MathDB

2

Part of 2017 District Olympiad

Problems(6)

two variables Diophantine with natural parameter

Source: Romanian District Olympiad 2012, Grade VII, Problem 2

10/9/2018
Let E(x,y)=xy+x+1y+1+x+2y+2. E(x,y)=\frac{x}{y} +\frac{x+1}{y+1} +\frac{x+2}{y+2} .
a) Solve in N2 \mathbb{N}^2 the equation E(x,y)=3. E(x,y)=3. b) Show that there are infinitely many natural numbers n n such that the equation E(x,y)=n E(x,y)=n has at least one solution in N2. \mathbb{N}^2.
diophantineequationsparameterization
diagonals of cube perpendicular to line joining midpoints of some sides

Source: Romanian District Olympiad 2017, Grade VIII, Problem 2

10/10/2018
Let ABCDA’B’C’D’ ABCDA’B’C’D’ a cube. M,P M,P are the midpoints of AB, AB, respectively, DD’. DD’.
a) Show that MP,A’C MP, A’C are perpendicular, but not coplanar. b) Calculate the distance between the lines above.
geometry3D geometry
two functions satisfying three properties

Source: Romanian District Ollympiad 2017, Grade XI, Problem 2

10/10/2018
a) Prove that there exist two functions f,g:R⟶R f,g:\mathbb{R}\longrightarrow\mathbb{R} having the properties: \text{(i)}  f\circ g=g\circ f \text{(ii)}  f\circ f=g\circ g \text{(iii)}  f(x)\neq g(x),   \forall x\in\mathbb{R}
b) Show that if there are two functions f1,g1:R⟶R f_1,g_1:\mathbb{R}\longrightarrow\mathbb{R} with the properties (i) \text{(i)} and (iii) \text{(iii)} from above, then (f1∘f1)(x)≠(g1∘g1)(x), \left( f_1\circ f_1\right)(x) \neq \left( g_1\circ g_1 \right)(x) , for all real numbers x. x.
function
mediator is the perpendicular through the midpoint

Source: Romanian District Olympiad 2017, Grade IX, Problem 2

10/10/2018
Let ABC ABC be a triangle in which O,I, O,I, are the circumcenter, respectively, incenter. The mediators of IA,IB,IC, IA,IB,IC, form a triangle A1B1C1. A_1B_1C_1. Show that OI→=OA1→+OA2→+OA3→. \overrightarrow{OI}=\overrightarrow{OA_1} +\overrightarrow{OA_2} +\overrightarrow{OA_3} .
geometrycircumcircleincentervectorial geometry
symmetric system: power of two + logarith in base 3 = perfect square

Source: Romanian District Olympiad 2017, Grade X, Problem 2

10/10/2018
Solve in Z \mathbb{Z} the system: {2x+log⁡3x=y22y+log⁡3y=x2. \left\{ \begin{matrix} 2^x+\log_3 x=y^2 \\ 2^y+\log_3 y=x^2 \end{matrix} \right. .
Systemequationslogarithms
x->x^{m+1},x^{n+1} are surjective and in End(G,G), then G abelian

Source: Romanian District Olympiad 2017, Grade XII, Problem 2

10/11/2018
Let be a group and two coprime natural numbers m,n. m,n. Show that if the applications G∋x↦xm+1,xn+1 G\ni x\mapsto x^{m+1},x^{n+1} are surjective endomorphisms, then the group is commutative.
group theorymorphismsabstract algebrasuperior algebra