MathDB

3

Part of MathLinks Contest 1st

Problems(7)

0113 number theory 1st edition Round 1 p3

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5/9/2021
Let x0=1x_0 = 1 and x1=2003x_1 = 2003 and define the sequence (xn)n≥0(x_n)_{n \ge 0} by: xn+1=xn2+1xn−1x_{n+1} =\frac{x^2_n + 1}{x_{n-1}} , ∀n≥1\forall n \ge 1 Prove that for every n≥2n \ge 2 the denominator of the fraction xnx_n, when xnx_n is expressed in lowest terms is a power of 20032003.
number theory1st edition
0133 number theory 1st edition Round 3 p3

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5/9/2021
Let (Ai)i≥1(A_i)_{i\ge 1} be sequence of sets of two integer numbers, such that no integer is contained in more than one AiA_i and for every AiA_i the sum of its elements is ii. Prove that there are infinitely many values of kk for which one of the elements of AkA_k is greater than 13k/713k/7.
number theory1st edition
0143 geometry 1st edition Round 4 p3

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5/9/2021
Find the triangle of the least area which can cover any triangle with sides not exceeding 11.
geometry1st edition
0123 geo inequality 1st edition Round 2 p3

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5/9/2021
Prove that in any acute triangle with sides a,b,ca, b, c circumscribed in a circle of radius RR the following inequality holds: 24<Rp2aR+bc<12\frac{\sqrt2}{4} <\frac{Rp}{2aR + bc} <\frac{1}{2} where pp represents the semi-perimeter of the triangle.
geometric inequality1st editiongeometryinequalities
0153 inequalities 1st edition Round 5 p3

Source:

5/9/2021
Prove that if the positive reals a,b,ca, b, c have sum 11 then the following inequality holds (ab)54+(bc)54+(ca)54<14.(ab)^{ \frac54} + (bc)^{\frac54} + (ca)^{\frac54} < \frac14 .
inequalities1st edition
0163 Fibonacci 1st edition Round 6 p3

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5/9/2021
Consider (fn)n≥0(f_n)_{n\ge 0} the Fibonacci sequence, defined by f0=0f_0 = 0, f1=1f_1 = 1, fn+1=fn+fn−1f_{n+1} = f_n + f_{n-1} for all positive integers nn. Solve the following equation in positive integers nfnfn+1=(fn+2−1)2.nf_nf_{n+1} = (f_{n+2} - 1)^2. .
number theory1st edition
0173 algebra 1st edition Round 7 p3

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5/9/2021
For a set SS, let ∣S∣|S| denote the number of elements in SS. Let AA be a set of positive integers with ∣A∣=2001|A| = 2001. Prove that there exists a set BB such that all of the following conditions are fulfilled: a) B⊆AB \subseteq A; b) ∣B∣≥668|B| \ge 668; c) for any x,y∈Bx, y \in B we have x+y∉Bx + y \notin B.
algebra1st edition