The non-zero natural number n is a perfect square. By dividing 2023 by n, we obtain the remainder 223−32⋅n. Find the quotient of the division.
2023 Romania National Olympiad
GRADE 5
We say that a natural number is called special if all of its digits are non-zero and any two adjacent digits in its decimal representation are consecutive (not necessarily in ascending order). a) Determine the largest special number m whose sum of digits is equal to 2023. b) Determine the smallest special number n whose sum of digits is equal to 2022.
Determine all natural numbers m and n such that n⋅(n+1)=3m+s(n)+1182, where s(n) represents the sum of the digits of the natural number n.
We say that a number n≥2 has the property (P) if, in its prime factorization, at least one of the factors has an exponent 3. a) Determine the smallest number N with the property that, no matter how we choose N consecutive natural numbers, at least one of them has the property (P). b) Determine the smallest 15 consecutive numbers a1,a2,…,a15 that do not have the property (P), such that the sum of the numbers 5a1,5a2,…,5a15 is a number with the property (P).
GRADE 6
Determine all sequences of equal ratios of the form a1a2=a3a4=a5a6=a7a8 which simultaneously satisfy the following conditions: ∙ The set {a1,a2,…,a8} represents all positive divisors of 24. ∙ The common value of the ratios is a natural number.
Determine all triples (a,b,c) of integers that simultaneously satisfy the following relations: a2+a=b+c,b2+b=a+c,c2+c=a+b.
Determine all positive integers n for which the number N=1n⋅(n+1)can be represented as a finite decimal fraction.
Let ABC be a triangle with ∠BAC=90∘ and ∠ACB=54∘. We construct bisector BD(D∈AC) of angle ABC and consider point E∈(BD) such that DE=DC. Show that BE=2⋅AD.
GRADE 7
For natural number n we define an={√n}−{√n+1}+{√n+2}−{√n+3}. a) Show that a1>0,2. b) Show that an<0 for infinity many values of n and an>0 for infinity values of natural numbers of n as well. ( We denote by {x} the fractional part of x.)
In the parallelogram ABCD, AC∩BD=O, and M is the midpoint of AB. Let P∈(OC) and MP∩BC=Q. We draw a line parallel to MP from O, which intersects line CD at point N. Show that A,N,Q are collinear if and only if P is the midpoint of OC.
We consider triangle ABC with ∠BAC=90∘ and ∠ABC=60∘. Let D∈(AC),E∈(AB), such that CD=2⋅DA and DE is bisector of ∠ADB. Denote by M the intersection of CE and BD, and by P the intersection of DE and AM. a) Show that AM⊥BD. b) Show that 3⋅PB=2⋅CM.
a) Show that there exist irrational numbers a, b, and c such that the numbers a+b⋅c, b+a⋅c, and c+a⋅b are rational numbers. b) Show that if a, b, and c are real numbers such that a+b+c=1, and the numbers a+b⋅c, b+a⋅c, and c+a⋅b are rational and non-zero, then a, b, and c are rational numbers.
GRADE 8
We consider real positive numbers a,b,c such that a+b+c=3. Prove that a2+b2+c2+a2b+b2c+c2a≥6.
Prove that: a) There are infinitely many pairs (x,y) of real numbers from the interval [0,√3] which satisfy the equation x√3−y2+y√3−x2=3. b) There do not exist any pairs (x,y) of rational numbers from the interval [0,√3] that satisfy the equation x√3−y2+y√3−x2=3.
We say that a natural number n is interesting if it can be written in the form n=⌊1a⌋+⌊1b⌋+⌊1c⌋,where a,b,c are positive real numbers such that a+b+c=1. Determine all interesting numbers. ( ⌊x⌋ denotes the greatest integer not greater than x.)
Let ABCD be a tetrahedron and M and N be the midpoints of AC and BD, respectively. Show that for every point P∈(MN) with P≠M and P≠N, there exist unique points X and Y on segments AB and CD, respectively, such that X,P,Y are collinear.
GRADE 9
We consider the equation x2+(a+b−1)x+ab−a−b=0, where a and b are positive integers with a≤b. a) Show that the equation has 2 distinct real solutions. b) Prove that if one of the solutions is an integer, then both solutions are non-positive integers and b<2a.
Determine functions f:R→R, with property that f(f(x))+y⋅f(x)≤x+x⋅f(f(y)), for every x and y are real numbers.
Let n≥2 be a natural number. We consider a (2n−1)×(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 2n2 rounds. Then, starting with Ana, each one forms a vector with origin at a red point and ending at a blue point, resulting in 2n2 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.
