MathDB

4

Part of 2018 Romania National Olympiad

Problems(6)

\sqrt{(20^n- 18^n)/ 19} is rational

Source: 2018 Romanian NMO grade VII P4

9/4/2024
Find the natural number nn for which 20n−18n19\sqrt{\frac{20^n- 18^n}{19}} is a rational number.
rationalnumber theory
BC\AB =BB' /BC+ BC/BB', planes angles, cube related (2018 Romanian NMO VIII P4)

Source:

6/3/2020
In the rectangular parallelepiped ABCDA′B′C′D′ABCDA'B'C'D' we denote by MM the center of the face ABB′A′ABB'A'. We denote by M1M_1 and M2M_2 the projections of MM on the lines B′CB'C and AD′AD' respectively. Prove that:
a) MM1=MM2MM_1 = MM_2
b) if (MM1M2)∩(ABC)=d(MM_1M_2) \cap (ABC) = d, then d∥ADd \parallel AD;
c) ∠(MM1M2),(ABC)=45o⇔BCAB=BB′BC+BCBB′\angle (MM_1M_2), (A B C)= 45^ o \Leftrightarrow \frac{BC}{AB}=\frac{BB'}{BC}+\frac{BC}{BB'}.
geometry3D geometrycubeanglesparallelequal segments
Romanian National Olympiad 2018 - Grade 9 - problem 4

Source: Romania NMO - 2018

4/12/2018
Let n∈N∗n \in \mathbb{N}^* and consider a circle of length 6n6n along with 3n3n points on the circle which divide it into 3n3n arcs: nn of them have length 1,1, some other nn have length 22 and the remaining nn have length 3.3. Prove that among these points there must be two such that the line that connects them passes through the center of the circle.
geometrycombinatorics
Romanian National Olympiad 2018 - Grade 10 - problem 4

Source: Romania NMO - 2018

4/12/2018
Let n∈N≥2.n \in \mathbb{N}_{\geq 2}. For any real numbers a1,a2,...,ana_1,a_2,...,a_n denote S0=1S_0=1 and for 1≤k≤n1 \leq k \leq n denote Sk=∑1≤i1<i2<...<ik≤nai1ai2...aikS_k=\sum_{1 \leq i_1 < i_2 < ... <i_k \leq n}a_{i_1}a_{i_2}...a_{i_k} Find the number of n−n-tuples (a1,a2,...an)(a_1,a_2,...a_n) such that (Sn−Sn−2+Sn−4−...)2+(Sn−1−Sn−3+Sn−5−...)2=2nSn.(S_n-S_{n-2}+S_{n-4}-...)^2+(S_{n-1}-S_{n-3}+S_{n-5}-...)^2=2^nS_n.
algebra
Romanian National Olympiad 2018 - Grade 11 - problem 4

Source: Romania NMO - 2018

4/7/2018
Let nn be an integer with n≥2n \geq 2 and let A∈Mn(C)A \in \mathcal{M}_n(\mathbb{C}) such that rank⁡A≠rank⁡A2.\operatorname{rank} A \neq \operatorname{rank} A^2. Prove that there exists a nonzero matrix B∈Mn(C)B \in \mathcal{M}_n(\mathbb{C}) such that AB=BA=B2=0AB=BA=B^2=0
Cornel Delasava
linear algebraMatricesrankmatrix
Romanian National Olympiad 2018 - Grade 12 - problem 4

Source: Romania NMO - 2018

4/7/2018
For any k∈Z,k \in \mathbb{Z}, define Fk=X4+2(1−k)X2+(1+k)2.F_k=X^4+2(1-k)X^2+(1+k)^2. Find all values k∈Zk \in \mathbb{Z} such that FkF_k is irreducible over Z\mathbb{Z} and reducible over Zp,\mathbb{Z}_p, for any prime p.p.
Marius Vladoiu
polynomialIrreduciblesuperior algebra