MathDB

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Part of 2018 Romania National Olympiad

Problems(6)

a^2 + b^2 + c^2 + d^2 = 2018, (2018 Romanian NMO grade VII P1)

Source:

9/4/2024
Find the distinct positive integers a,b,c,da, b, c,d, such that the following conditions hold:
(1) exactly three of the four numbers are prime numbers; (2) a2+b2+c2+d2=2018.a^2 + b^2 + c^2 + d^2 = 2018.
number theorySum of Squares
Romanian National Olympiad 2018 - Grade 9 - problem 1

Source: Romania NMO - 2018

4/12/2018
Prove that if in a triangle the orthocenter, the centroid and the incenter are collinear, then the triangle is isosceles.
geometry
sum of squares of any three elements is perfect square

Source: 2018 Romanian NMO grade VIII P1

9/3/2024
Prove that there are infinitely many sets of four positive integers so that the sum of the squares of any three elements is a perfect square.
number theoryPerfect SquaresPerfect SquareSum of Squares
Romanian National Olympiad 2018 - Grade 10 - problem 1

Source: Romania NMO - 2018

4/12/2018
Let n∈N≥2n \in \mathbb{N}_{\geq 2} and a1,a2,…,an∈(1,∞).a_1,a_2, \dots , a_n \in (1,\infty). Prove that f:[0,∞)→Rf:[0,\infty) \to \mathbb{R} with f(x)=(a1a2...an)x−a1x−a2x−...−anxf(x)=(a_1a_2...a_n)^x-a_1^x-a_2^x-...-a_n^x is a strictly increasing function.
Romanian National Olympiad 2018 - Grade 11 - problem 1

Source: Romania NMO - 2018

4/7/2018
Let n≥2n \geq 2 be a positive integer and, for all vectors with integer entries X=(x1x2⋮xn)X=\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} let δ(X)≥0\delta(X) \geq 0 be the greatest common divisor of x1,x2,…,xn.x_1,x_2, \dots, x_n. Also, consider A∈Mn(Z).A \in \mathcal{M}_n(\mathbb{Z}). Prove that the following statements are equivalent: <spanclass=′latex−bold′>i)</span><span class='latex-bold'>i) </span> ∣det⁡A∣=1|\det A | = 1 <spanclass=′latex−bold′>ii)</span><span class='latex-bold'>ii) </span> δ(AX)=δ(X),\delta(AX)=\delta(X), for all vectors X∈Mn,1(Z).X \in \mathcal{M}_{n,1}(\mathbb{Z}).
Romeo Raicu
greatest common divisorvectorlinear algebramatrix
Romanian National Olympiad 2018 - Grade 12 - problem 1

Source: Romania NMO - 2018

4/7/2018
Let AA be a finite ring and a,b∈A,a,b \in A, such that (ab−1)b=0.(ab-1)b=0. Prove that b(ab−1)=0.b(ab-1)=0.
superior algebraRing Theory