## Subject III. (20 points)

At a training session, an athlete climbs a set of stairs according to the rule: he climbs 4 steps and then descends 1 step, then climbs 5 steps and descends 3 steps, and then starts over.

a) On which step is the athlete after 2014 steps?

b) After how many steps does he reach step 2014?

(Consider a step as either climbing or descending one step)

Prof. Vasile Şerdean, Şcoala Gimnazială nr. 1 Gherla
## Subject III.

The rule of climbing requires $(4+1)+(5+3)=13$ steps.

In 13 steps, the athlete advances $(4-1)+(5-3)=5$ steps.
a) $2014=13 \cdot 154+12$, the athlete applies the climbing rule 154 times, reaching step 770 and has 12 steps left to perform (climbs 4 steps and descends 1 step, climbs 5 steps and descends 2 steps), advancing 6 more steps. After 2014 steps, he is on step 776.
b) $2014=5 \cdot 402+4$, the athlete reaches step 2010 after $402 \cdot 13=5226$ steps. With 4 more steps needed, he reaches step 2014 after 5230 steps.