MathDB

4

Part of 2013 District Olympiad

Problems(6)

AE = BD, <BAE =15^o inside square (2013 Romania District VII P4)

Source:

5/20/2020
Consider the square ABCDABCD and the point EE inside the angle CABCAB, such that ∠BAE=15o\angle BAE =15^o, and the lines BEBE and BDBD are perpendicular. Prove that AE=BDAE = BD.
geometrysquareequal segments
x_nx_{n+1} \le 2(x_1 + x_2 + ... + x_n) NT inequality

Source: 2013 Romania District VIII p4

9/1/2024
For a given a positive integer nn, find all integers x1,x2,...,xnx_1, x_2,... , x_n subject to 0<x1<x2<...<xn<xn+10 < x_1 < x_2 < ...< x_n < x_{n+1} and xnxn+1≤2(x1+x2+...+xn).x_nx_{n+1} \le 2(x_1 + x_2 + ... + x_n).
number theoryinequalitiesalgebra
cos

Source: Romania District Olympiad 2013,grade X(problem 4)

3/14/2013
Let n∈N∗n\in {{\mathbb{N}}^{*}}. Prove that 22ncos⁡(narccos⁡24)2\sqrt{{{2}^{n}}}\cos \left( n\arccos \frac{\sqrt{2}}{4} \right) is an odd integer.
trigonometrymodular arithmeticinductionalgebra proposedalgebra
problem

Source: Romania District Olympiad 2013,grade IX(problem 4)

3/14/2013
At the top of a piece of paper is written a list of distinctive natural numbers. To continue the list you must choose 2 numbers from the existent ones and write in the list the least common multiple of them, on the condition that it isn’t written yet. We can say that the list is closed if there are no other solutions left (for example, the list 2, 3, 4, 6 closes right after we add 12). Which is the maximum numbers which can be written on a list that had closed, if the list had at the beginning 10 numbers?
least common multiplealgebra proposedalgebra
continuous

Source: Romania District Olympiad 2013,grade XI(problem 4)

3/14/2013
Letf:R→Rf:\mathbb{R}\to \mathbb{R}be a monotone function. a) Prove thatff have side limits in each point x0∈R{{x}_{0}}\in \mathbb{R}. b) We define the function g:R→Rg:\mathbb{R}\to \mathbb{R}, g(x)=lim⁡t↗x f(t)g\left( x \right)=\underset{t\nearrow x}{\mathop{\lim }}\,f\left( t \right)( g(x)g\left( x \right) with limit at at left in xx). Prove that if the gg function is continuous, than the function ff is continuous.
functionlimitcalculuscalculus computations
ring problem

Source: Romania District Olympiad 2013,grade XII(problem 4)

3/14/2013
Problem 4. Let(A,+,⋅)\left( A,+,\cdot \right) be a ring with the property that x=0x=0 is the only solution of the x2=0,x∈A{{x}^{2}}=0,x\in Aecuation. Let B={a∈A∣a2=1}B=\left\{ a\in A|{{a}^{2}}=1 \right\}. Prove that: (a) ab−ba=bab−aab-ba=bab-a, whatever would be a∈Aa\in A and b∈Bb\in B. (b) (B,⋅)\left( B,\cdot \right) is a group
superior algebrasuperior algebra unsolved