## 7th and 8th Grade, March, 2019

C.1241. In an isosceles triangle the angle between the two congruent sides is 120 degrees. Draw the bisecting perpendicular of both of the congruent sides so that they would intercept the base of the triangle. What size of pieces will they cut the 24 cm long base into?

C.1242. Can you find such x and y whole numbers for which xy(x+y) = 15015?

C.1243. In a TV Game Show the show host poses questions to 10 people, who each have to push a button (all at the same time when they are asked to do so) representing (in their opinion) the correct answer. Those who answered the question correctly will get as many points for it as the number of people who did not answer this question correctly. After the first 5 questions, the 10 players all together received a total of 116 points, out of which Rebecca received 30 points. How many questions did Rebecca answer correctly?

C.1244. What is the greatest 2-digit number that can be written as the sum of 2 different primes in only 2 different ways?

C.1245. A 3-digit number has all different digits. Using these digits, we create all the possible 2-digit numbers with different digits. The sum of all these 2-digit numbers is the original 3-digit number. Find all such 3-digit numbers.

C.1246. A Math teacher has two classes. One has 16 students, the other class has 11 students in it. Both groups have their Math classes in the same class room. The school bought 20 calculators for the two groups, so that they all could use them. The teacher gave some of the calculators to individual students, who can take them home, but have to bring them to every Math class. The rest of the calculators stay in the class room, and those who did not receive their own calculators will get one for each Math class. (Only one student may use the same calculator in class.)

a) How many students may get a calculator to take home in each class if their teacher wants to give a calculator to the same number of students in each class, so that there would be enough left over in the class room so that everybody could use one in class?

b) How many students can get a calculator in each class if the teacher does not mind that different number of students got one in each class?

C.1247. The 23.6 cm long shorter diagonal of a quadrilateral is perpendicular to the longer diagonal, and bisects it. Each angle between the longer diagonal and the sides of the quadrilateral is 30 degrees. How many cm is the perimeter of this quadrilateral?

C.1248. In a store they put some 1.5-liter sodas on sale for \$1.20. They put a sticker on each of the bottles on sale that says: “+20% for free”. What was the regular price of a bottle of these sodas? (The price of the soda is directly proportional to the quantity of the soda.)