Chính phương và chính phương mod p


Một số chính phương đương nhiên là chính phương modulo mọi số nguyên tố, nhưng ngược lại có đúng không? Dưới đây là một số link liên quan:

1) http://mathoverflow.net/questions/135820/does-there-exist-a-non-square-number-which-is-the-quadratic-residue-of-every-pri

2) http://www.artofproblemsolving.com/community/c6h64322

3) http://artofproblemsolving.com/community/c6h388566p2158819

4) http://www.artofproblemsolving.com/community/c7h34935p239819

 

Real roots of a polynomial


Cách đây vài tháng tôi có đưa lên blog này một số bài toán về nghiệm thực của đa thức, cụ thể ở các link sau:

Dưới đây là file pdf tổng hợp các bài trên sau khi nhận được sự góp ý từ các bạn đồng nghiệp và các em học sinh.

Continue reading “Real roots of a polynomial”

Lagrange’s interpolation polynomial (4)


Part 3’s link https://nttuan.org/2016/11/14/topic-833/


Problem 17. Prove that, for infinitely many positive integers n, there exists a polynomial P of degree n with real coefficients such that P(1),P(2),\cdots, P(n+2) are different whole powers of 2.

Problem 18. Suppose q_{0}, \, q_{1}, \, q_{2}, \ldots \; \, is an infinite sequence of integers satisfying the following two conditions:

(i)  m-n \, divides q_{m}-q_{n} for m > n \geq 0,

(ii) there is a polynomial P such that |q_{n}| < P(n) \, for all n

Prove that there is a polynomial Q such that q_{n}= Q(n) for all n.

Problem 19. Let P\in\mathbb{R}[x] such that for infty of integer number c : Equation P(x)=c has more than one integer root. Prove that P(x)=Q((x-a)^{2}), where a\in\mathbb R and Q is a polynomial.

Problem 20. Find all the polynomials P(x) with odd degree such that

P(x^{2}-2)=P^{2}(x)-2.

Problem 21. Suppose p(x) is a polynomial  with integer coefficients assumes at n distinct integral values of x that are different form 0 and in absolute value less than  \dfrac{(n-[\frac n2])!}{2^{n-[\frac n2]}} . Prove that p(x) is irreducible.

Prove that the bound may be replaced by (\dfrac d2)^{n-[\frac n2]}(n-[\frac n2])! where d is minimum distance between any two of the n integral values of x where p(x) assumes the integral values considered.

Continue reading “Lagrange’s interpolation polynomial (4)”

Lagrange’s interpolation polynomial (3)


Part 2’s link https://nttuan.org/2016/11/13/topic-832/


Problem 9. Let t and n be fixed integers each at least 2. Find the largest positive integer m for which there exists a polynomial P, of degree n and with rational coefficients, such that the following property holds: exactly one of  \displaystyle\frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}}  is an integer for each k = 0,1, ..., m.

Problem 10. Let f\left ( x \right )=x^{n}+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+...+a_{1}x+a_{0} be a polynomial. Prove that we have an \displaystyle i\in \left \{ 1,2,...,n \right \}\mid  \left | f\left ( i \right ) \right |\geq \frac{n!}{\binom{n}{i}}.

Problem 11.  Let (F_n)_{n\geq 1} be the Fibonacci sequence F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1), and P(x) the polynomial of degree 990 satisfying

P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982. Prove that P(1983) = F_{1983} - 1.

Continue reading “Lagrange’s interpolation polynomial (3)”

Lagrange’s interpolation polynomial (2)


Part 1’s link https://nttuan.org/2016/01/01/topic-733/


Problem 1. Let P be a polynomial of degree at most n satisfying \displaystyle P(k)=\frac{1}{C^k_{n+1}}\,\,\forall k=\overline{0,n}. Determine P(n+1).

Problem 2. A polynomial P(x) has degree at most 2k, where k = 0, 1,2,\cdots. Given that for an integer i, the inequality -k \le i \le k implies |P(i)| \le 1, prove that for all real numbers x, with -k \le x \le k, the following inequality holds |P(x)| \leq 2^{2k}.

Problems 3. Prove that at least one of the numbers |f(1)|,|f(2)|,\cdots, |f(n+1)| is greater than or equal to \dfrac{n!}{2n}. Where

f(x) = x^n + a_1x^{n-1} + \cdots+ a_n\,\,  (\quad a_i \in  \mathbb R, \quad  i = 1, \ldots , n,n\in\mathbb{N}.)

Problem 4. Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree n with real coefficients is the average of two monic polynomials of degree n with n real roots.

Problem 5. Let p be a prime number and f an integer polynomial of degree d such that f(0) = 0,f(1) = 1 and f(n) is congruent to 0 or 1 modulo p for every integer n. Prove that d\geq p - 1.

Problem 6. Let P be a polynomial of degree n\in\mathbb{N} satisfying P(k)=2^k\,\,\forall k=\overline{0,n}. Prove that P(n+1)=2^{n+1}-1.

Problem 7.  P(x) is a polynomial of degree 3n\,\, (n\in\mathbb{N}) such that

P(0) = P(3) = \cdots = P(3n) = 2,\,\,\, P(1) = P(4) = \cdots = P(3n-2) = 1,

P(2) = P(5) = \cdots = P(3n-1) = 0, \quad\text{and}\quad P(3n+1) = 730.

Determine n.

Continue reading “Lagrange’s interpolation polynomial (2)”

VMO training 2017 – Part 1


Đây là một số bài toán tôi dùng trong đợt luyện VMO 2017,  gửi các đồng nghiệp tham khảo.

Bài 1. Tìm tất cả các cặp số nguyên dương (a, b) sao cho 7^a - 3^b chia hết a^4 + b^2.

Bài 2. Cho số nguyên dương k. Chứng minh rằng có vô hạn số chính phương dạng n\cdot 2^k - 7, ở đây n là số nguyên dương.

Bài 3. Tìm tất cả các cặp số nguyên dương (m,n) sao cho \dfrac {n^3+1}{mn-1} là số nguyên.

Bài 4. Cho a,b là các số nguyên dương lẻ sao cho 2ab+1 \mid a^2 + b^2 + 1. Chứng minh a=b.

Continue reading “VMO training 2017 – Part 1”