MathDB

2

Part of 2021 Romania National Olympiad

Problems(6)

System of three equations

Source: Romania NMO 2021 grade 7

4/25/2021
Solve the system in reals: 4−ab=5−ba=10a2+b2\frac{4-a}{b}=\frac{5-b}{a}=\frac{10}{a^2+b^2}.
algebra
Easy refinement of (a+b+c)(1/a+1/b+1/c)\ge 9

Source: Romanian NMO 2021 grade 8 P2

4/15/2023
Prove that for all positive real numbers a,b,ca,b,c the following inequality holds: (a+b+c)(1a+1b+1c)≥2(a2+b2+c2)ab+bc+ca+7(a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\ge\frac{2(a^2+b^2+c^2)}{ab+bc+ca}+7 and determine all cases of equality.
Lucian Petrescu
inequalities
maximum length of vector

Source: Romanian NMO 2021 grade 9 P2

4/15/2023
Let P0,P1,…,P2021P_0, P_1,\ldots, P_{2021} points on the unit circle of centre OO such that for each n∈{1,2,…,2021}n\in \{1,2,\ldots, 2021\} the length of the arc from Pn−1P_{n-1} to PnP_n (in anti-clockwise direction) is in the interval [π2,π]\left[\frac{\pi}2,\pi\right]. Determine the maximum possible length of the vector: OP0→+OP1→+…+OP2021→.\overrightarrow{OP_0}+\overrightarrow{OP_1}+\ldots+\overrightarrow{OP_{2021}}.
Mihai Iancu
vectorgeometrytrigonometry
floor of (an+b)/(cn+d) is surjective

Source: Romanian NMO 2021 grade 10 P2

4/15/2023
Let a,b,c,d∈Z≥0a,b,c,d\in\mathbb{Z}_{\ge 0}, d≠0d\ne 0 and the function f:Z≥0→Z≥0f:\mathbb{Z}_{\ge 0}\to\mathbb Z_{\ge 0} defined by f(n)=⌊an+bcn+d⌋ for all n∈Z≥0.f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}. Prove that the following are equivalent:
[*] ff is surjective; [*] c=0c=0, b<db<d and 0<a≤d0<a\le d.
Tiberiu Trif
floor functionalgebrafunction
Romania National Olympiad Grade 11 P2

Source:

4/28/2021
Let n≥2n \ge 2 and a1,a2,…,an a_1, a_2, \ldots , a_n , nonzero real numbers not necessarily distinct. We define matrix A=(aij)1≤i,j≤n∈Mn(R)A = (a_{ij})_{1 \le i,j \le n} \in M_n( \mathbb{R} ) , ai,j=max{ai,aj}a_{i,j} = max \{ a_i, a_j \}, ∀i,j∈{1,2,…,n}\forall i,j \in \{ 1,2 , \ldots , n \} . Show that rank(A)\mathbf{rank}(A) = card\mathbf{card} {ak∣k=1,2,…n}\{ a_k | k = 1,2, \ldots n \}
linear algebra
Sum of elements in ring is invertible

Source: Romanian NMO 2021 grade 12 P2

4/15/2023
Determine all non-trivial finite rings with am unit element in which the sum of all elements is invertible.
Mihai Opincariu
superior algebraRings