5. Ari is watching several frogs jump straight towards him from the same place. Some frogs are larger, and some are smaller. When the larger frog makes 7 equal jumps, it is still $3 \mathrm{~cm}$ away from Ari. When the smaller frog makes 10 equal jumps, it is still $1 \mathrm{~cm}$ away from Ari. If the frogs are initially more than $2 \mathrm{~m}$ away from Ari, what is the smallest possible length of the smaller frog's jump? The length of each jump in centimeters is a natural number.

No pocket calculators or any reference materials are allowed.

## Ministry of Science and Education of the Republic of Croatia Agency for Education and Education Croatian Mathematical Society

## County Competition in Mathematics February 26, 2024.  5th grade - elementary school

Each task is worth 10 points. In addition to the final result, the procedure is also scored. To earn all points, it is necessary to find all solutions and confirm that there are no others, write down the procedure, and justify your conclusions.
Another solution.

The frogs are more than $2 \mathrm{~m}=200 \mathrm{~cm}$ away from Ari.

$1 \mathrm{~POINT}$

$200: 7=28$ with a remainder of 4.

$7 \cdot 28+3=199$ which is less than 200, so the smallest possible jump length for the larger frogs is $29 \mathrm{~cm}$.

2 POINTS

Let the jump length of the larger frogs be $v$, and the jump length of the smaller frogs be $m$, expressed in centimeters.

The distance of the frog from Ari is $7 \cdot v+3$.

$1 \mathrm{~POINT}$

Let's calculate the possible distances for $v>28$.

Since the distance of the frog from Ari is also equal to $10 \cdot m+1$, we subtract 1 from the obtained distances and divide the difference by 10.

The desired jump length of the smaller frogs $m$ is the first quotient in the sequence without a remainder when divided by 10.

| $v=29$ | $7 \cdot 29+3=206$  $206-1=205$  205 is not a multiple of 10. | $v=30$ | $7 \cdot 30+3=213$  $213-1=212$  212 is not a multiple of 10. | $v=31$ | $7 \cdot 31+3=220$  $220-1=219$  219 is not a multiple of 10. |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $1 \mathrm{~POINT}$ |  | $1 \mathrm{~POINT}$ |  | $1 \mathrm{~POINT}$ |  |
| $v=32$ | $7 \cdot 32+3=227$  $227-1=226$  226 is not a multiple of 10. | $v=33$ | $7 \cdot 33+3=234$  $234-1=233$  233 is not a multiple of 10. | $v=34$ | $7 \cdot 34+3=241$  $241-1=240$  $240: 10=24$  $\boldsymbol{m}=\mathbf{2 4} \mathbf{~ c m}$ |
|  | $1 \mathrm{~POINT}$ |  | $1 \mathrm{~POINT}$ |  | $1 \mathrm{~POINT}$ |

Note: If students do not explain why they start from the number 29, they must check all possibilities for the value of $v$ from 1 to 28 and for this they get 2 POINTS.

TOTAL 10 POINTS

## COUNTY COMPETITION IN MATHEMATICS