MathDB

1

Part of 2023 Romania National Olympiad

Problems(8)

Romania NMO 2023 Grade 5 P1

Source: Romania National Olympiad 2023

4/14/2023
The non-zero natural number n is a perfect square. By dividing 20232023 by nn, we obtain the remainder 223−32⋅n223- \frac{3}{2} \cdot n. Find the quotient of the division.
number theoryDivisibility
Romania NMO 2023 Grade 6 P1

Source: Romania National Olympiad 2023

4/14/2023
Determine all sequences of equal ratios of the form
a1a2=a3a4=a5a6=a7a8 \frac{a_1}{a_2} = \frac{a_3}{a_4} = \frac{a_5}{a_6} = \frac{a_7}{a_8}
which simultaneously satisfy the following conditions:
∙\bullet The set {a1,a2,…,a8}\{ a_1, a_2, \ldots , a_8 \} represents all positive divisors of 2424.
∙\bullet The common value of the ratios is a natural number.
Fractionsalgebra
Romania NMO 2023 Grade 7 P1

Source: Romania National Olympiad 2023

4/14/2023
For natural number nn we define
an={n}−{n+1}+{n+2}−{n+3}. a_n = \{ \sqrt{n} \} - \{ \sqrt{n + 1} \} + \{ \sqrt{n + 2} \} - \{ \sqrt{n + 3} \}.
a) Show that a1>0,2a_1 > 0,2.
b) Show that an<0a_n < 0 for infinity many values of nn and an>0a_n > 0 for infinity values of natural numbers of nn as well. ( We denote by {x}\{ x \} the fractional part of x.x.)
fractional partalgebraInequality
Romania NMO 2023 Grade 8 P1

Source: Romania National Olympiad 2023

4/14/2023
We consider real positive numbers a,b,ca,b,c such that a+b+c=3.a + b + c = 3.
Prove that a2+b2+c2+a2b+b2c+c2a≥6.a^2 + b^2 + c^2 + a^2b + b^2 c + c^2 a \ge 6.
inequalities
Romania NMO 2023 Grade 9 P1

Source: Romania National Olympiad 2023

4/14/2023
We consider the equation x2+(a+b−1)x+ab−a−b=0x^2 + (a + b - 1)x + ab - a - b = 0, where aa and bb are positive integers with a≤ba \leq b.
a) Show that the equation has 22 distinct real solutions.
b) Prove that if one of the solutions is an integer, then both solutions are non-positive integers and b<2a.b < 2a.
algebraquadratic equation
Romania NMO 2023 Grade 10 P1

Source: Romania National Olympiad 2023

4/14/2023
Solve the following equation for real values of xx:
2(5x+6x−3x)=7x+9x. 2 \left( 5^x + 6^x - 3^x \right) = 7^x + 9^x.
exponentialExponential equationalgebra
Romania NMO 2023 Grade 11 P1

Source: Romania National Olympiad 2023

4/14/2023
Determine twice differentiable functions f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} which verify relation
(f′(x))2+f′′(x)≤0,∀x∈R. \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
real analysisdifferentiable functions
Romania NMO 2023 Grade 12 P1

Source: Romania National Olympiad 2023

4/14/2023
Let (G,⋅)(G, \cdot) a finite group with order n∈N∗,n \in \mathbb{N}^{*}, where n≥2.n \geq 2. We will say that group (G,⋅)(G, \cdot) is arrangeable if there is an ordering of its elements, such that
G={a1,a2,…,ak,…,an}={a1⋅a2,a2⋅a3,…,ak⋅ak+1,…,an⋅a1}. G = \{ a_1, a_2, \ldots, a_k, \ldots , a_n \} = \{ a_1 \cdot a_2, a_2 \cdot a_3, \ldots, a_k \cdot a_{k + 1}, \ldots , a_{n} \cdot a_1 \}.
a) Determine all positive integers nn for which the group (Zn,+)(Z_n, +) is arrangeable.
b) Give an example of a group of even order that is arrangeable.
abstract algebragroup theorynumber theory