MathDB

4

Part of 2009 District Olympiad

Problems(6)

Natural functional inverse-like and symmetric functional equality

Source: Romanian District Olympiad 2009, Grade IX, Problem 4

10/7/2018
Fin the functions f:N⟶N f:\mathbb{N}\longrightarrow\mathbb{N} such that: \frac{f(x+y)+f(x)}{2x+f(y)} =\frac{2y+f(x)}{f(x+y)+f(y)} , \forall x,y\in\mathbb{N} .
functionalgebrafunctional equation
angle chasing, isosceles, equilateral (2009 Romania District VII P4)

Source:

5/19/2020
Let ABCABC be an equilateral ABCABC. Points M,N,PM, N, P are located on the sides AC,AB,BCAC, AB, BC, respectively, such that ∠CBM=12∠AMN=13∠BNP\angle CBM= \frac{1}{2} \angle AMN = \frac{1}{3} \angle BNP and ∠CMP=90o\angle CMP = 90 ^o. a) Show that △NMB\vartriangle NMB is isosceles. b) Determine ∠CBM\angle CBM.
anglesgeometryisoscelesEquilateral
(a^2- 9b^2)^2 - 33b = 1 diophantine

Source: 2009 Romania District VIII p4

8/16/2024
Positive integer numbers a and b satisfy (a2−9b2)2−33b=1(a^2- 9b^2)^2 - 33b = 1. a) Prove ∣a−3b∣≥1|a -3b|\ge 1. b) Find all pairs of positive integers (a,b)(a, b) satisfying the equality.
number theoryDiophantine equation
Subpoint a) is canonic; roots of unity that don’t resemble an equilateral triang

Source: Romanian District Olympiad 2009, Grade X, Problem 4

10/8/2018
a) Let z1,z2,z3 z_1,z_2,z_3 be three complex numbers of same absolute value, and 0=z1+z2+z3. 0=z_1+z_2+z_3. Show that these represent the affixes of an equilateral triangle.
b) Find all subsets formed by roots of the same unity that have the property that any three elements of every such, doesn’t represent the vertices of an equilateral triangle.
complex numbersgeometryabsolute value
Romania District Olympiad 2009 - Grade XI

Source:

4/10/2011
a) Prove that the function F:R→R, F(x)=2⌊x⌋−cos⁡(3π{x})F:\mathbb{R}\rightarrow \mathbb{R},\ F(x)=2\lfloor x\rfloor-\cos(3\pi\{x\}) is continuous over R\mathbb{R} and for any y∈Ry\in \mathbb{R}, the equation F(x)=yF(x)=y has exactly three solutions.
b) Let kk a positive even integer. Prove that there is no function f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} such that ff is continuous over R\mathbb{R} and that for any y∈Im fy\in \text{Im}\ f, the equation f(x)=yf(x)=y has exactly kk solutions (Im f=f(R))(\text{Im}\ f=f(\mathbb{R})).
functionfloor functionreal analysisreal analysis unsolved
Romanian District Olympiad 2009 - Grade 12 - Problem 4

Source: Romanian District Olympiad 2009 - Grade 12 - Problem 4

8/17/2024
Let KK be a finite field with qq elements and let n≥qn \ge q be an integer. Find the probability that by choosing an nn-th degree polynomial with coefficients in K,K, it doesn't have any root in K.K.
superior algebraPolynomialsfinite fieldprobability