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4

Part of 2024 Romania National Olympiad

Problems(4)

Romanian National Olympiad 2024 - Grade 10 - Problem 4

Source: Romanian National Olympiad 2024 - Grade 10 - Problem 4

4/6/2024
We consider an integer n≥3,n \ge 3, the set S={1,2,3,…,n}S=\{1,2,3,\ldots,n\} and the set F\mathcal{F} of the functions from SS to S.S. We say that G⊂F\mathcal{G} \subset \mathcal{F} is a generating set for H⊂F\mathcal{H} \subset \mathcal{F} if any function in H\mathcal{H} can be represented as a composition of functions from G.\mathcal{G}.
a) Let the functions a:S→S,a:S \to S, a(n−1)=n,a(n-1)=n, a(n)=n−1a(n)=n-1 and a(k)=ka(k)=k for k∈S∖{n−1,n}k \in S \setminus \{n-1,n\} and b:S→S,b:S \to S, b(n)=1b(n)=1 and b(k)=k+1b(k)=k+1 for k∈S∖{n}.k \in S \setminus \{n\}. Prove that {a,b}\{a,b\} is a generating set for the set B\mathcal{B} of bijective functions of F.\mathcal{F}. b) Prove that the smallest number of elements that a generating set of F\mathcal{F} has is 3.3.
functionalgebrafunctionspermutationsbasiscombinatorics
Romanian National Olympiad 2024 - Grade 9 - Problem 4

Source: Romanian National Olympiad 2024 - Grade 9 - Problem 4

4/6/2024
Let aa be a given positive integer. We consider the sequence (xn)n≥1(x_n)_{n \ge 1} defined by xn=11+na,x_n=\frac{1}{1+na}, for every positive integer n.n. Prove that for any integer k≥3,k \ge 3, there exist positive integers n1<n2<…<nkn_1<n_2<\ldots<n_k such that the numbers xn1,xn2,…,xnkx_{n_1},x_{n_2},\ldots,x_{n_k} are consecutive terms in an arithmetic progression.
algebraArithmetic ProgressionSequence
Romanian National Olympiad 2024 - Grade 11 - Problem 4

Source: Romanian National Olympiad 2024 - Grade 11 - Problem 4

4/5/2024
Let f,g:R→Rf,g:\mathbb{R}\to\mathbb{R} be functions with g(x)=2f(x)+f(x2),g(x)=2f(x)+f(x^2), for all x∈R.x \in \mathbb{R}.
a) Prove that, if ff is bounded in a neighbourhood of the origin and gg is continuous in the origin, then ff is continuous in the origin. b) Provide an example of function ff, discontinuous in the origin, for which the function gg is continuous in the origin.
functionreal analysiscontinuity
x^n=(x^2+1)^n if and only if ...

Source: Romanian National Olympiad 2024 - Grade 12 - Problem 4

4/3/2024
Let L\mathbb{L} be a finite field with qq elements. Prove that:
a) If q≡3(mod4)q \equiv 3 \pmod 4 and n≥2n \ge 2 is a positive integer divisible by q−1,q-1, then xn=(x2+1)nx^n=(x^2+1)^n for all x∈L×.x \in \mathbb{L}^{\times}.
b) If there exists a positive integer n≥2n \ge 2 such that xn=(x2+1)nx^n=(x^2+1)^n for all x∈L×,x \in \mathbb{L}^{\times}, then q≡3(mod4)q \equiv 3 \pmod 4 and q−1q-1 divides n.n.
finite fieldsabstract algebra