MathDB

3

Part of 2008 District Olympiad

Problems(6)

School and friends

Source: Romanian DMO 7th grade p3

3/1/2008
In a school there are 10 10 rooms. Each student from a room knows exactly one student from each one of the other 9 9 rooms. Prove that the rooms have the same number of students (we suppose that if A A knows B B then B B knows A A).
pigeonhole principlecombinatorics proposedcombinatorics
collinear in 3D wanted, cube related, perpendicular on plane

Source: 2008 Romania District VIII P3, Gazeta Matematica, 2007

5/18/2020
Let ABCDA′B′C′D′ABCDA' B' C' D ' be a cube , MM the foot of the perpendicular from AA on the plane (A′CD)(A'CD), NN the foot of the perpendicular from BB on the diagonal A′CA'C and PP is symmetric of the point DD with respect to CC. Show that the points M,N,PM, N, P are collinear.
geometry3D geometrycollinearperpendicularSymmetric
Fraction is power of two

Source: Romanian District MO 2008, Grade 9, Problem 3

4/30/2008
Prove that if n≥4 n\geq 4, n∈Z n\in\mathbb Z and ⌊2nn⌋ \left \lfloor \frac {2^n}{n} \right\rfloor is a power of 2, then n n is also a power of 2.
floor functionnumber theory proposednumber theory
Romania District Olympiad 2008 - Grade XI

Source:

4/10/2011
Let (xn)n≥1(x_n)_{n\ge 1} and (yn)n≥1(y_n)_{n\ge 1} a sequence of positive real numbers, such that:
xn+1≥xn+yn2, yn+1≥xn2+yn22, (∀)n∈N∗x_{n+1}\ge \frac{x_n+y_n}{2},\ y_{n+1}\ge \sqrt{\frac{x_n^2+y_n^2}{2}},\ (\forall)n\in \mathbb{N}^*
a) Prove that the sequences (xn+yn)n≥1(x_n+y_n)_{n\ge 1} and (xnyn)n≥1(x_ny_n)_{n\ge 1} have limit.
b) Prove that the sequences (xn)n≥1(x_n)_{n\ge 1} and (yn)n≥1(y_n)_{n\ge 1} have limit and that their limits are equal.
real analysisreal analysis unsolved
Nice divisibility in an odd ring

Source: RMO 2008 - District Round - 12th grade - Problem 3

3/5/2008
Let A A be a commutative unitary ring with an odd number of elements. Prove that the number of solutions of the equation x^2 \equal{} x (in A A) divides the number of invertible elements of A A.
group theoryabstract algebrasuperior algebrasuperior algebra unsolved
Periodic function from R to R^2

Source: RMO District Round, Bucharest 2008, Grade 10, Problem 3

1/27/2008
For any real a a define fa:R→R2 f_a : \mathbb{R} \rightarrow \mathbb{R}^2 by the law f_a(t) \equal{} \left( \sin(t), \cos(at) \right). a) Prove that fπ f_{\pi} is not periodic. b) Determine the values of the parameter a a for which fa f_a is periodic. Remark. L. Euler proved in 1737 1737 that π \pi is irrational.
functionparameterizationEuleralgebra proposedalgebra