of positive integers, such that the sequence defined by 
, 
and 
for 
has only finite number of composite terms.[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.22.4"][et_pb_column type="4_4" _builder_version="3.22.4"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.23.3" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Bulgarian Mathematical Olympiad 2002[/et_pb_accordion_item][et_pb_accordion_item title="Topic" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Number theory[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Medium[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Problem Solving Strategies by Arthur Engel[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]The sequence cannot be decreasing because it is a sequence of positive integers. Hence there exists (a smallest) $latex k$ such that $latex a_k\le a_{k+1}$. If $latex a_k=a_{k+1}$ then the sequence becomes constant from the $latex k$th term onwards (we shall treat this case later). Otherwise $latex 3a_{k+1}-2a_k>a_k$ hence $latex a_{k+2}>a_{k+1}$. This implies that the sequence becomes increasing from the $latex k$th term onwards. Also, $latex a_{n+2}=3a_{n+1}-2a_n$ for $latex n\ge k$. This difference equation has the characteristic equation $latex \lambda^2-3\lambda +2=0$ (see the link in hint 1) which has the solutions $latex \lambda = 2,1$. Thus, $latex a_{n+k}=2^nA+B$ for $latex A,B$ satisfying $latex A+B=a_k, 2A+B=a_{k+1}$. Take any prime divisor $latex p$ of $latex A+B$. By Fermat's little theorem, $latex 2^{m(p-1)} \equiv 1 \; (\text{mod}\; p)$ for every positive integer $latex m$. Thus $latex 2^{m(p-1)}A+B\equiv A+B\equiv 0\; (\text{mod}\; p)$. Hence the sequence contains infinitely many composites. This cannot be allowed, so the sequence cannot be strictly increasing at any point.
The above discussion shows that, for any permissible sequence, there exists a (smallest) $latex j$ and a prime $latex q$ such that $latex a_n=q$ for all $latex n\ge j$. For $latex n<j$, the sequence is decreasing. Note that, either $latex q=a_{j+1}=3a_j-2a_{j-1}=3q-2a_{j-1}$ or $latex q =2a_{j-1}-3q$. Hence, either $latex a_{j-1}=q$ or $latex a_{j-1}=2q$. The first one can happen only if $latex j=1$ because otherwise the minimality of $latex j$ is violated. In that case, the sequence is constant and $latex b=c=q$. If $latex a_{j-1}=2q$ then either $latex j=2$ (in which case $latex b=2q, c=q$) or $latex q=|6q-2a_{j-2}|$ hence $latex a_{j-2}=3q\pm\frac{q}{2}$. The last equality forces $latex q$ to be 2. Thus $latex a_{j-2}=6\pm 1$. If $latex j>3$ then $latex 4=a_{j-1}=|18\pm 3 - 2a_{j-3}|$ which is absurd as $latex 2a_{j-3}$ cannot be an odd number. Hence $latex j=3$ in this case and $latex c=4,b=5,7$.
[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.[/et_pb_blurb][et_pb_button button_url="https://cheenta.com/matholympiad/" url_new_window="on" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3" background_layout="dark"][/et_pb_button][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px"]
