## 5th and 6th Grade, February, 2020

B.1289. Fill in the blanks in the following diagram, so that in every row and every column there would be exactly one 1, 2, 3, 4, and 5.

B.1290. We made the following 8 identical triangles using 16 toothpicks. Take away 4 toothpicks so that the number of remaining identical triangles is:

a) 6

b) 5

c) 4.

Every toothpick that remains on the diagram must belong to at least one of the triangles, and each remaining diagram must contain only triangles.

B.1291. For how many 3-digit numbers are at least one of the following 2 statements true?

– There is a digit 5 at the hundreds place.

– There is a digit 3 at the units place.

B.1292. There are only truth tellers and liars in a village. The truth tellers always tell the truth, the liars always lie. Albert, Ben, Charlie and Dan live in this village. One day they had the following conversation:

Albert: We are all liars.

Ben: You are the only liar amongst us.

Charlie: You are both liars.

Who are the liars and who are the truth tellers in this group?

B.1293. There is a very strange analog clock in my room. It looks like a regular 12-hour clock, but it takes 4 seconds (instead of a minute) for its second hand to go around once. The other two hands follow the movement of the second hand as they do on a regular clock. At exactly 9 am, my strange clock shows 4 hr 20 min 0 seconds. What time does my strange clock show at 9:50 am on the same day?

B.1294. Find the greatest and the smallest positive whole number in which the product of the digits is 120.

B.1295. Find all those 3-digit numbers in which both: the number created from the first two digits, and the number created from the last two digits, are square numbers.

B.1296. Kate added six positive whole numbers, and the sum turned out to be 2019. Then she multiplied the same six numbers, and realized that the last digit of the product is 3. She is sure about her additions. Could her multiplications be all correct, also?