MathDB

4

Part of 2018 District Olympiad

Problems(6)

Inifinitely Many Increasing Functions

Source: Romanian District Olympiad 2018 - Grade IX - Problem 4

3/10/2018
Let f:R→Rf:\mathbb{R} \to\mathbb{R} be a function. For every a∈Za\in\mathbb{Z} consider the function fa:R→Rf_a : \mathbb{R} \to\mathbb{R}, fa(x)=(x−a)f(x)f_a(x) = (x - a)f(x). Prove that if there exist infinitely many values a∈Za\in\mathbb{Z} for which the functions faf_a are increasing, then the function ff is monotonic.
functionromania
angle bisector, 80-70-30 triangle (2018 Romania District VII p4)

Source:

5/21/2020
Let ABCABC be a triangle with ∠A=80o\angle A = 80^o and ∠C=30o\angle C = 30^o. Consider the point MM inside the triangle ABCABC so that ∠MAC=60o\angle MAC= 60^o and ∠MCA=20o\angle MCA = 20^o. If NN is the intersection of the lines BMBM and ACAC to show that a MNMN is the bisector of the angle ∠AMC\angle AMC.
geometryangle bisectorangles
a_1^2+a_2^2+...+ a_{n-1}^2=a_n^2, a_1 < a_2 < ...< a_{n-1} < a_n

Source: 2018 Romania District VII p4

9/1/2024
a) Consider the positive integers a,b,ca, b, c so that a<b<ca < b < c and a2+b2=c2a^2+b^2 = c^2. If a1=a2a_1 = a^2, a2=aba_2 = ab, a3=bca_3 = bc, a4=c2a_4 = c^2, prove that a12+a22+a32=a42a_1^2+a_2^2+a_3^2=a_4^2 and a1<a2<a3<a4a_1 < a_2 < a_3 < a_4.
b) Show that for any n∈Nn \in N, n≥3n\ge 3, there exist the positive integers a1,a2,...,ana_1, a_2,..., a_n so that a12+a22+...+an−12=an2a_1^2+a_2^2+...+ a_{n-1}^2=a_n^2 and a1<a2<...<an−1<ana_1 < a_2 < ...< a_{n-1} < a_n
number theoryinequalities
Complex Numbers and Two Constraints

Source: Romanian District Olympiad 2018 - Grade X - Problem 4

3/10/2018
Let n≥2n\ge 2 be a natural number. Find all complex numbers zz which simultaneously satisfy the relations
a) zn+zn−1+…+z2+∣z∣=n;\text{a)}\ z^n + z^{n - 1} + \ldots + z^2 + |z| = n;
b) ∣z∣n−1+∣z∣n−2+…+∣z∣2+z=nzn.\text{b)}\ |z|^{n- 1} + |z|^{n - 2} + \ldots + |z|^2 + z = n z^n.
complex numbers
Continuous Function on Open Interval

Source: Romanian District Olympiad 2018 - Grade XI - Problem 4

3/10/2018
Let a<ba < b be real numbers and let f:(a,b)→Rf : (a, b) \to \mathbb{R} be a function such that the functions g:(a,b)→Rg : (a, b) \to \mathbb{R}, g(x)=(x−a)f(x)g(x) = (x - a) f(x) and h:(a,b)→Rh : (a, b) \to \mathbb{R}, h(x)=(x−b)f(x)h(x) = (x - b) f(x) are increasing. Show that the function ff is continuous on (a,b)(a, b).
functioncontinuityreal analysis
Finite Field with $q \ne 1 (mod 4)$ Elements

Source: Romanian District Olympiad 2018 - Grade XII - Problem 4

3/10/2018
Let nn and qq be two natural numbers, n≥2n\ge 2, q≥2q\ge 2 and q≢1(mod 4)q\not\equiv 1 (\text{mod}\ 4) and let KK be a finite field which has exactly qq elements. Show that for any element aa from KK, there exist xx and yy in KK such that a=x2n+y2na = x^{2^n} + y^{2^n}. (Every finite field is commutative).
finite fieldssuperior algebra