MathDB

P1

Part of 2023 District Olympiad

Problems(4)

Vector geometry with excenter

Source: Romanian District Olympiad 2023 9.1

3/11/2023
Consider the triangle ABCABC{} and let IAI_A{} be its AA{}-excenter. Let M,NM,N and PP{} be the projections of IAI_A{} onto the lines AC,BCAC,BC{} and ABAB{} respectively. Prove that if IAM→+IAP→=IAN→\overrightarrow{I_AM}+\overrightarrow{I_AP}=\overrightarrow{I_AN} then ABCABC{} is an equilateral triangle.
geometryvector geometry
Exponential equation

Source: Romanian District Olympiad 2023 10.1

3/11/2023
Determine all real numbers xx{} satisfying 2x−1+21/x=32^{x-1}+2^{1/\sqrt{x}}=3.
algebraequation
FE with continuity

Source: Romanian District Olympiad 2023 11.1

3/11/2023
Determine all continuous functions f:R→Rf:\mathbb{R}\to\mathbb{R} for which f(1)=ef(1)=e and f(x+y)=e3xy⋅f(x)f(y),f(x+y)=e^{3xy}\cdot f(x)f(y),for all real numbers xx{} and yy{}.
functional equationcontinuityalgebra
Integral inequality

Source: Romanian District Olympiad 2023 12.1

3/11/2023
Let f:[−π/2,π/2]→Rf:[-\pi/2,\pi/2]\to\mathbb{R} be a twice differentiable function which satisfies (f′′(x)−f(x))⋅tan⁡(x)+2f′(x)⩾1,\left(f''(x)-f(x)\right)\cdot\tan(x)+2f'(x)\geqslant 1,for all x∈(−π/2,π/2)x\in(-\pi/2,\pi/2). Prove that ∫−π/2π/2f(x)⋅sin⁡(x) dx⩾π−2.\int_{-\pi/2}^{\pi/2}f(x)\cdot \sin(x) \ dx\geqslant \pi-2.
real analysisIntegralinequalities