MathDB

4

Part of 2023 OMpD

Problems(3)

x^n+y^n and x^m+y^m integer implies x+y integer

Source: 2023 4th OMpD L3 P4 - Brazil - Olimpíada Matemáticos por Diversão

9/21/2023
Are there integers m,n≥2m, n \geq 2 such that the following property is always true? ‘‘For any real numbers x,y, if xm+ym and xn+yn are integers, then x+y is an integer".``\text{For any real numbers } x, y, \text{ if } x^m + y^m \text{ and } x^n + y^n \text{ are integers, then } x + y \text{ is an integer}".
analysisIntegerreal numberalgebra
Cute geometry problem

Source: 2023 4th OMpD L2 P4 - Brazil - Olimpíada Matemáticos por Diversão

9/21/2023
Let ABCABC be a scalene acute triangle with circumcenter OO. Let KK be a point on the side BC‾\overline{BC}. Define MM as the second intersection of OK↔\overleftrightarrow{OK} with the circumcircle of BOCBOC. Let LL be the reflection of KK by AC↔\overleftrightarrow{AC}. Show that the circumcircles of the triangles LCMLCM and ABCABC are tangent if, and only if, AK‾⊥BC‾\overline{AK} \perp \overline{BC}.
geometrycircumcircletangenttangent circles
Integral inequality

Source: 2023 4th OMpD LU P4 - Brazil - Olimpíada Matemáticos por Diversão

9/21/2023
Let n≥0n \geq 0 be an integer and f:[0,1]→Rf: [0, 1] \rightarrow \mathbb{R} an integrable function such that: ∫01f(x)dx=∫01xf(x)dx=∫01x2f(x)dx=…=∫01xnf(x)dx=1\int^1_0f(x)dx = \int^1_0xf(x)dx = \int^1_0x^2f(x)dx = \ldots = \int^1_0x^nf(x)dx = 1 Prove that: ∫01f(x)2dx≥(n+1)2\int_0^1f(x)^2dx \geq (n+1)^2
calculusinequalitiesintegrationfunction