Bài 1. Cho [You must be registered and logged in to see this image.] là một hàm số liên tục. Một điểm [You must be registered and logged in to see this image.] được gọi là điểm đen nếu tồn tại [You must be registered and logged in to see this image.] với [You must be registered and logged in to see this image.] sao cho [You must be registered and logged in to see this image.]. Cho [You must be registered and logged in to see this image.] là các số thực sao cho
• Tất cả các điểm thuộc khoảng [You must be registered and logged in to see this image.] là điểm đen.
• [You must be registered and logged in to see this image.] không phải là điểm đen.
Chứng minh rằng
• a. [You must be registered and logged in to see this image.];
• b. [You must be registered and logged in to see this image.].
Problem 1. Let [You must be registered and logged in to see this image.] be a continuous function. A point [You must be registered and logged in to see this image.] is called a shadow point if there is a point [You must be registered and logged in to see this image.] with [You must be registered and logged in to see this image.] such that [You must be registered and logged in to see this image.]. Let [You must be registered and logged in to see this image.] be real numbers and suppose that
• all points in [You must be registered and logged in to see this image.] are shadow points;
• [You must be registered and logged in to see this image.] are not shadow points.
Prove that
• a) [You must be registered and logged in to see this image.];
• b) [You must be registered and logged in to see this image.].
Bài 2. Tồn tại hay không một ma trận thực [You must be registered and logged in to see this image.] cấp [You must be registered and logged in to see this image.] sao cho [You must be registered and logged in to see this image.] và [You must be registered and logged in to see this image.]?
Problem 2. Does there exist a real [You must be registered and logged in to see this image.] matrix [You must be registered and logged in to see this image.] such that [You must be registered and logged in to see this image.] and [You must be registered and logged in to see this image.]?
Bài 3. Cho [You must be registered and logged in to see this image.] là một số nguyên tố. Ta gọi số nguyên [You must be registered and logged in to see this image.] là tốt nếu [You must be registered and logged in to see this image.] với [You must be registered and logged in to see this image.].
• a) Chứng minh rằng số [You must be registered and logged in to see this image.] là tốt.
• b) Tìm [You must be registered and logged in to see this image.] sao cho [You must be registered and logged in to see this image.] là số tốt nhỏ nhất.
Problem 3. Let [You must be registered and logged in to see this image.] be a prime number. Call a positive integer [You must be registered and logged in to see this image.] interesting if
[You must be registered and logged in to see this image.] for some polynomials [You must be registered and logged in to see this image.]..
• a) Prove that the number [You must be registered and logged in to see this image.] is interesting.
• b) For which [You must be registered and logged in to see this image.] is [You must be registered and logged in to see this image.] the minimal interesting number?
Bài 4. Cho [You must be registered and logged in to see this image.] là những tập hợp hữu hạn khác rỗng. Ta xác định
[You must be registered and logged in to see this image.]
Chứng minh rằng [You must be registered and logged in to see this image.] là không giảm trên [0,1].

Problem 4. Let [You must be registered and logged in to see this image.] be finite, nonempty sets. Define the function
[You must be registered and logged in to see this image.]
Prove that [You must be registered and logged in to see this image.] is nondecreasing on [0,1].
Bài 5. Cho [You must be registered and logged in to see this image.] là số nguyên dương và [You must be registered and logged in to see this image.] là một không gian vectơ [You must be registered and logged in to see this image.]-chiều trên trường chỉ có hai phần tử. Chứng minh rằng với mọi vecto [You must be registered and logged in to see this image.], luôn tồn tại một dãy [You must be registered and logged in to see this image.] sao cho [You must be registered and logged in to see this image.].

Problem 5. Let [You must be registered and logged in to see this image.] be a positive integer and let [You must be registered and logged in to see this image.] be a [You must be registered and logged in to see this image.]-dimensional vector space over the field with two elements. Prove that for arbitrary vectors [You must be registered and logged in to see this image.], there exists a sequence [You must be registered and logged in to see this image.] of indices such that [You must be registered and logged in to see this image.].

Nguồn [You must be registered and logged in to see this link.]