Found problems: 4776
2004 Regional Olympiad - Republic of Srpska, 3
Given a sequence $(a_n)$ of real numbers such that the set $\{a_n\}$ is finite.
If for every $k>1$ subsequence $(a_{kn})$ is periodic, is it true that the sequence $(a_n)$ must be periodic?
2000 Moldova Team Selection Test, 11
Let $S$ be a finite set with $n{}$ $(n>1)$ elements, $M{}$ the set of all subsets of $S{}$ and a function $f:M\rightarrow\mathbb{R}$, that verifies the relation $f(A\cap B)=\min\{f(A),f(B)\}, \forall A,B\in M$. Show that $$\sum_{A\in M} (-1)^{n-|A|}\cdot f(A)=f(S)-\max\{f(A)|A\in M, A\neq S\},$$ where$|A|$ is the number of elements of subset $A{}$.
2010 Today's Calculation Of Integral, 578
Find the range of $ k$ for which the following inequality holds for $ 0\leq x\leq 1$.
\[ \int_0^x \frac {dt}{\sqrt {(3 \plus{} t^2)^3}}\geq k\int _0^x \frac {dt}{\sqrt {3 \plus{} t^2}}\]
If necessary, you may use $ \ln 3 \equal{} 1.10$.
2003 Miklós Schweitzer, 8
Let $f_1, f_2, \ldots$ be continuous real functions on the real line. Is it true that if the series $\sum_{n=1}^{\infty} f_n(x)$ is divergent for every $x$, then this holds also true for any typical choice of the signs in the sum (i.e. the set of those $\{ \epsilon _n\}_{n=1}^{\infty} \in \{ +1, -1\}^{\mathbb{N}}$ sequences, for which there series $\sum_{n=1}^{\infty} \epsilon_nf_n(x)$ is convergent at least at one point $x$, forms a subset of first category within the set $\{+1,-1\}^{\mathbb{N}} $)?
(translated by L. Erdős)
KoMaL A Problems 2017/2018, A. 706
Find all positive integer $k$s for which such $f$ exists and unique:
$f(mn)=f(n)f(m)$ for $n, m \in \mathbb{Z^+}$
$f^{n^k}(n)=n$ for all $n \in \mathbb{Z^+}$ for which $f^x (n)$ means the n times operation of function $f$(i.e. $f(f(...f(n))...)$)
2019 Jozsef Wildt International Math Competition, W. 25
Let $x_i$, $y_i$, $z_i$, $w_i \in \mathbb{R}^+, i = 1, 2,\cdots n$, such that$$\sum \limits_{i=1}^nx_i=nx,\ \sum \limits_{i=1}^ny_i=ny,\ \sum \limits_{i=1}^nw_i=nw $$ $$\Gamma \left(z_i\right)\geq \Gamma \left(w_i\right),\ \sum \limits_{i=1}^n\Gamma \left(z_i\right)=n\Gamma^* (z)$$Then$$\sum \limits_{i=1}^n \frac{\left(\Gamma \left(x_i\right)+\Gamma \left(y_i\right)\right)^2}{\Gamma \left(z_i\right)-\Gamma \left(w_i\right)}\geq n\frac{\left(\Gamma \left(x\right)+\Gamma \left(y\right)\right)^2}{\Gamma^* \left(z\right)-\Gamma \left(w\right)}$$
2014 NIMO Problems, 8
Define the function $\xi : \mathbb Z^2 \to \mathbb Z$ by $\xi(n,k) = 1$ when $n \le k$ and $\xi(n,k) = -1$ when $n > k$, and construct the polynomial \[ P(x_1, \dots, x_{1000}) = \prod_{n=1}^{1000} \left( \sum_{k=1}^{1000} \xi(n,k)x_k \right). \]
(a) Determine the coefficient of $x_1x_2 \dots x_{1000}$ in $P$.
(b) Show that if $x_1, x_2, \dots, x_{1000} \in \left\{ -1,1 \right\}$ then $P(x_1,x_2,\dots,x_{1000}) = 0$.
[i]Proposed by Evan Chen[/i]
2014-2015 SDML (High School), 6
Let $f\left(x\right)=x^2-14x+52$ and $g\left(x\right)=ax+b$, where $a$ and $b$ are positive. Find $a$, given that $f\left(g\left(-5\right)\right)=3$ and $f\left(g\left(0\right)\right)=103$.
$\text{(A) }2\qquad\text{(B) }5\qquad\text{(C) }7\qquad\text{(D) }10\qquad\text{(E) }17$
2010 Polish MO Finals, 2
Positive rational number $a$ and $b$ satisfy the equality
\[a^3 + 4a^2b = 4a^2 + b^4.\]
Prove that the number $\sqrt{a}-1$ is a square of a rational number.
