Bài giảng toán học

Đáp án đề thi toán Hà nội mở rộng (HOMC) năm học 2016-2017 dành cho lớp 8 và lớp 10

Giáo viên: Trung tâm Edufly

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Dưới đây là một số câu


Lớp 8. 

Question 5. Let a,b,c be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number a + b, and the sum of all digits of b + c are all equal to 2. The largest value of a + b + c is

(A): 1099 (B): 2099 (C): 1199 (D): 2199 (E): None of the above.
Solution. The choice is (E).

Lớp 10.

Question 9. Cut off a square carton by a straight line into two pieces, then cut one of two pieces into two small pieces by a straight line, ect. By cutting 2017 times we obtain 2018 pieces. We write number 2 in every triangle, number 1 in every quadrilateral, and 0 in the polygons. Is the sum of all inserted numbers always greater than 2017? 

Solution. After 2017 cuts, we obtain 2018 n-convex polygons with n ≥ 3. After each cut the total of all sides of those n-convex polygons increases at most 4. We deduce that the total number of sides of 2018 pieces is not greater than 4×2018. If the side of a piece is kj, then the number inserted on it is greater or equal to 5−kj. Therefore, the total of all inserted numbers on the pieces is greater or equal to X j (5−kj) = 5×2018−X j kj ≥ 5×2018−4×2018 = 2018 > 2017. The answer is positive.


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