MathDB

3

Part of 2023 Romania National Olympiad

Problems(8)

Romania NMO 2023 Grade 5 P3

Source: Romania National Olympiad 2023

4/14/2023
Determine all natural numbers mm and nn such that
n⋅(n+1)=3m+s(n)+1182, n \cdot (n + 1) = 3^m + s(n) + 1182,
where s(n)s(n) represents the sum of the digits of the natural number nn.
number theorysum of digits
Romania NMO 2023 Grade 6 P3

Source: Romania National Olympiad 2023

4/14/2023
Determine all positive integers nn for which the number
N=1n⋅(n+1) N = \frac{1}{n \cdot (n + 1)} can be represented as a finite decimal fraction.
algebraperiodic decimal number
Romania NMO 2023 Grade 7 P3

Source: Romania National Olympiad 2023

4/14/2023
We consider triangle ABCABC with ∠BAC=90∘\angle BAC = 90^{\circ} and ∠ABC=60∘.\angle ABC = 60^{\circ}. Let D∈(AC),E∈(AB), D \in (AC) , E \in (AB), such that CD=2⋅DACD = 2 \cdot DA and DEDE is bisector of ∠ADB.\angle ADB. Denote by MM the intersection of CECE and BDBD, and by PP the intersection of DEDE and AMAM.
a) Show that AM⊥BDAM \perp BD.
b) Show that 3⋅PB=2⋅CM3 \cdot PB = 2 \cdot CM.
geometryMetric Relationperpendicularity
Romania NMO 2023 Grade 8 P3

Source: Romania National Olympiad 2023

4/14/2023
We say that a natural number nn is interesting if it can be written in the form
n=⌊1a⌋+⌊1b⌋+⌊1c⌋, n = \left\lfloor \frac{1}{a} \right\rfloor + \left\lfloor \frac{1}{b} \right\rfloor + \left\lfloor \frac{1}{c} \right\rfloor, where a,b,ca,b,c are positive real numbers such that a+b+c=1.a + b + c = 1.
Determine all interesting numbers. ( ⌊x⌋\lfloor x \rfloor denotes the greatest integer not greater than xx.)
floor functionalgebra
Romania NMO 2023 Grade 9 P3

Source: Romania National Olympiad 2023

4/14/2023
Let n≥2n \geq 2 be a natural number. We consider a (2n−1)×(2n−1)(2n - 1) \times (2n - 1) table.Ana and Bob play the following game: starting with Ana, the two of them alternately color the vertices of the unit squares, Ana with red and Bob with blue, in 2n22n^2 rounds. Then, starting with Ana, each one forms a vector with origin at a red point and ending at a blue point, resulting in 2n22n^2 vectors with distinct origins and endpoints. If the sum of these vectors is zero, Ana wins. Otherwise, Bob wins. Show that Bob has a winning strategy.
combinatoricsVectorsgame
Romania NMO 2023 Grade 10 P3

Source: Romania National Olympiad 2023

4/14/2023
We consider triangle ABCABC and variables points MM on the half-line BCBC, NN on the half-line CACA, and PP on the half-line ABAB, each start simultaneously from B,CB,C and respectively AA, moving with constant speeds v1,v2,v3>0 v_1, v_2, v_3 > 0 , where v1v_1, v2v_2, and v3v_3 are expressed in the same unit of measure.
a) Given that there exist three distinct moments in which triangle MNPMNP is equilateral, prove that triangle ABCABC is equilateral and that v1=v2=v3v_1 = v_2 = v_3.
b) Prove that if v1=v2=v3v_1 = v_2 = v_3 and there exists a moment in which triangle MNPMNP is equilateral, then triangle ABCABC is also equilateral.
geometrycomplex numbers
Romania NMO 2023 Grade 11 P3

Source: Romania National Olympiad 2023

4/14/2023
Let nn be a natural number n≥2n \geq 2 and matrices A,B∈Mn(C),A,B \in M_{n}(\mathbb{C}), with property A2B=A.A^2 B = A.
a) Prove that (AB−BA)2=On.(AB - BA)^2 = O_{n}.
b) Show that for all natural number kk, k≤n2k \leq \frac{n}{2} there exist matrices A,B∈Mn(C)A,B \in M_{n}(\mathbb{C}) with property stated in the problem such that rank(AB−BA)=k.rank(AB - BA) = k.
linear algebraNilpotentrank
Romania NMO 2023 Grade 12 P3

Source: Romania National Olympiad 2023

4/14/2023
Let a,b∈Ra,b \in \mathbb{R} with a<b,a < b, 2 real numbers. We say that f:[a,b]→Rf: [a,b] \rightarrow \mathbb{R} has property (P)(P) if there is an integrable function on [a,b][a,b] with property that
f(x)−f(x+a2)=f(x+b2)−f(x),∀x∈[a,b]. f(x) - f \left( \frac{x + a}{2} \right) = f \left( \frac{x + b}{2} \right) - f(x) , \forall x \in [a,b].
Show that for all real number tt there exist a unique function f:[a,b]→Rf:[a,b] \rightarrow \mathbb{R} with property (P),(P), such that ∫abf(x)dx=t.\int_{a}^{b} f(x) \text{dx} = t.
real analysisintegrationfunctionriemann integral