1996 Romania National Olympiad, 2
a) Let $f_1,f_2,\ldots,f_n: \mathbb{R} \to \mathbb{R}$ be periodic functions such that the function $f: \mathbb{R} \to \mathbb{R},$ $f=f_1+f_2+\ldots+f_n$ has finite limit at $\infty.$ Prove that $f$ is constant.
b) If $a_1,a_2,a_3$ are real numbers such that $a_1 \cos(a_1x) + a_2 \cos (a_2x) + a_3 \cos(a_3x) \ge 0$ for every $x \in \mathbb{R},$ then $a_1a_2a_3=0.$
2019 India IMO Training Camp, P3
Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.
1970 Putnam, B5
Let $u_n$ denote the ramp function
$$ u_n (x) =\begin{cases}
-n \;\; \text{for} \;\; x \leq -n, \\
\; x \;\;\; \text{for} \;\; -n \leq x \leq n,\\
\;n \;\; \; \text{for} \;\; n \leq x,
\end{cases}$$
and let $f$ be a real function of a real variable. Show that $f$ is continuous if and only if $u_n \circ f$ is continuous for all $n.$
2006 Switzerland Team Selection Test, 3
Find all the functions $f : \mathbb{R} \to \mathbb{R}$ satisfying for all $x,y \in \mathbb{R}$ $f(f(x)-y^2) = f(x)^2 - 2f(x)y^2 + f(f(y))$.
2009 Finnish National High School Mathematics Competition, 2
A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$. For how many $x$ we might have $P(x)=x$?
1996 IMO Shortlist, 6
Let $ n$ be an even positive integer. Prove that there exists a positive inter $ k$ such that
\[ k \equal{} f(x) \cdot (x\plus{}1)^n \plus{} g(x) \cdot (x^n \plus{} 1)\]
for some polynomials $ f(x), g(x)$ having integer coefficients. If $ k_0$ denotes the least such $ k,$ determine $ k_0$ as a function of $ n,$ i.e. show that $ k_0 \equal{} 2^q$ where $ q$ is the odd integer determined by $ n \equal{} q \cdot 2^r, r \in \mathbb{N}.$
Note: This is variant A6' of the three variants given for this problem.
2016 Balkan MO Shortlist, A6
Prove that there is no function from positive real numbers to itself, $f : (0,+\infty)\to(0,+\infty)$ such that:
$f(f(x) + y) = f(x) + 3x + yf(y)$ ,for every $x,y \in (0,+\infty)$
by Greece, Athanasios Kontogeorgis (aka socrates)
1959 Putnam, A3
Find all complex-valued functions $f$ of a complex variable such that $$f(z)+zf(1-z)=1+z$$
for all $z\in \mathbb{C}$.
2017 OMMock - Mexico National Olympiad Mock Exam, 5
Let $k$ be a positive real number. Determine all functions $f:[-k, k]\rightarrow[0, k]$ satisfying the equation
$$f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2$$
for any $x, y\in[-k, k]$ such that $x+y\in[-k, k]$.
[i]Proposed by Maximiliano Sánchez[/i]
2010 Today's Calculation Of Integral, 557
Find the folllowing limit.
\[ \lim_{n\to\infty} \frac{(2n\plus{}1)\int_0^1 x^{n\minus{}1}\sin \left(\frac{\pi}{2}x\right)dx}{(n\plus{}1)^2\int_0^1 x^{n\minus{}1}\cos \left(\frac{\pi}{2}x\right)dx}\ \ (n\equal{}1,\ 2,\ \cdots).\]
PEN I Problems, 19
Let $a, b, c$, and $d$ be real numbers. Suppose that $\lfloor na\rfloor +\lfloor nb\rfloor =\lfloor nc\rfloor +\lfloor nd\rfloor $ for all positive integers $n$. Show that at least one of $a+b$, $a-c$, $a-d$ is an integer.
2016 Iran Team Selection Test, 6
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called [i]$k$-good[/i] if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.
[i]Proposed by James Rickards, Canada[/i]
2007 Baltic Way, 20
Let $a$ and $b$ be positive integers, $b<a$, such that $a^3+b^3+ab$ is divisible by $ab(a-b)$. Prove that $ab$ is a perfect cube.
2010 Stanford Mathematics Tournament, 4
If $x^2+\frac{1}{x^2}=7,$ find all possible values of $x^5+\frac{1}{x^5}.$
2006 AIME Problems, 15
Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\ge 1,$ find the minimum possible value of $|x_1+x_2+\cdots+x_{2006}|$.
2007 Gheorghe Vranceanu, 4
Let be a sequence $ \left( a_n \right)_{n\geqslant 1} $ of real numbers defined recursively as
$$ a_n=2007+1004n^2-a_{n-1}-a_{n-2}-\cdots -a_2-a_1. $$ Calculate:
$$ \lim_{n\to\infty} \frac{1}{n}\int_1^{a_n} e^{1/\ln t} dt $$