Found problems: 4776
2011 IMC, 5
Let $n$ be a positive integer and let $V$ be a $(2n-1)$-dimensional vector space over the two-element field. Prove that for arbitrary vectors $v_1,\dots,v_{4n-1} \in V,$ there exists a sequence $1\leq i_1<\dots<i_{2n}\leq 4n-1$ of indices such that $v_{i_1}+\dots+v_{i_{2n}}=0.$
2012 Math Prize For Girls Problems, 19
Define $L(x) = x - \frac{x^2}{2}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by
\[
a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr),
\]
where there are $n$ iterations of $L$. For example,
\[
a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr).
\]
As $n$ approaches infinity, what value does $n a_n$ approach?
2012 APMO, 5
Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then
\[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \]
must hold.
2003 VJIMC, Problem 4
Let $f,g:[0,1]\to(0,+\infty)$ be two continuous functions such that $f$ and $\frac gf$ are increasing. Prove that
$$\int^1_0\frac{\int^x_0f(t)\text dt}{\int^x_0g(t)\text dt}\text dx\le2\int^1_0\frac{f(t)}{g(t)}\text dt.$$
2023 Brazil Team Selection Test, 3
Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$.
(For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)
1997 Greece National Olympiad, 1
Let $P$ be a point inside or on the boundary of a square $ABCD$. Find the minimum and maximum values of $f(P ) = \angle ABP + \angle BCP + \angle CDP + \angle DAP$.
1955 Miklós Schweitzer, 3
[b]3.[/b] Let the density function $f(x)$ of the random variable $\xi$ bean even function; let further $f(x)$ be monotonically non-increasing for $x>0$. Suppose that
$D^{2}= \int_{-\infty }^{\infty }x^{2}f(x) dx$
exists. Prove that for every non negative $\lambda $
$P(\left |\xi \right |\geq \lambda D)\leq \frac{1}{1+\lambda ^{2}}$. [b](P. 7)[/b]
2012 ELMO Shortlist, 7
Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant.
[i]Victor Wang.[/i]
1983 AIME Problems, 6
Let $a_n = 6^n + 8^n$. Determine the remainder on dividing $a_{83}$ by 49.
2000 Taiwan National Olympiad, 3
Define a function $f:\mathbb{N}\rightarrow\mathbb{N}_0$ by $f(1)=0$ and
\[f(n)=\max_j\{ f(j)+f(n-j)+j\}\quad\forall\, n\ge 2 \]
Determine $f(2000)$.
2016 Peru IMO TST, 9
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]
2015 Switzerland Team Selection Test, 7
Find all finite and non-empty sets $A$ of functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that for all $f_1, f_2 \in A$, there exists $g \in A$ such that for all $x, y \in \mathbb{R}$
$$f_1 \left(f_2 (y)-x\right)+2x=g(x+y)$$
1976 Miklós Schweitzer, 10
Suppose that $ \tau$ is a metrizable topology on a set $ X$ of cardinality less than or equal to continuum. Prove that there exists a separable and metrizable topology on $ X$ that is coarser that $ \tau$.
[i]L. Juhasz[/i]
1996 Bosnia and Herzegovina Team Selection Test, 2
$a)$ Let $m$ and $n$ be positive integers. If $m>1$ prove that $ n \mid \phi(m^n-1)$ where $\phi$ is Euler function
$b)$ Prove that number of elements in sequence $1,2,...,n$ $(n \in \mathbb{N})$, which greatest common divisor with $n$ is $d$, is $\phi\left(\frac{n}{d}\right)$
2002 Korea - Final Round, 2
Find all functions $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(x-y)=f(x)+xy+f(y)$ for every $x \in \mathbb{R}$ and every $y \in \{f(x) \mid x\in \mathbb{R}\}$, where $\mathbb{R}$ is the set of real numbers.
2010 Postal Coaching, 3
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that
$\boxed{1} \ f(1) = 1$
$\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}$
2021 Science ON grade XII, 3
Define $E\subseteq \{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$ such that $E$ posseses the following properties:\\
$\textbf{(i)}$ If $\int_0^1 f(x)g(x) dx = 0$ for $f\in E$ with $\int_0^1f^2(t)dt \neq 0$, then $g\in E$; \\
$\textbf{(ii)}$ There exists $h\in E$ with $\int_0^1 h^2(t)dt\neq 0$.\\
Prove that $E=\{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$.
\\
[i](Andrei Bâra)[/i]
2005 AMC 12/AHSME, 20
For each $ x$ in $ [0,1]$, define
\[ f(x)=\begin{cases}2x, &\text { if } 0 \leq x \leq \frac {1}{2}; \\
2 - 2x, &\text { if } \frac {1}{2} < x \leq 1. \end{cases}
\]Let $ f^{[2]}(x) = f(f(x))$, and $ f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $ n \geq 2$. For how many values of $ x$ in $ [0,1]$ is $ f^{[2005]}(x) = \frac {1}{2}$?
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2005 \qquad \textbf{(C)}\ 4010 \qquad \textbf{(D)}\ 2005^2 \qquad \textbf{(E)}\ 2^{2005}$
2009 Today's Calculation Of Integral, 423
Let $ f(x)\equal{}x^2\plus{}3$ and $ y\equal{}g(x)$ be the equation of the line with the slope $ a$, which pass through the point $ (0,\ f(0))$ .
Find the maximum and minimum values of $ I(a)\equal{}3\int_{\minus{}1}^1 |f(x)\minus{}g(x)|\ dx$.
2008 Harvard-MIT Mathematics Tournament, 2
Let $ f(n)$ be the number of times you have to hit the $ \sqrt {\ }$ key on a calculator to get a number less than $ 2$ starting from $ n$. For instance, $ f(2) \equal{} 1$, $ f(5) \equal{} 2$. For how many $ 1 < m < 2008$ is $ f(m)$ odd?
2004 Thailand Mathematical Olympiad, 12
Let $n$ be a positive integer and define $A_n = \{1, 2, ..., n\}$. How many functions $f : A_n \to A_n$ are there such that for all $x, y \in A_n$, if $x < y$ then $f(x) \ge f(y)$?
2024 Indonesia TST, N
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$,
$$\{ x,f(x),\cdots f^{p-1}(x) \} $$
is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$.
[i]Proposed by IndoMathXdZ[/i]
2018 VJIMC, 4
Determine all possible (finite or infinite) values of
\[\lim_{x \to -\infty} f(x)-\lim_{x \to \infty} f(x),\]
if $f:\mathbb{R} \to \mathbb{R}$ is a strictly decreasing continuous function satisfying
\[f(f(x))^4-f(f(x))+f(x)=1\]
for all $x \in \mathbb{R}$.
PEN M Problems, 22
Let $\, a$, and $b \,$ be odd positive integers. Define the sequence $\{f_n\}_{n\ge 1}$ by putting $\, f_1 = a,$ $f_2 = b, \,$ and by letting $\, f_n \,$ for $\, n \geq 3 \,$ be the greatest odd divisor of $\, f_{n-1} + f_{n-2}$. Show that $\, f_n \,$ is constant for sufficiently large $\, n \,$ and determine the eventual value as a function of $\, a \,$ and $\, b$.
2005 Iran Team Selection Test, 2
Suppose there are $n$ distinct points on plane. There is circle with radius $r$ and center $O$ on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move $O$ to the baricenter of the point in the circle. Prove that location of $O$ is constant after some steps.