Question 7 of the 4th International Mathematical Olympiad? 100

This forward proof is quite simple, using symmetry, directly find the distance from the point to the straight line, and the distance is equal and tangent at the same time. On the other hand, choose a face first, the three sides tangent spherical projection is the heart, the heart pushes out the inscribed projection point on the angular bisector, the tangent projection point, and the upper and bottom points are three points collinear, and at the same time in the vertical plane of the bottom surface. In the same way, if you change a side, it is also a three-point collinear in the vertical plane.

It can be launched with an inscribed spherical center and an inscribed spherical center, and the upper and bottom points are three-point collinear and the intersection of two vertical surfaces. Then take the top and bottom points as the endpoints and take out the three edges, and the inscribed ball and the inscribed ball have three intersections with these three edges respectively, and form two triangles. Since the first three points are collinear, it can be proved that the two triangles are similar and parallel.

The projection is made on this parallel plane, and the inscribed and inscribed spherical projection is externally centered and coincides. According to the three-point collinear, the upper vertex projection is also outward. It has also been proved that the side inner is connected, so the inner projection of the side and the outer center are bisectoral, and the two sides of the angular bisector are similar, so the five hearts are one.

Then it is concluded that the five hearts are one, and the new triangle where the projection plane is located is an equilateral triangle. It is concluded that the three sides of the tetrahedron with the same vertex are equal in pairs. Then take a projection of one side of the original tetrahedron, and take two inscribed sphere projections at the same time.

The two inscribed spheres are projected on the bisector of the two angles, and the two angles are equal and colateral, so the tangent points of the two spheres coincide and the radius is the same. Then take the triangle formed by the non-common side and the tangent point to be similar, and push out the two adjacent sides of the projection surface equally. So regular tetrahedron.

I didn't read the answer, and I don't know if it's right or not.

Key question: 1. The middle of the 5 spheres is tangent to every edge of each tetrahedron, so the ball intersects all four faces of the tetrahedron in a circle, and this circle is an inscribed circle of each face.

2. Each of the other four spheres, except for the middle sphere, is tangent to three edges of the tetrahedron and tangent to the extension line of the other three edges, and this sphere intersects one of the faces of the tetrahedron in a circle, which is the inscribed circle of the triangle; Three circles intersect the planes of the other three faces, i.e., three side-tangent circles.

3. Transform the three-dimensional three-dimensional figure into a two-dimensional figure, consider the inscribed circle and the side-cut circle in these two-dimensional, mainly with the help of the tangent length formula, and can find that all edges of the tetrahedron are equal in length, that is, the tetrahedron is a regular tetrahedron.

Based on my three tips, and then study the answer carefully, you should understand.

In 1989 and 2008, the Chinese team won the first prize in the International Olympiad. This is one of the few awards won by the Chinese team, and in order to obtain such a competition ranking, the participating team members have put in a lot of hard work and sacrifice in exchange for a collective honor.

In the international Olympiad, the world's top athletes are gathered, and everyone hopes to be able to win the honor in such competitions, not only for the affirmation of their own ability, but more importantly, for the national honor of the country. Because in such a competition, it is not only the ability of a player to test, but also the level of a country's education, so many players have paid a lot for this goal, but the Chinese team members rely on their own talent and their continuous hard work, so they can have such a good ranking.

Looking at the development of the country's education career, I think it is very difficult for the country to have such a level of education. Because the country's education started relatively late, especially for the construction of talents and technology in higher mathematics, which has always belonged to the stage of exploration and exploration, the country can send a team to enter the International Mathematical Olympiad and win the first prize of the team, which is enough to show that the country has achieved initial results in the development of education. I hope that everyone can think about education, which will bring great changes to our lives, and at the same time, we must also realize the importance of learning, only in this way can we better develop ourselves steadily, otherwise we will let ourselves become idle because of wasted time.

Summary: The Mathematics Olympiad is not only a test for a person, but more importantly, it has a good enlightening effect on a country's education, because while popularizing basic education, our country should also develop high-quality education, only in this way can we communicate with the world's education powers on an equal footing, learn from each other's strengths and weaknesses and find their own shortcomings.

In 2008, the Chinese team won the first place in the total team score with 217 points, and the Chinese team gave birth to 2 super players, and 6 players won gold medals, among which Chinese players Mu Xiaosheng and Wei Dongyi won full scores in their respective events.

In 2008. The 49th International Mathematical Olympiad was held in Madrid, Spain, with 549 students from 103 countries participating, and the Chinese team won the first place in the total team score with 217 points.

In the 49th International Mathematical Olympiad in 2008, the Chinese team won the first prize, with a total of three perfect scores and five gold medals.

Suppose the race is equal to 4, Ao can only be an even number 2, and the gram can only be equal to 8, and then set the forest = x, horse = y, then there is 4 (10x + y) + 3 = 10y + x, simplified to 13y-2x + 1 = 0, easy to solve x = 7, y = 1So it's 2178*4=8712

To sum up, it can only be 2178*4=8712

According to international practice, the World Mathematical Olympiad in China is held once a year for children aged 10 to 16, that is, seven grade groups from the third grade of primary school to the third grade of junior high school. The nature of the competition is a social welfare activity, the purpose of the activity is to cultivate the enthusiasm and interest of the majority of children to learn mathematics and love mathematics, and the activity is organized into three parts:

1. The regional competitions mainly reflect the wide participation, and encourage and stimulate the interest of most participating students in learning mathematics through a wide range of award ratios, so as to realize the wide social significance of the competition.

2. The annual global finals mainly reflect the high-end elite selection of the event, and gather the players with excellent results in the competitions in the sub-divisions across the country for competition, display, and other related exchange activities. Contestants who have won the bronze medal or above will be eligible to form a representative pair of China to participate in the World Finals of the World Mathematical Olympiad to show themselves and win glory for the country.

Summary. The National Mathematics Olympiad is run by local education bureaus, including primary, junior and senior high schools. It's not the same as the National Junior High School Mathematics Competition.

National Junior High School Mathematics Competition:

Is the Mathematics Olympiad a nationwide competition?

The National Mathematics Olympiad is run by local education bureaus, including primary, junior and senior high schools. It's not the same as the National Junior High School Mathematics Competition. National Junior High School Sou Hao Mathematics Competition:

Is the 90th grade Mathematics Olympiad a national competition?

Yes, nationwide.

After so many years, Yula will never forget the skater for the rest of his life, only remember that the difficulty is quite high, and the invigilator also believes that the false potato has emphasized many times: It is not ordinary difficulty, it is difficult, especially special difficulty.

National,The difficulty must be high.,This Olympiad competition is the lowest is national.,Generally the former rock is an international competition.,This can win the award.,Mathematics can't be wise.,It's very honorable.。

I was young at the time, I didn't understand, yes, I must have been very simple-minded at that time.

In the '90s, many years had passed.

I really can't forget that Mathematics Olympiad.

It's okay to be able to participate, and it's good to remember such an unforgettable thing all the time