B4. a) How many different pizzas can you make if you have four different toppings and each pizza must have at least two toppings?

b) An ice cream shop offers 9 different ice cream flavors. A group of children came to the ice cream shop and each of them bought two scoops. No child chose scoops of the same flavor, and all possible flavor combinations were chosen. How many children bought ice cream?

c) During the Christmas season, Jana ate 100 cookies in five days. Each day she ate 6 more cookies than the previous day. How many cookies did she eat on the first day?

## Solutions to Problems and Scoring

A contestant who arrives at the solution by any correct method (even if not anticipated by the scoring) receives all possible points.

A correct method is one that

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

A contestant who has only partially solved the problem, from otherwise correct steps in solving but does not show the path to the final solution of the problem, cannot receive more than half of the possible points.

## PART I

The correct answers are recorded in the table. A correct answer by the contestant is scored with 2 points, an incorrect answer with -1 point, and a blank field in the table with 0 points.

| Problem | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | A10 |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Answer | $\mathrm{D}$ | $\mathrm{B}$ | $\mathrm{C}$ | $\mathrm{D}$ | $\mathrm{D}$ | $\mathrm{B}$ | $\mathrm{D}$ | $\mathrm{C}$ | $\mathrm{B}$ | $\mathrm{E}$ |
B4. a) We can make 6 pizzas with two toppings, 4 pizzas with three toppings, and 1 pizza with four toppings, totaling 11 pizzas.

b) When choosing the first ice cream flavor, we can still choose from the remaining eight flavors; when choosing the second ice cream flavor, seven flavors remain for selection .... There were $8+7+6+5+4+3+2+1=36$ children in the group.

c) Let $x$ be the number of cookies Jana ate on the first day. We write the equation $x+(x+6)+(x+12)+(x+18)+(x+24)=100$ and solve it. Jana ate $x=8$ cookies on the first day.

Scoring: Total: 5 points

a) We can make 11 different pizzas.

b) 36 children bought ice cream.

c) Jana ate 8 cookies on the first day.