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9.6

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(3)

radicals simplify -- VI Soros Olympiad 1999-00 Round 1 9.6

Source:

5/21/2024
For all valid values of aa and bb, simplify the expression 4b−a2+2ab+4+a4ab−10b2−8+b.\frac{\sqrt{4b-a^2+2ab+4}+a}{\sqrt{4ab-10b^2-8}+b}.
algebra
battleship game, revisited (VI Soros Olympiad 1990-00 R1 9.6)

Source:

5/27/2024
On the "battleship" field (a square of 10×1010\times 10 cells), 1010 "ships" are placed in the following sequence: first one "ship" of size 1×41\times 4, then two - of size 1×31\times 3, three - of size 1×21\times 2, and, finally, four - 1×11\times 1. The rules do not allow "ships" to touch each other even with their tops. Can it happen that when part of the "ships" have already been displayed, there is nowhere to place the next one?
combinatorics
recurrence relation, parity of a_{n+1}a_n

Source: VI Soros Olympiad 1990-00 R3 9.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
The sequence of integers a1,a2,a3,..a_1,a_2,a_3 ,.. such that a1=1a_1 = 1, a2=2a_2 = 2 and for every natural n≥1n \ge 1
an+2={2001an+1−1999an,  if  the  product  an+1anis  an  even  number/an+1−an,  if  the  product  an+1anis  an  odd  numbera_{n+2}=\begin{cases} 2001a_{n+1} - 1999a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,even\,\,number} /\\ a_{n+1}-a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,odd\,\,number} \end{cases}
Is there such a natural mm that am=2000a_m= 2000?
algebranumber theory