MathDB

1

Part of 2016 District Olympiad

Problems(6)

Diophantine: x+y=\sqrt x+\sqrt y+\sqrt{xy}

Source: Romanian District Olympiad 2016, Grade VII, Problem 1

10/4/2018
Solve in N2: \mathbb{N}^2: x+y=x+y+xy. x+y=\sqrt x+\sqrt y+\sqrt{xy} .
equationsDiophantine equationalgebra
pyramid with perpendicular opposite faces

Source: Romanian District Olympiad, Grade VIII, Problem 1

10/4/2018
Let be a pyramid having a square as its base and the projection of the top vertex to the base is the center of the square. Prove that two opposite faces are perpendicular if and only if the angle between two adjacent faces is 120∘. 120^{\circ } .
geometry3D geometrypyramid
i hope I translated it ok (didn´t solve it)

Source: Romanian District Olympiad 2016, Grade IX, Problem 1

10/4/2018
Let ABCD ABCD be a sqare and E E be a point situated on the segment BD, BD, but not on the mid. Denote by H H and K K the orthocenters of ABE, ABE, respectively, ADE. ADE. Show that BH→=KD→. \overrightarrow{BH}=\overrightarrow{KD} .
geometryvector
1=\cos\left( \pi\log_3 (x+6)\right)\cdot\cos\left( \pi\log_3 (x-2)\right)

Source: Romanian District Olympiad 2016, Grade X, Problem 1

10/4/2018
Solve in the interval (2,∞) (2,\infty ) the following equation: 1=cos⁡(πlog⁡3(x+6))⋅cos⁡(πlog⁡3(x−2)). 1=\cos\left( \pi\log_3 (x+6)\right)\cdot\cos\left( \pi\log_3 (x-2)\right) .
equationalgebralogarithms
|A²+A+I|=|A²-A+I|=3 implies A²(A²+I)=2I

Source: Romanian District Olympiad 2016, Grade XI, Problem 1

10/5/2018
Let A∈M2(C) A\in M_2\left( \mathbb{C}\right) such that det⁡(A2+A+I2)=det⁡(A2−A+I2)=3. \det \left( A^2+A+I_2\right) =\det \left( A^2-A+I_2\right) =3. Prove that A2(A2+I2)=2I2. A^2\left( A^2+I_2\right) =2I_2.
linear algebraalgebraMatrices
Another rings with isomorphism between its multiplicative and additive groups

Source: Romanian District Olympiad 2016, Grade XII, Problem 1

10/5/2018
A ring A A has property (P), if A A is finite and there exists ({0}≠R,+)≤(A,+) (\{ 0\}\neq R,+)\le (A,+) such that (U(A),⋅)≅(R,+). (U(A),\cdot )\cong (R,+) . Show that:
a) If a ring has property (P), then, the number of its elements is even. b) There are infinitely many rings of distinct order that have property (P).
superior algebraRing Theoryabstract algebragroup theoryFTA