MathDB

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Part of 2015 District Olympiad

Problems(7)

Romanian District Olympiad 2015, Grade V, Problem 1

Source:

9/25/2018
Determine all natural numbers ab‾ \overline{ab} with a<b a<b which are equal with the sum of all the natural numbers between a a and b, b, inclusively.
number theoryarithmeticbase 10 arithmetic
Romanian District Olympiad 2015, Grade VI, Problem 1

Source:

9/25/2018
On a blackboard there are written the numbers 11 11 and 13. 13. One step means to sum two written numbers and write it. Show that:
a) after any number of steps, the number 86 86 will not be written. b) after some number of steps, the number 2015 2015 may be written.
arithmeticromania
Romanian District Olympiad 2015, Grade VII, Problem 1

Source:

9/25/2018
a) Show that the number 9−77⋅2⋅(11−7)⋅(9+77) \sqrt{9-\sqrt{77}}\cdot\sqrt {2}\cdot\left(\sqrt{11}-\sqrt{7}\right)\cdot\left( 9+\sqrt{77}\right) is natural.
b) Consider two real numbers x,y x,y such that xy=6 xy=6 and x,y>2. x,y>2. Show that x+y<5. x+y<5.
arithmetic
vectorial geometry

Source: Romanian District Olympiad 2015, Grade IX, Problem 1

9/25/2018
Consider the parallelogram ABCD, ABCD, whose diagonals intersect at O. O. The bisector of the angle ∠DAC \angle DAC and that of ∠DBC \angle DBC intersect each other at T. T. Moreover, TD→+TC→=TO→. \overrightarrow{TD} +\overrightarrow{TC} =\overrightarrow{TO} .
Find the angles of the triangle ABT. ABT.
geometryparallelogramVectors
unorthodox inequality

Source: Romanian District Olympiad 2015, Grade X, Problem 1

9/25/2018
For any n≥2 n\ge 2 natural, show that the following inequality holds: ∑i=2n1(2i)!≥n−12n+2. \sum_{i=2}^n\frac{1}{\sqrt{(2i)!}}\ge\frac{n-1}{2n+2} .
inequalitiesinductionfactorial
epsilon-delta problem

Source: Romanian District Olympiad 2015, Grad XI, Problem 1

9/25/2018
Let f:[0,1]⟶[0,1] f:[0,1]\longrightarrow [0,1] a function with the property that, for all y∈[0,1] y\in [0,1] and ε>0, \varepsilon >0, there exists a x∈[0,1] x\in [0,1] such that ∣f(x)−y∣<ε. |f(x)-y|<\varepsilon .
a) Prove that if f∣[0,1] \left. f\right|_{[0,1]} is continuos, then f f is surjective. b) Give an example of a function with the given property, but which isn´t surjective.
functionreal analysis
modular equation

Source: Romanian District Olympiad 2015, Grade XII, Problem 1

9/26/2018
a) Solve the equation x2−x+2≡0(mod7). x^2-x+2\equiv 0\pmod 7. b) Determine the natural numbers n≥2 n\ge 2 for which the equation x2−x+2≡0(modn) x^2-x+2\equiv 0\pmod n has an unique solution modulo n. n.
abstract algebragroup theorymodular arithmeticnumber theory