MathDB

2

Part of 2023 Romania National Olympiad

Problems(8)

Romania NMO 2023 Grade 5 P2

Source: Romania National Olympiad 2023

4/14/2023
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 mm whose sum of digits is equal to 20232023.
b) Determine the smallest special number nn whose sum of digits is equal to 20222022.
algebrasum of digits
Romania NMO 2023 Grade 6 P2

Source: Romania National Olympiad 2023

4/14/2023
Determine all triples (a,b,c)(a,b,c) of integers that simultaneously satisfy the following relations:
\begin{align*} a^2 + a = b + c, \\ b^2 + b = a + c, \\ c^2 + c = a + b. \end{align*}
algebraequationnumber theory
Romania NMO 2023 Grade 7 P2

Source: Romania National Olympiad 2023

4/14/2023
In the parallelogram ABCDABCD, AC∩BD=OAC \cap BD = { O }, and MM is the midpoint of ABAB. Let P∈(OC)P \in (OC) and MP∩BC=QMP \cap BC = { Q }. We draw a line parallel to MPMP from OO, which intersects line CDCD at point NN. Show that A,N,QA,N,Q are collinear if and only if PP is the midpoint of OCOC.
geometrysimilar trianglesMenelaus
Romania NMO 2023 Grade 8 P2

Source: Romania National Olympiad 2023

4/14/2023
Prove that:
a) There are infinitely many pairs (x,y)(x,y) of real numbers from the interval [0,3][0,\sqrt{3}] which satisfy the equation x3−y2+y3−x2=3x\sqrt{3-y^2}+y\sqrt{3-x^2}=3.
b) There do not exist any pairs (x,y)(x,y) of rational numbers from the interval [0,3][0,\sqrt{3}] that satisfy the equation x3−y2+y3−x2=3x\sqrt{3-y^2}+y\sqrt{3-x^2}=3.
algebra
Romania NMO 2023 Grade 9 P2

Source: Romania National Olympiad 2023

4/14/2023
Determine functions f:R→R,f : \mathbb{R} \rightarrow \mathbb{R}, with property that
f(f(x))+y⋅f(x)≤x+x⋅f(f(y)), f(f(x)) + y \cdot f(x) \le x + x \cdot f(f(y)), for every xx and yy are real numbers.
algebrafunction
Romania NMO 2023 Grade 10 P2

Source: Romania National Olympiad 2023

4/14/2023
Determine the largest natural number kk such that there exists a natural number nn satisfying:
sin⁡(n+1)<sin⁡(n+2)<sin⁡(n+3)<…<sin⁡(n+k). \sin(n + 1) < \sin(n + 2) < \sin(n + 3) < \ldots < \sin(n + k).
algebratrigonometry
Romania NMO 2023 Grade 11 P2

Source: Romania National Olympiad 2023

4/14/2023
Let A,B∈Mn(R).A,B \in M_{n}(\mathbb{R}). Show that rank(A)=rank(B)rank(A) = rank(B) if and only if there exist nonsingular matrices X,Y,Z∈Mn(R)X,Y,Z \in M_{n}(\mathbb{R}) such that
AX+YB=AZB. AX + YB = AZB.
matrixlinear algebraNon singular matrices
Romania NMO 2023 Grade 12 P2

Source: Romania National Olympiad 2023

4/14/2023
Let pp be a prime number, nn a natural number which is not divisible by pp, and K\mathbb{K} is a finite field, with char(K)=p,∣K∣=pn,1Kchar(K) = p, |K| = p^n, 1_{\mathbb{K}} unity element and 0^=0K.\widehat{0} = 0_{\mathbb{K}}. For every m∈N∗m \in \mathbb{N}^{*} we note m^=1K+1K+…+1K⏟m times \widehat{m} = \underbrace{1_{\mathbb{K}} + 1_{\mathbb{K}} + \ldots + 1_{\mathbb{K}}}_{m \text{ times}} and define the polynomial
fm=∑k=0m(−1)m−k(mk)^Xpk∈K[X]. 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 f1f_1 are {k^∣k∈{0,1,2,…,p−1}} \left\{ \widehat{k} | k \in \{0,1,2, \ldots , p - 1 \} \right\}.
b) Let m∈N∗.m \in \mathbb{N}^{*}. Determine the set of roots from K\mathbb{K} of polynomial fm.f_{m}.
polynomialfieldfinite fieldssuperior algebra