Let r and s be real numbers in the interval [1,∞) such that for all positive integers a and b with a∣b⟹⌊ar⌋ divides ⌊bs⌋. a) Prove that sr is a natural number. b) Show that both r and s are natural numbers. Here, ⌊x⌋ denotes the greatest integer that is less than or equal to x.
GRADE 10
Solve the following equation for real values of x: 2(5x+6x−3x)=7x+9x.
Determine the largest natural number k such that there exists a natural number n satisfying: sin(n+1)<sin(n+2)<sin(n+3)<…<sin(n+k).
We consider triangle ABC and variables points M on the half-line BC, N on the half-line CA, and P on the half-line AB, each start simultaneously from B,C and respectively A, moving with constant speeds v1,v2,v3>0, where v1, v2, and v3 are expressed in the same unit of measure. a) Given that there exist three distinct moments in which triangle MNP is equilateral, prove that triangle ABC is equilateral and that v1=v2=v3. b) Prove that if v1=v2=v3 and there exists a moment in which triangle MNP is equilateral, then triangle ABC is also equilateral.
In an art museum, n paintings are exhibited, where n≥33. In total, 15 colors are used for these paintings such that any two paintings have at least one common color, and no two paintings have exactly the same colors. Determine all possible values of n≥33 such that regardless of how we color the paintings with the given properties, we can choose four distinct paintings, which we can label as T1,T2,T3, and T4, such that any color that is used in both T1 and T2 can also be found in either T3 or T4.
GRADE 11
Determine twice differentiable functions f:R→R which verify relation (f′(x))2+f″
Let A,B \in M_{n}(\mathbb{R}). Show that rank(A) = rank(B) if and only if there exist nonsingular matrices X,Y,Z \in M_{n}(\mathbb{R}) such that AX + YB = AZB.
Let n be a natural number n \geq 2 and matrices A,B \in M_{n}(\mathbb{C}), with property A^2 B = A. a) Prove that (AB - BA)^2 = O_{n}. b) Show that for all natural number k, k \leq \frac{n}{2} there exist matrices A,B \in M_{n}(\mathbb{C}) with property stated in the problem such that rank(AB - BA) = k.
We consider a function f:\mathbb{R} \rightarrow \mathbb{R} for which there exist a differentiable function g : \mathbb{R} \rightarrow \mathbb{R} and exist a sequence (a_n)_{n \geq 1} of real positive numbers, convergent to 0, such that g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n}, \forall x \in \mathbb{R}. a) Give an example of such a function f that is not differentiable at any point x \in \mathbb{R}. b) Show that if f is continuous on \mathbb{R}, then f is differentiable on \mathbb{R}.
GRADE 12
Let (G, \cdot) a finite group with order n \in \mathbb{N}^{*}, where n \geq 2. We will say that group (G, \cdot) is arrangeable if there is an ordering of its elements, such that 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 n for which the group (Z_n, +) is arrangeable. b) Give an example of a group of even order that is arrangeable.
Let p be a prime number, n a natural number which is not divisible by p, and \mathbb{K} is a finite field, with char(K) = p, |K| = p^n, 1_{\mathbb{K}} unity element and \widehat{0} = 0_{\mathbb{K}}. For every m \in \mathbb{N}^{*} we note \widehat{m} = \underbrace{1_{\mathbb{K}} + 1_{\mathbb{K}} + \ldots + 1_{\mathbb{K}}}_{m \text{ times}} and define the polynomial f_m = \sum_{k = 0}^{m} (-1)^{m - k} \widehat{\binom{m}{k}} X^{p^k} \in \mathbb{K}[X]. a) Show that roots of f_1 are \left\{ \widehat{k} | k \in \{0,1,2, \ldots , p - 1 \} \right\}. b) Let m \in \mathbb{N}^{*}. Determine the set of roots from \mathbb{K} of polynomial f_{m}.
Let a,b \in \mathbb{R} with a < b, 2 real numbers. We say that f: [a,b] \rightarrow \mathbb{R} has property (P) if there is an integrable function on [a,b] with property that 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 t there exist a unique function f:[a,b] \rightarrow \mathbb{R} with property (P), such that \int_{a}^{b} f(x) \text{dx} = t.
Let f:[0,1] \rightarrow \mathbb{R} a non-decreasing function, f \in C^1, for which f(0) = 0. Let g:[0,1] \rightarrow \mathbb{R} a function defined by g(x) = f(x) + (x - 1) f'(x), \forall x \in [0,1]. a) Show that \int_{0}^{1} g(x) \text{dx} = 0. b) Prove that for all functions \phi :[0,1] \rightarrow [0,1], convex and differentiable with \phi(0) = 0 and \phi(1) = 1, the inequality holds \int_{0}^{1} g( \phi(t)) \text{dt} \leq 0.