Toán Tiếng Anh:
1. How many boxes are crossed by a diagonal in a rectangular table formed by 199 x 991 small squares?
2. The son of a professor's father is talking to the father of the professor's son, and the professor does not take part in the conversation. Is this possible?
How many boxes are crossed by a diagonal in a rectangular table formed by 199 x 991 small squares?
Another alternative explanation.
Mark the leftmost square crossed of each row as ‘r’, and the topmost crossed square of each column with ‘c’. Thus, each square can be marked either ‘r’ or ‘c’ or ‘both r and c’ or ‘neither r nor c’. We’ll examine each case.
For a square to be marked both ‘r’ and ‘c’, the diagonal must pass through the upper left corner of the square.
For square to be marked ‘r’, diagonal should pass through its upper edge.
For square to be marked ‘c’, diagonal must pass through its left edge.
For square to be marked neither ‘r’ nor ‘c’, diagonal must pass through it’s upper as well as left edge, which is not possible. Therefore, no triangles are unmarked.
Now, no. of squares crossed = no. of squares marked ‘r’ + no. of squares marked ‘c’ - no. of squares marked both ‘r’ and ‘c’
Now, no. of r’s = no. of rows (only 1 leftmost crossed square in each row)
no. of c’s = no. of columns (only 1 topmost crossed square in each column)
all rows and columns are crossed by the diagonal.
Therefore, squares crossed = rows + columns - (no. of squares marked both ‘r’ and ‘c’)
Now, only 1 square is marked both ‘r’ and ‘c’ as 199 and 991 are coprime.
Therefore squares crossed = 199 + 991 - 1 = 1189
Look at this video if you want a clearer visual explanation:
How many boxes are crossed by a diagonal in a rectangular table formed by 199 x 991 small squares?
Tham khảo:
Another alternative explanation.
Mark the leftmost square crossed of each row as ‘r’, and the topmost crossed square of each column with ‘c’. Thus, each square can be marked either ‘r’ or ‘c’ or ‘both r and c’ or ‘neither r nor c’. We’ll examine each case.
For a square to be marked both ‘r’ and ‘c’, the diagonal must pass through the upper left corner of the square.
For square to be marked ‘r’, diagonal should pass through its upper edge.
For square to be marked ‘c’, diagonal must pass through its left edge.
For square to be marked neither ‘r’ nor ‘c’, diagonal must pass through it’s upper as well as left edge, which is not possible. Therefore, no triangles are unmarked.
Now, no. of squares crossed = no. of squares marked ‘r’ + no. of squares marked ‘c’ - no. of squares marked both ‘r’ and ‘c’
Now, no. of r’s = no. of rows (only 1 leftmost crossed square in each row)
no. of c’s = no. of columns (only 1 topmost crossed square in each column)
all rows and columns are crossed by the diagonal.
Therefore, squares crossed = rows + columns - (no. of squares marked both ‘r’ and ‘c’)
Now, only 1 square is marked both ‘r’ and ‘c’ as 199 and 991 are coprime.
Therefore squares crossed = 199 + 991 - 1 = 1189
For a rectangle of size m×nm×n with sizes being coprime:
when a diagonal leaves the starting corner, it goes through the first square;and before it reaches the opposite corner it crosses n−1n−1, say, horizontal lines of the grid, and m−1m−1 vertical lines, each time entering a new square.So the diagonal visits m+n−1m+n−1 unit squares.
For sizes not coprime let d=gcd(m,n)d=gcd(m,n). Then we can reduce the problem to dd rectangles of size md×ndmd×nd which makes a result of
d⋅(md+nd−1)=m+n−dd⋅(md+nd−1)=m+n−d=m+n−gcd(m,n)=m+n−gcd(m,n)
And we can see, that the former result is a special case of the latter: the 'minus one' term is a 'minus GCD of sizes', since a GCD of coprime numbers is 11.
Chúc bạn học tốt nhớ cho mình 1 like nhé !
How many boxes are crossed by a diagonal in a rectangular table formed by 199 x 991 small squares?
Có bao nhiêu ô được gạch chéo bởi một đường chéo trong một bảng hình chữ nhật được tạo thành bởi 199 x 991 ô vuông nhỏ
Giải bằng tiếng việt nha
The son of a professor's father is talking to the father of the professor's son, and the professor does not take part in the conversation. Is this possible?
Có bao nhiêu số có hai chữ số N có các tài sản mà tổng của N và số lượng được hình thành bằng cách đảo ngược các chữ số của N là một hình vuông hoàn hảo?
