Track / Stage 5 / 191 of 400 #791 of 2000

Problem 791

AIME late

Number theory Difficulty 5.5 Find the answer

7. A frog is located in the plane at the point with coordinates $(0,0)$ . It can make three different types of jumps:

- from $(x, y)$ to $(x, y-5)$ ;
- from $(x, y)$ to $(x-2, y+3)$ ;
- from $(x, y)$ to $(x+4, y+9)$ .

Further away, there are three juicy morsels that the frog would like to eat: a worm at $(2013,2018)$ , a beetle at $(2018,2019)$ , and a snail at $(2018,2023)$ .

Which of these morsels can the frog reach?
A) the worm and the snail
B) the beetle and the snail
D) only the beetle
E) only the snail
C) the worm and the beetle

Your answer

Official solution

A7. E) only the snail Regardless of the jumps the frog makes, the $x$ -coordinate of the frog always remains even. A jump of type 1 does not change the $x$ -coordinate at all, and a jump of type 2 or 3 adds -2 or 4, respectively. The frog cannot reach the worm. If we now look at $x-y$ , we see that for the points the frog can reach, it is always a multiple of five. Indeed, at the beginning, $x-y=0$ , and with the three types of jumps, $x-y$ changes by $0-(-5)=5$ , $-2-3=-5$ , and $4-9=-5$ , respectively. The beetle cannot be reached either.

The snail can be reached. It is useful to combine a jump of type 1 and type 3. Together, they move the frog from position $(x, y)$ to $(x+4, y+4)$ . By performing this combination 505 times, the frog lands at position $(2020,2020)$ . Finally, by making one jump of type 2, the frog ends up at $(2018,2023)$ , where the snail is.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.