# Task 5. Maximum 20 points

In the city of Eifyadl, runic stones are sold. It is known that the first merchant offers a fixed discount of $\mathrm{n} \%$ for every 5th stone purchased, while the second merchant increases the discount by $1 \%$ for each subsequent stone purchased (0% for the 1st stone, 3% for the 4th stone, and so on), but not exceeding $20 \%$. An order for 100 runic stones has been placed. Assume that all stones must be purchased from only one merchant.

a) Find the minimum discount rate for the first merchant so that it is more advantageous for the buyer to purchase all the stones from him.

b) What is the maximum number of runic stones that would be more advantageous to buy from the first merchant with the discount found in part a)?

c) Suppose now that the buyer has increased the order to 175 stones. Upon learning this, the second merchant increased the discount limit to $30 \%$ and the discount increment to $2 \%$. Can the first merchant compete with the second?
# Solution and Evaluation Criteria:

a) Let's assume the cost of one rune stone without a discount is 1 unit of currency. We find the average cost of a rune stone from the first merchant:

$(20(1-n)+80) / 100=(100-20 n) / 100$

For the second merchant:

$(1+0.99+0.98+\ldots+0.81+0.8 * 80) / 100=82.1 / 100$

Then:

$100-20 n>82.1$

$17.9 / 20 \quad n>0.895$

Thus, $n=90\%$

Answer: $90\%$.

## Evaluation Criteria:

Logically justified solution - 5 points, logically justified solution with arithmetic error - 0 points

b) We find the average cost of a rune stone from the second merchant:

$(1+0.98+\ldots+0.72+0.7 * 159) / 175=125.06 / 175$

And from the first merchant:

$(35 *(1-n)+140) / 175=(175-35 n) / 175$

Then:

$125.06>175-35 n$

$n>(175-125.06) / 35$

$n>49.94 / 35$

$n>1.427$, which is impossible, as the maximum discount is $100\%$, and paying extra for the purchase of a stone significantly affects the merchant's costs. Answer - no, $n>100\%$

## Evaluation Criteria:

Logically justified solution - 7 points,

justified solution with arithmetic error - 3.5 points, incorrect answer - 0 points

c) Let the number of stones be X. We write the general equation for the average cost of a stone from the first and second merchants:

$1: (x / 5 * (1-0.9) + (x - x / 5)) / x$

$2: (18.1 + 0.8 * (x - 20)) / x$

We get:

$0.1 x / 5 + x - x / 5 < 18.1 + 0.8 * (x - 20)$

$x - 0.18 x - 0.8 x < 2.1$

$0.02 x < 2.1$

$x < 105$

Then: $x=104$ stones.

Answer: 104.

Evaluation Criteria:

Logically justified solution - 8 points,

solution with arithmetic error - 0 points, unsolved part a) - 0 points.