Các SAMF vuông có diện tích 169. Nó chứa hai qua các quảng trường vỗ. Nhỏ hơn của những sqares có một khu vực mà là một phần tư lớn hơn của họ, và các erea của chồng lên nhau của họ là 4. erea của vùng bóng mờ là gì?
Trong sơ đồ, mỗi số nguyên từ 1 đến 9 là phải được đặt trong một vòng tròn, cả chiều ngang và đường chéo, thêm vào 18. 6 và 1 đã được điền vào như hình vẽ. Xác định giá trị của các số đại diện bởi x trong biểu đồ. Biện minh cho câu trả lời của bạn
Bao nhiêu chữ số không số năm 2013 có?
How many natural numbers not greater than 1665 which are divisible by 5?
Answer: numbers
The sum of age of father and son is 60 years old and son is 30 years younger than father. Father is years old.
bài dưới:
Tuổi cha là:
(60+30):2=45(tuổi)
Đáp số:45 tuổi
tiếng anh là:
Father's age:
(60 + 30): 2 = 45 (years old)
Answer: 45 years old
Read the passage, and choose the correct answer A, B, C or D for each question.
Son Doong Cave
Located in the limestone mountains of Phong Nha - Ke Bang National Park. Son Doong Cave was first found in 1991 by Ho Khanh, a local man. In 2009, the cave was explored and published by a group of scientists from British Cave Research Association.
Son Doong Cave was formed from 2 to 5 million years ago, when the river flowed through the limestone mountain area and was buried. The water eroded limestone and created an underground tunnel. In soft limestone areas, the collapse of tunnel ceiling formed large holes which became giant domes afterward. The cave is about 200m high, 200m wide and at least 8 5km long. Especially, Son Doong consists of two entrances, which is unique among explored caves in Phong Nha - Ke Bang National Park. Exploring the cave, visitors were surprised with spectacular scenery of numerous stalactites of which some giant stalagmites are more than 70 metres high. Deep inside the cave exists a grandiose tropical jungle called “Garden of Eden" by the explorers. The jungle is home to a diversifiedsystem of fauna and flora. Besides, the cave possesses a 2.5-kilometre underground river. Not far from the “Garden of Eden” lies an enormous “pearl collection” consisting of tens of thousands of small stone pieces in dry ponds, contributing to magnificent beauty of the cave.
The reason why Son Doong Cave was formed is that ………..
A. an underground tunnel was buried in the area
B. the limestone areas were soft
C.it only happened from 2 to 5 million years ago
D. the river was buried in the limestone mountains
Đáp án: D
Thông tin: Son Doong Cave was formed from 2 to 5 million years ago, when the river flowed through the limestone mountain area and was buried.
Dịch: Hang Sơn Đoòng được hình thành từ 2 đến 5 triệu năm trước, khi dòng sông chảy qua khu vực núi đá vôi và bị chôn vùi.
Read the passage, and choose the correct answer A, B, C or D for each question.
Son Doong Cave
Located in the limestone mountains of Phong Nha - Ke Bang National Park. Son Doong Cave was first found in 1991 by Ho Khanh, a local man. In 2009, the cave was explored and published by a group of scientists from British Cave Research Association.
Son Doong Cave was formed from 2 to 5 million years ago, when the river flowed through the limestone mountain area and was buried. The water eroded limestone and created an underground tunnel. In soft limestone areas, the collapse of tunnel ceiling formed large holes which became giant domes afterward. The cave is about 200m high, 200m wide and at least 8 5km long. Especially, Son Doong consists of two entrances, which is unique among explored caves in Phong Nha - Ke Bang National Park. Exploring the cave, visitors were surprised with spectacular scenery of numerous stalactites of which some giant stalagmites are more than 70 metres high. Deep inside the cave exists a grandiose tropical jungle called “Garden of Eden" by the explorers. The jungle is home to a diversifiedsystem of fauna and flora. Besides, the cave possesses a 2.5-kilometre underground river. Not far from the “Garden of Eden” lies an enormous “pearl collection” consisting of tens of thousands of small stone pieces in dry ponds, contributing to magnificent beauty of the cave.
The reason why Son Doong Cave was formed is that ………..
A. an underground tunnel was buried in the area
B. the limestone areas were soft
C.it only happened from 2 to 5 million years ago
D. the river was buried in the limestone mountains
Đáp án: D
Giải thích: Thông tin ở đoạn 2: Son Doong Cave was formed from 2 to 5 million years ago, when the river flowed through the limestone mountain area and was buried.
Dịch: Hang Sơn Đoòng được hình thành từ 2 đến 5 triệu năm trước, khi dòng sông chảy qua khu vực núi đá vôi và bị chôn vùi.
