MathDB

1

Part of 2002 District Olympiad

Problems(6)

180=x+y+z, x,y,z proportional to 3 consecutive 2002 Romania District VII p1

Source:

8/15/2024
Find the number of representations of the number 180180 in the form 180=x+y+z180 =x+y+z, where x,y,zx, y, z are positive integers that are proportional with some three consecutive positive integers
combinatoricsnumber theory
\sqrt{ prod ( x^2+1/y^2) }=prod (x+y) - 2002 Romania District VIII p1

Source:

8/15/2024
Let x,y,zx, y, z be positive real numbers such that xyz(x+y+z)=1xyz(x+y+z) = 1. Show that the following equality holds: (x2+1y2)(y2+1z2)(z2+1x2)=(x+y)(y+z)(z+x)\sqrt{\left( x^2+\frac{1}{y^2}\right)\left( y^2+\frac{1}{z^2}\right)\left( z^2+\frac{1}{x^2}\right)}=(x+y)(y+z)(z+x) Find some numbers x,y,zx ,y ,z which satisfy the given property.
algebra
a formula involving floors

Source: Districy Olympiad 2002, Grade IX, Problem 1

4/5/2019
Prove the identity \left[ \frac{3+x}{6} \right] -\left[ \frac{4+x}{6} \right] +\left[ \frac{5+x}{6} \right] =\left[ \frac{1+x}{2} \right] -\left[ \frac{1+x}{3} \right] , \forall x\in\mathbb{R} , where [] [] is the integer part.
C. Mortici
algebrafractional part
Find sequence satisfying a long equality

Source: Romanian District Olympiad 2002, Grade X, Problem 1

10/7/2018
Determine the sequence of complex numbers (xn)n≥1 \left( x_n\right)_{n\ge 1} for which 1=x1, 1=x_1, and for any natural number n, n, the following equality is true: 4(x1xn+2x2xn−1+3x3xn−2+⋯+nxnx1)=(1+n)(x1x2+x2x3+⋯+xn−1xn+xnxn+1). 4\left( x_1x_n+2x_2x_{n-1}+3x_3x_{n-2}+\cdots +nx_nx_1\right) =(1+n)\left( x_1x_2+x_2x_3+\cdots +x_{n-1}x_n +x_nx_{n+1}\right) .
Sequencesalgebra
Romania District Olympiad 2002 - Grade XI

Source:

3/18/2011
a) Evaluate
lim⁡n→∞a+a+…+a+b⏟n square roots\lim_{n\to \infty} \underbrace{\sqrt{a+\sqrt{a+\ldots+\sqrt{a+\sqrt{b}}}}}_{n\ \text{square roots}}
with a,b>0a,b>0.
b)Let (an)n≥1(a_n)_{n\ge 1} and (xn)n≥1(x_n)_{n\ge 1} such that an>0a_n>0 and
xn=an+an−1+…+a2+a1, ∀n∈N∗x_n=\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}},\ \forall n\in \mathbb{N}^*
Prove that:
1) (xn)n≥1(x_n)_{n\ge 1} is bounded if and only if (an)n≥1(a_n)_{n\ge 1} is bounded. 2) (xn)n≥1(x_n)_{n\ge 1} is convergent if and only if (an)n≥1(a_n)_{n\ge 1} is convergent.
Valentin Matrosenco
limitinductionreal analysisreal analysis unsolved
Sufficient conditions for certain elements to have finite order

Source: Romanian District Olympiad 2002, Grade XII, Problem 1

10/7/2018
Let A A be a ring, a∈A, a\in A, and let n,k≥2 n,k\ge 2 be two natural numbers such that n⋮char(A) n\vdots\text{char} (A) and 1+a=ak. 1+a=a^k. Show that the following propositions are true:
a) \forall s\in\mathbb{N}  \exists p_0,p_1,\ldots ,p_{k-1}\in\mathbb{Z}_{\ge 0}  a^s=\sum_{i=0}^{k-1} p_ia^{i} . b) ord(a)≠∞. \text{ord} (a)\neq\infty .
superior algebragroup theory