MathDB

2

Part of 2003 District Olympiad

Problems(6)

PM = DN,MN // BC ,AH_|_BC, right triangle (2003 Romania District VII P2)

Source:

5/24/2020
In the right triangle ABCABC ( ∠A=90o\angle A = 90^o), DD is the intersection of the bisector of the angle AA with the side (BC)(BC), and PP and QQ are the projections of the point DD on the sides (AB),(AC)(AB),(AC) respectively . If BQ∩DP={M}BQ \cap DP=\{M\}, CP∩DQ={N}CP \cap DQ=\{N\}, BQ∩CP={H}BQ\cap CP=\{H\}, show that:
a) PM=DNPM = DN b) MN∥BCMN \parallel BC c) AH⊥BCAH \perp BC.
geometryequal segmentsparallelperpendicularright triangle
function f has property P, f(x) = ax + b

Source: 2003 Romania District VIII p2

8/15/2024
Let M⊂RM \subset R be a finite set containing at least two elements. We say that the function ff has property PP if f:M→Mf : M \to M and there are a∈R∗a \in R^* and b∈Rb \in R such that f(x)=ax+bf(x) = ax + b. (a) Show that there is at least a function having property PP. (b) Show that there are at most two functions having property PP. (c) If MM has 20032003 elements with sum 00 and if there are two functions with property PP, prove that 0∈M0 \in M.
algebra
Digits

Source: RMO 2003, District Round

5/29/2006
Find n∈N\displaystyle n \in \mathbb N, n≥2\displaystyle n \geq 2, and the digits a1,a2,…,an\displaystyle a_1,a_2,\ldots,a_n such that a1a2…an‾−a1a2…an−1‾=an. \displaystyle \sqrt{\overline{a_1 a_2 \ldots a_n}} - \sqrt{\overline{a_1 a_2 \ldots a_{n-1}}} = a_n .
quadraticsalgebraquadratic formula
Famous recurrence

Source: RMO 2003, District Round

5/29/2006
Find all functions f:N∗→M\displaystyle f : \mathbb N^\ast \to M such that 1+f(n)f(n+1)=2n2(f(n+1)−f(n)), ∀n∈N∗, \displaystyle 1 + f(n) f(n+1) = 2 n^2 \left( f(n+1) - f(n) \right), \, \forall n \in \mathbb N^\ast , in each of the following situations: (a) M=N\displaystyle M = \mathbb N; (b) M=Q\displaystyle M = \mathbb Q. Dinu Şerbănescu
functionalgebra proposedalgebra
Romania District Olympiad 2003 - Grade XI

Source:

3/18/2011
Let f:[0,1]→[0,1]f:[0,1]\rightarrow [0,1] a continuous function in 00 and in 11, which has one-side limits in any point and f(x−0)≤f(x)≤f(x+0), (∀)x∈(0,1)f(x-0)\le f(x)\le f(x+0),\ (\forall)x\in (0,1). Prove that:
a)for the set A={x∈[0,1] ∣ f(x)≥x}A=\{x\in [0,1]\ |\ f(x)\ge x\}, we have sup⁡A∈A\sup A\in A. b)there is x0∈[0,1]x_0\in [0,1] such that f(x0)=x0f(x_0)=x_0.
Mihai Piticari
functionreal analysisreal analysis unsolved
Sequence of integral of ratio between two powers of functions

Source: Romanian District Olympiad 2002, Grade XII, Problem 2

10/7/2018
Let be two distinct continuous functions f,g:[0,1]⟶(0,∞) f,g:[0,1]\longrightarrow (0,\infty ) corelated by the equality ∫01f(x)dx=∫01g(x)dx, \int_0^1 f(x)dx =\int_0^1 g(x)dx , and define the sequence (xn)n≥0 \left( x_n \right)_{n\ge 0} as xn=∫01(f(x))n+1(g(x))ndx. x_n=\int_0^1 \frac{\left( f(x) \right)^{n+1}}{\left( g(x) \right)^n} dx .
a) Show that ∞=lim⁡n→∞xn. \infty =\lim_{n\to\infty} x_n. b) Demonstrate that the sequence (xn)n≥0 \left( x_n \right)_{n\ge 0} is monotone.
functionSequencesIntegralcalculusintegrationratio