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2

Part of MathLinks Contest 3rd

Problems(7)

0342 number theory with sequence 3rd edition Round 4 p2

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5/9/2021
The sequence {xn}n≥1\{x_n\}_{n\ge1} is defined by x1=7x_1 = 7, xn+1=2xn2−1x_{n+1} = 2x^2_n - 1, for all positive integers nn. Prove that for all positive integers nn the number xnx_n cannot be divisible by 20032003.
number theory3rd edition
0312 inequalities 3rd edition Round 1 p2

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5/9/2021
Prove that for all positive reals a,b,ca, b, c the following double inequality holds: a+b+c3≥(a+b)(b+c)(c+a)83≥ab+bcca3\frac{a+b+c}{3}\ge \sqrt[3]{\frac{(a+b)(b+c)(c+a)}{8}}\ge \frac{\sqrt{ab}+\sqrt{bc}\sqrt{ca}}{3}
3rd editioninequalitiesalgebra
0322 geo inequality 3rd edition Round 2 p2

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5/9/2021
Let ABCABC be a triangle with semiperimeter ss and inradius rr. The semicircles with diameters BC,CA,ABBC, CA, AB are drawn on the outside of the triangle ABCABC. The circle tangent to all three semicircles has radius tt. Prove that s2<t≤s2+(1−32)r.\frac{s}{2} < t \le \frac{s}{2} + \left( 1 - \frac{\sqrt3}{2} \right)r.
inequalities3rd edition
0332 inequalities 3rd edition Round 3 p2

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5/9/2021
Let n be a positive integer and let a1,a2,...,an,b1,b2,...,bn,c2,c3,...,c2na_1, a_2, ..., a_n, b_1, b_2, ... , b_n, c_2, c_3, ... , c_{2n} be 4n−14n - 1 positive real numbers such that ci+j2≥aibjc^2_{i+j} \ge a_ib_j , for all 1≤i,j≤n1 \le i, j \le n. Also let m=max⁡2≤i≤2ncim = \max_{2 \le i\le 2n} c_i. Prove that (m+c2+c3+...+c2n2n)2≥(a1+a2+...+ann)(b1+b2+...+bnn)\left(\frac{m + c_2 + c_3 +... + c_{2n}}{2n} \right)^2 \ge \left(\frac{a_1+a_2 + ... +a_n}{n}\right)\left(\frac{ b_1 + b_2 + ...+ b_n}{n}\right)
inequalitiesalgebra3rd edition
0352 floor function sums 3rd edition Round 5 p2

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5/9/2021
Let k≥1k \ge 1 be an integer and a1,a2,...,ak,b1,b2,...,bka_1, a_2, ... , a_k, b1, b_2, ..., b_k rational numbers with the property that for any irrational numbers xi>1x_i >1, i=1,2,...,ki = 1, 2, ..., k, there exist the positive integers n1,n2,...,nk,m1,m2,...,mkn_1, n_2, ... , n_k, m_1, m_2, ..., m_k such that a1⌊x1n1⌋+a2⌊x2n2⌋+...+ak⌊xknk⌋=b1⌊x1m1⌋+21⌊x2m2⌋+...+bk⌊xkmk⌋a_1\lfloor x^{n_1}_1\rfloor + a_2 \lfloor x^{n_2}_2\rfloor + ...+ a_k\lfloor x^{n_k}_k\rfloor=b_1\lfloor x^{m_1}_1\rfloor +2_1\lfloor x^{m_2}_2\rfloor+...+b_k\lfloor x^{m_k}_k\rfloor Prove that ai=bia_i = b_i for all i=1,2,...,ki = 1, 2, ... , k.
algebrafloor function3rd edition
0372 functional in Z 3rd edition Round 7 p2

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5/9/2021
Find all functions f:{1,2,...,n,...}→Zf : \{1, 2, ... , n,...\} \to Z with the following properties (i) if a,ba, b are positive integers and a∣ba | b, then f(a)≥f(b)f(a) \ge f(b); (ii) if a,ba, b are positive integers then f(ab)+f(a2+b2)=f(a)+f(b)f(ab) + f(a^2 + b^2) = f(a) + f(b).
functionalfunctional equation3rd editionalgebranumber theory
0362 number theory 3rd edition Round 6 p2

Source:

5/9/2021
Let a1,a2,...,a2004a_1, a_2, ..., a_{2004} be integer numbers such that for all positive integers nn the number An=a1n+a2n+...+a2004nA_n = a^n_1 + a^n_2 + ...+ a^n_{2004} is a perfect square. What is the minimal number of zeros within the 20042004 numbers?
number theory3rd edition