MathDB

4

Part of 2016 Romania National Olympiad

Problems(6)

All functions f with f(a^2) - f(b^2) \leq (f(a) + b)(a - f(b))

Source: France JBMO TST 2017, Exam 1, Problem 1

6/25/2018
Determine all functions f:R→Rf: \mathbb R \to \mathbb R which satisfy the inequality f(a2)−f(b2)≤(f(a)+b)(a−f(b)),f(a^2) - f(b^2) \leq \left( f(a) + b\right)\left( a - f(b)\right), for all a,b∈Ra,b \in \mathbb R.
functioninequalities
isosceles and perpendicular, right isosceles (2016 Romanian NMO grade VII P4)

Source:

6/1/2020
Consider the isosceles right triangle ABCABC, with ∠A=90o\angle A = 90^o and the point M∈(BC)M \in (BC) such that ∠AMB=75o\angle AMB = 75^o. On the inner bisector of the angle MACMAC take a point FF such that BF=ABBF = AB. Prove that:
a) the lines AMAM and BFBF are perpendicular; b) the triangle CFMCFM is isosceles.
geometryisoscelesperpendicularanglesright triangle
x_1x_2 ...x_n = x_1 + 2x_2 + 3x_3 +...+ nx_n

Source: 2016 Romanian NMO grade VIII P4

9/3/2024
For n∈N∗n \in N^* we will say that the non-negative integers x1,x2,...,xnx_1, x_2, ... , x_n have property (P)(P) if x1x2...xn=x1+2x2+3x3+...+nxn.x_1x_2 ...x_n = x_1 + 2x_2 + 3x_3 + ...+ nx_n.
a) Show that for every n∈N∗n \in N^* there exists nn positive integers with property (P)(P).
b) Find all integers n≥2n \ge 2 so that there exists nn positive integers x1,x2,...,xnx_1, x_2, ... , x_n with x1<x2<x3<...<xnx_1< x_2<x_3< ... <x_n, having property (P)(P).
number theory
Num. of words of a language in func. of an alphabet, their length and a property

Source: Romanian National Olympiad, grade x, p. 4

8/25/2019
In order to study a certain ancient language, some researchers formatted its discovered words into expressions formed by concatenating letters from an alphabet containing only two letters. Along the study, they noticed that any two distinct words whose formatted expressions have an equal number of letters, greater than 2, 2, differ by at least three letters. Prove that if their observation holds indeed, then the number of formatted expressions that have n≥3 n\ge 3 letters is at most [2nn+1]. \left[ \frac{2^n}{n+1} \right] .
combinatoricscountingdiscrete maths
Romanian National Olympiad 2016 (correction)

Source: Romanian National Olympiad 2016, grade 12, problem 4

12/29/2023
Let KK be a finite field with qq elements, q≥3.q \ge 3. We denote by MM the set of polynomials in K[X]K[X] of degree q−2q-2 whose coefficients are nonzero and pairwise distinct. Find the number of polynomials in MM that have q−2q-2 distinct roots in K.K.
Marian Andronache
abstract algebraField theory
all f s.t. (f&sup2;)'=f

Source: Romanian National Olympiad 2016, grade xi, p.1

8/25/2019
Find all functions, f:R⟶R, f:\mathbb{R}\longrightarrow\mathbb{R} , that have the properties that f2 f^2 is differentiable and f=(f2)′. f=\left( f^2 \right)' .
real analysisFind all functionsdifferentiabilityDarbouxfunction