exactly one of these three numbers?

6.6.35 Combinatorics

2.54 $

Problem: Find the coefficients for \( a=x^{3} \cdot y^{2} \cdot z^{2}, b=x^{2} \cdot y^{2} \cdot z^{2} \) and \( c=x^{4} \cdot z^{4} \) in the expansion \( \left(2 x+3 y+5 z^{2}\right)^{6} \).

6.6.36 Combinatorics

2.03 $

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