B.1265. Olga is from a country where they write August 2nd as 8/2. She was vacationing in a country where they write that same date as 2/8. (8/2 in that country means February 8th.) She noticed that due to this difference, her arrival date in the foreign country was written there the same way as the date of her return day was written in her home country. How many days could Olga vacation for the most and for the least, if the calendar year did not change during her trip, and she was away for more than a day? (The day of departure and the day of arrival back to her home country are considered to be parts of the trip.)
B.1266. The dancing robot on the following picture is made of 2 kinds of rectangles and 2 kinds of squares. Based on the measurements on the picture, how tall is the robot?
B.1267. By using unit cubes, we glued together a big solid cube with no holes in it. We paint the whole big cube red, and then we cut it up again to unit cubes. We found 72 unit cubes with 2 red sides. How many unit cubes did we use to make the big cube?
B.1268. After the 4th round of a soccer league, Pete’s favorite team scored 3 goals and gave up one goal. We know that for a victory a team gets 3 points, for a tie each team gets 1 point, and no point is awarded for a game lost. How many points could Pete’s team have after 4 rounds? (List all possible numbers of points.)
B.1269. Gluing three unit-cubes together by their sides, we make L-shaped solids. Can you make a 3x3x3 cube using these L-shaped solids only?
B.1270. You divide the 2019-digit number containing all digit 2’s by 3. How many digit 7’s are there in the quotient?
B.1271. Starting at the same time, Peter and Paul are traveling towards each other from the opposite ends of a 30-km-long road. They both travel with a constant speed. Peter would travel the whole distance in 5 hours, Paul would do it in 6 hours. After both of them traveled for 2 hours, Paul sent a text message to Peter saying: “I got tired, I will rest for an hour.” Peter kept on going. How far apart will they be when Paul starts moving again?
B.1272. We build a big solid cube using 27 smaller cubes of identical sizes. Now we take out the center small cube from the top, bottom, left and right facing sides, and then paint the surface of the solid with the remaining 23 cubes red. Now take the big cube apart and count how many small cubes have 0, 1, 2, 3, 4, 5, and 6 red sides!
Please send your solutions here.
The Sharma Kamala Educational Trust is sponsoring the participation of students from India. So, if you are a student living in India, please, send your solutions to: Group B from India