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Scanner on squarefree integers
Assassino9931   2
N 29 minutes ago by Assassino9931
Source: Bulgaria National Olympiad 2025, Day 2, Problem 5
Let $n$ be a positive integer. Prove that there exists a positive integer $a$ such that exactly $\left \lfloor \frac{n}{4} \right \rfloor$ of the integers $a + 1, a + 2, \ldots, a + n$ are squarefree.
2 replies
Assassino9931
Yesterday at 1:54 PM
Assassino9931
29 minutes ago
J
H
Poly with sequence give infinitely many prime divisors
Assassino9931   5
N 36 minutes ago by Assassino9931
Source: Bulgaria National Olympiad 2025, Day 1, Problem 3
Let $P(x)$ be a non-constant monic polynomial with integer coefficients and let $a_1, a_2, \ldots$ be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence $b_n = P(n)^{a_n} + 1$.
5 replies
Assassino9931
Yesterday at 1:51 PM
Assassino9931
36 minutes ago
J
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sequence in which sum of terms equals product of the same terms
CatalinBordea   6
N 2 hours ago by Levieee
Source: Romanian District Olympiad 2012, Grade XI, Problem 1
Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfying the equation
$$ x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} . $$Show that this sequence is convergent and find its limit.
6 replies
CatalinBordea
Oct 9, 2018
Levieee
2 hours ago
J
H
Romanian National Olympiad 2018 - Grade 12 - problem 2
Catalin   4
N 4 hours ago by Levieee
Source: Romania NMO - 2018
Let $\mathcal{F}$ be the set of continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R}$$For $f \in \mathcal{F},$ let $$I(f)=\int_0^ef(x) dx$$Determine $\min_{f \in \mathcal{F}}I(f).$

Liviu Vlaicu
4 replies
Catalin
Apr 7, 2018
Levieee
4 hours ago
J
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f(x)<=f(a) for all a and all x in a left neighbour of a implies monotony if cont
CatalinBordea   6
N Today at 10:37 AM by Rohit-2006
Source: Romanian District Olympiad 2012, Grade XI, Problem 4
A function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has property $ \mathcal{F} , $ if for any real number $ a, $ there exists a $ b<a $ such that $ f(x)\le f(a), $ for all $ x\in (b,a) . $

a) Give an example of a function with property $ \mathcal{F} $ that is not monotone on $ \mathbb{R} . $
b) Prove that a continuous function that has property $ \mathcal{F} $ is nondecreasing.
6 replies
CatalinBordea
Oct 9, 2018
Rohit-2006
Today at 10:37 AM
J
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limsup a_n/n^4
EthanWYX2009   0
Today at 10:01 AM
Source: 2023 Aug taca-15
Let \( M_n = \{ A \mid A \text{ is an } n \times n \text{ real symmetric matrix with entries from } \{0, \pm1, \pm2\} \} \). Define \( a_n \) as the average of all \( \text{tr}(A^6) \) for \( A \in M_n \). Determine the value of \[ a = \lim_{k \to \infty} \sup_{n \geq k} \frac{a_n}{n^4} .\]
0 replies
EthanWYX2009
Today at 10:01 AM
0 replies
J
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polynomial with real coefficients
Peter   8
N Yesterday at 3:55 PM by mqoi_KOLA
Source: IMC 1998 day 1 problem 5
Let $P$ be a polynomial of degree $n$ with only real zeros and real coefficients.
Prove that for every real $x$ we have $(n-1)(P'(x))^2\ge nP(x)P''(x)$. When does equality occur?
8 replies
Peter
Nov 1, 2005
mqoi_KOLA
Yesterday at 3:55 PM
J
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real analysis
ay19bme   4
N Yesterday at 1:55 PM by ay19bme
.............
4 replies
ay19bme
Apr 7, 2025
ay19bme
Yesterday at 1:55 PM
J
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Unique global minimum points
chirita.andrei   1
N Yesterday at 9:39 AM by chirita.andrei
Source: Own. Proposed for Romanian National Olympiad 2025.
Let $f\colon[0,1]\rightarrow \mathbb{R}$ be a continuous function. Suppose that for each $t\in(0,1)$, the function \[f_t\colon[0,1-t]\rightarrow\mathbb{R}, f_t(x)=f(x+t)-f(x)\]has an unique global minimum point, which we will denote by $g(t)$. Prove that if $\lim\limits_{t\to 0}g(t)=0$, then $g$ is constant zero.
1 reply
chirita.andrei
Apr 2, 2025
chirita.andrei
Yesterday at 9:39 AM
J
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Simple limit with standard recurrence
AndreiVila   2
N Yesterday at 7:33 AM by Tung-CHL
Source: Romanian District Olympiad 2025 11.1
Consider the sequence $(a_n)_{n\geq 1}$ given by $a_1=1$ and $a_{n+1}=\frac{a_n}{1+\sqrt{1+a_n}}$, for all $n\geq 1$. Show that $$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} = \lim_{n\rightarrow\infty}\sum_{k=1}^n \log_2(1+a_k)=2.$$Mathematical Gazette
2 replies
AndreiVila
Mar 8, 2025
Tung-CHL
Yesterday at 7:33 AM
J
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real analysis
ay19bme   1
N Apr 7, 2025 by paxtonw
...........
1 reply
ay19bme
Apr 7, 2025
paxtonw
Apr 7, 2025
J
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Continuous functions
joybangla   3
N Apr 7, 2025 by Rohit-2006
Source: Romanian District Olympiad 2014, Grade 11, P2
[list=a]
[*]Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be a function such that
$g\colon\mathbb{R}\rightarrow\mathbb{R}$, $g(x)=f(x)+f(2x)$, and
$h\colon\mathbb{R}\rightarrow\mathbb{R}$, $h(x)=f(x)+f(4x)$, are continuous
functions. Prove that $f$ is also continuous.
[*]Give an example of a discontinuous function $f\colon\mathbb{R} \rightarrow\mathbb{R}$, with the following property: there exists an interval
$I\subset\mathbb{R}$, such that, for any $a$ in $I$, the function $g_{a} \colon\mathbb{R}\rightarrow\mathbb{R}$, $g_{a}(x)=f(x)+f(ax)$, is continuous.[/list]
3 replies
joybangla
Jun 15, 2014
Rohit-2006
Apr 7, 2025
J
a