[3rd grade].
Fill in the eight numbers 1 8 in the diagram on the right, so that the sum of the five numbers on the two great circles is equal to 21.
Analysis] The middle two numbers are overlapping numbers, and the number of overlaps is 1 time, so the sum of the two overlapping numbers is.
Of the eight known numbers, the sum of two numbers is 6, only 1 and 5, and 2 and 4. The sum of the other three numbers on each great circle is 21-6=15.
If the two overlapping numbers are 1 and 5, then the remaining six numbers 2, 3, 4, 6, 7, and 8 are divided into two groups, and the sum of the three numbers in each group is 15.
2+6+7=15 and 3+4+8=15, so there is a filling method in the lower left figure.
If the two overlapping numbers are 2 and 4, then the same can be found in the upper right figure.
4th grade] TV station should ** a 30-episode TV series, if the number of episodes scheduled to be broadcast each day is not equal, the TV series can be broadcast for a maximum of a few days
Analysis] Since it is desirable to broadcast as many days as possible, the number of episodes per day should be as small as possible under the condition that the number of episodes broadcast per day is not equal to each other.
We know that 1+2+3+4+5+6+7=28. If the number of episodes aired on each day is 1, 2, 3, 4, 5, 6, and 7, then a total of 28 episodes can be aired in seven days, and there are 2 episodes left unaired. Since there have been cases where two episodes have been broadcast in one day, the remaining two episodes can no longer be broadcast on a separate day, but they have to be divided into previous days and the problem can be solved by changing the number of episodes broadcast on one or two days. For example, it is okay to schedule the number of episodes to be broadcast on each day as 1, 2, 3, 4, 5, 7, 8 or 1, 2, 3, 4, 5, 6, 9.
So it can be broadcast for up to 7 days.
Grade 5] The ABCD in the figure is rectangular, and the area of the triangle EFD is 6 square centimeters larger than the area of the triangle ABF, and the length of the ED is found.
Analysis] Because the area of the triangle EFD is 6 square centimeters larger than the area of the triangle ABF, the area of the triangle BCE is 6 square centimeters larger than the area of the rectangular ABCD. The area of the triangular BCE is 6 4 6 = 30 square centimeters, and the length of the EC is 30 2 6 = 10 centimeters. Therefore, the length of the ed is 10 4 = 6 cm.
Grade 6] As shown in the figure, the triangle ABC is a right triangle, DE is parallel to AC, and it is known that BD=16 cm, CF=10 cm, so what is the area of the triangle BEF?
Analysis].
Primary School Mathematics Olympiad