Problems for 5th grade

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In this issue of Quaent we
We are limiting the publication of the most interesting
tasks from the new trial training
nickname for 5th grade.

1233. Out of 8 coins, one is counterfeit
      (easier). How to identify false
      sewing a coin by two weighers
      on a scale with two cups without a scale?

1234. Among 20 coins there is one
      fake, different in weight
      from real ones. Using three weights
      sewing on scales with cups without
      weights to detect a counterfeit coin
      and establish whether it is lighter or heavier than the real ones.

1235. I only have wall ones
      the hours that are left. I’m from-
      I’m going to see my friend, cha-
      whose sons walk faithfully, being
      he has some time and, having returned-
      When I get home, I set my watch correctly.
      How did I manage to do this if I didn’t first
      it was known how much time
      is the road long?

1236. For word page numbering
      6869 digits were required.

How many pages does the dictionary have?

1238. What is the divisor of the number
      3⁶·5⁴?

1239. When dividing the number N by 2 in
      the remainder is 1, and when dividing
      3 leaves a remainder of 2. Which one?
      the remainder will be obtained when dividing the number
      N for 6?

1240. If from the planned three-valued
      If you subtract 7 from a given number, then it will be divided by 7, but if you subtract 8 from it, then it will be divided by 8, and if you subtract 9, then it will be divided by 9. What number was intended?

1241. What digit is the number 66⁶⁶? 33³³? 7⁷?

1242. Prove that the number 7⁷⁷+1 is not divisible by 5.

1243. Prove that the sum of any two consecutive odd numbers is divisible by 4.

1244. Some six-digit numbers begin with the number 7. By dropping this number next and adding it to the right, we get a number that is 5 times less than the first initial one. Find the original number.

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1251. If you enter a zero between the digits of a two-digit number, then the resulting three-digit number will be 9 times larger than the first by 7, 11 and 13.

1252. In a six-digit number, the first digit coincides with the fourth, the second with the fifth and the third with the sixth. Prove that this number is divisible by 7, 11 and 13.

1253. Add one digit to the right and left of the number 43 so that the number is divisible by 45.

1254. Can the sum of four consecutive natural numbers be a prime number? What about the sum of three?

1255. Pupils A, B and C lined up in a 100m race. When A completed his run, B was 10m behind him. When B completed his run, C was 10m behind him. How many meters was C behind A when A finished his run?

1256. Show that any amount of money can be paid if the seller and buyer have only three- and five-kopeck coins.

1257. The father instructed his son to measure the length of the yard in steps. It was winter. To check, the father measured the same length of the yard with his steps. The father walked from the same place as the son and walked in the same direction, so that in some cases the tracks of father and son coincided. There are 61 footprints in the snow. What is the length of the yard if the son’s step is 0.54 m and the father’s step is 0.72 m?

1258. Prove theorems:

1. If the product of three integers is an odd number, then their sum is an odd number;
2. If the sum of two integers is an odd number, then their product is an even number.

1268. The edges of a right parallelepidea are expressed by natural numbers, and its volume is expressed by a prime number greater than 2. Prove that the sum of the lengths of two unequal edges is an even number.

1269. Can there exist a right paralleleped whose edge lengths are natural numbers and whose surface area is a prime number?

1270. Divide 7 apples into 12 people equally, without cutting any apple into 12 or more equal parts.

1271. Reduce fractions:
      $$\frac{37\ 373\ 737}{81\ 818\ 181}; \quad \frac{609\ 609\ 609}{205\ 205\ 205}.$$

1272. Find the meaning of the expression:
      $$\frac{1}{2\cdot3} + \frac{1}{3\cdot4} + \frac{1}{5\cdot6} + \dots + \frac{1}{19\cdot20}.$$

1273. Prove that
      $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots + \frac{1}{199} - \frac{1}{200} = \frac{1}{101} + \frac{1}{102} + \frac{1}{103} + \dots + \frac{1}{200}.$$

1274. Find the smallest natural number that, when divided by each of the fractions $\frac{28}{297}$ and $\frac{35}{396}$, produces integers.

1275. The train covers this distance in 10 numbers. If he increased his speed by 10 km/h, he would travel 8 numbers. Find the speed and distance.

1276. The car drove half the way at a speed of 50 km/h, and the second half at a speed of 30 km/h. At what average speed did the automatic travel the entire distance?

1277. From Gorky to Astrakhan the shuttle takes 5 days, and back 7 days. How many days does it take to travel from Gorky to Astrakhan?

1278. I drank $1/6$ a cup of black coffee and topped it up with milk. Then I drank $1/3$ cups and topped it up with milk again. Then I drank a cup of it. Did I drink more coffee or milk?

1279. The cook was preparing soup and putting little salt in it. I had to get fed up-

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cook the finished soup. The next time the cook recognized this mistake and added 2 times more salt to the same amount of soup than the first time. This also turned out to be not enough, but it was enough with half the amount of salt. How much salt did the cook put into the soup for the first time?

1283. Senya paid the canteen cash register for 3 dishes, and Sasha for 2 (all dishes were stacked equally). Kostya joined them at the table, and they were the second to eat these 5 dishes. When calculating, it turned out that Kostya had to pay his comrades 50 kopecks. How much money should Senya receive from this amount, and how much should Sasha?

1284. Front wheel purchases are automatically repaired after 25,000 km, and rear wheels after 15,000 km. When is it advisable to swap purchases so that they cost the same? (Assume that the front and rear purchases were replaced only once, although in practice drivers make this replacement more often.)

1285. Several chocolate lovers had breakfast together and had to pay 1 ruble. 75 kop. It turned out that the two did not have money with them, so each of the remaining ones had to pay 10 kopecks. more than everyone had to share. How many chocolate lovers had breakfast?

1286. The sum of three additions is equal to 1. The difference between the first and second additions is equal to the third. The sum of the first two dobies is 5 times greater than the third dobie. Find these dobies.

1287. The difference of two non-negative additions is $2/9$. The numeral of one dobi is 4 times greater than the other, and the denominator of this dobi is 3 times greater than the other. Find the dobie.

1288. Several natural numbers are given. Each of them was divided by the sum of all given numbers. What is the sum of all quotients?

1289. I thought of a three-digit number. If you make up all possible two-digit numbers from the digits of this number and then add them, then the required number is equal to the intended number. Find the target number.

1290. Mom bought apples for her children Vanya, Nina and Misha. The children had to divide the apples equally among themselves. Vanya came home first, counted the apples, took the third part and left. Then Nina came and, believing that she had arrived first, counted the remaining apples, took her share and left the same way. Finally, Misha came and took himself a third of the remaining apples. After all this, there are 8 apples left. How many apples does mother have for children?

1291. Two brothers would need to go to the railway station, which is located at a distance of 4 km from their house. In order not to be late for the train, there was only one option left - to ride a bicycle. But the older brothers have a certain weight. If you walk, you will be 10 minutes late. However, both brothers got to the station at the same time and 10 minutes before the train departed. Tell me, what did they do if walking is slower than riding a bicycle and if two people cannot ride a bicycle?

1292. What number has such a property that if you add an arbitrary number to it, then subtract the same arbitrary number from the product and divide the resulting difference by it, the result will be the same arbitrary number taken?