Found problems: 5802
Between any two cities of a country there is only one one-way road. Show that there is a city from that every other city can be reached directly or by going over only one intermediate city.
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I'm sure it was posted before but couldn't find it.
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Let $a_1, a_2, \dots, a_{2^{2016}}$ be positive integers not bigger than $2016$. We know that for each $n \leq 2^{2016}$, $a_1a_2 \dots a_{n} +1 $ is a perfect square. Prove that for some $i $ , $a_i=1$.
Let $S = \left\{ 1,2,\dots,n \right\}$, where $n \ge 1$. Each of the $2^n$ subsets of $S$ is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set $T \subseteq S$, we then write $f(T)$ for the number of subsets of $T$ that are blue.
Determine the number of colorings that satisfy the following condition: for any subsets $T_1$ and $T_2$ of $S$, \[ f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2). \]
Find $f_n(x)$ such that $f_1(x)=x,\ f_n(x)=\int_0^x tf_{n-1}(x-t)dt\ (n=2,\ 3,\ \cdots).$
Let $ k\equal{}2008^2\plus{}2^{2008}$. What is the units digit of $ k^2\plus{}2^k$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 8$
Given integer $n\geq 2$ and real numbers $x_1,x_2,\cdots, x_n$ in the interval $[0,1]$. Prove that there exist real numbers $a_0,a_1,\cdots,a_n$ satisfying the following conditions:
(1) $a_0+a_n=0$;
(2) $|a_i|\leq 1$, for $i=0,1,\cdots,n$;
(3) $|a_i-a_{i-1}|=x_i$, for $i=1,2,\cdots,n$.
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
Let $n>1$ be an integer. Prove that there exists an integer $n-1 \ge m \ge \left \lfloor \frac{n}{2} \right \rfloor$ such that the following equation has integer solutions with $a_m>0:$
$$\frac{a_{m}}{m+1}+\frac{a_{m+1}}{m+2}+ \cdots + \frac{a_{n-1}}{n}=\frac{1}{\textrm{lcm}\left ( 1,2, \cdots , n \right )}$$
[i]Proposed by Navid Safaei[/i]
We denote $N_{2010}=\{1,2,\cdots,2010\}$
[b](a)[/b]How many non empty subsets does this set have?
[b](b)[/b]For every non empty subset of the set $N_{2010}$ we take the product of the elements of the subset. What is the sum of these products?
[b](c)[/b]Same question as the [b](b)[/b] part for the set $-N_{2010}=\{-1,-2,\cdots,-2010\}$.
Albanian National Mathematical Olympiad 2010---12 GRADE Question 2.
Let $ \, a_{0}, a_{1}, a_{2},\ldots\,$ be a sequence of positive real numbers satisfying $ \, a_{i\minus{}1}a_{i\plus{}1}\leq a_{i}^{2}\,$ for $ i \equal{} 1,2,3,\ldots\; .$ (Such a sequence is said to be [i]log concave[/i].) Show that for each $ \, n > 1,$
\[ \frac{a_{0}\plus{}\cdots\plus{}a_{n}}{n\plus{}1}\cdot\frac{a_{1}\plus{}\cdots\plus{}a_{n\minus{}1}}{n\minus{}1}\geq\frac{a_{0}\plus{}\cdots\plus{}a_{n\minus{}1}}{n}\cdot\frac{a_{1}\plus{}\cdots\plus{}a_{n}}{n}.\]
There are $n \ge 3$ positive integers written on a board. A [i]move[/i] consists of choosing three numbers $a, b, c$ written from the board such that there exists a non-degenerate non-equilateral triangle with sides $a, b, c$ and replacing those numbers with $a + b - c, b + c - a$ and $c + a - b$.
Prove that a sequence of moves cannot be infinite.
Find all integers $x,y,z$ such that:
$7^x+13^y=2^z$
Let $n$ be a given positive integer. Say that a set $K$ of points with integer coordinates in the plane is connected if for every pair of points $R, S\in K$, there exists a positive integer $\ell$ and a sequence $R=T_0,T_1, T_2,\ldots ,T_{\ell}=S$ of points in $K$, where each $T_i$ is distance $1$ away from $T_{i+1}$. For such a set $K$, we define the set of vectors
\[\Delta(K)=\{\overrightarrow{RS}\mid R, S\in K\}\]
What is the maximum value of $|\Delta(K)|$ over all connected sets $K$ of $2n+1$ points with integer coordinates in the plane?
[i]Grigory Chelnokov, Russia[/i]
Let $ A$ be the subset of the set of positive integers, having the following $ 2$ properties:
1) If $ a$ belong to $ A$,than all of the divisors of $ a$ also belong to $ A$;
2) If $ a$ and $ b$, $ 1 < a < b$, belong to $ A$, than $ 1 \plus{} ab$ is also in $ A$;
Prove that if $ A$ contains at least $ 3$ positive integers, than $ A$ contains all positive integers.
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
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Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
For a finite set $ X$ of positive integers, let $ \Sigma(X) \equal{} \sum_{x \in X} \arctan \frac{1}{x}.$ Given a finite set $ S$ of positive integers for which $ \Sigma(S) < \frac{\pi}{2},$ show that there exists at least one finite set $ T$ of positive integers for which $ S \subset T$ and $ \Sigma(S) \equal{} \frac{\pi}{2}.$
[i]Kevin Buzzard, United Kingdom[/i]
Let $N$ be an integer greater than $1$ and let $T_n$ be the number of non empty subsets $S$ of $\{1,2,.....,n\}$ with the property that the average of the elements of $S$ is an integer.Prove that $T_n - n$ is always even.
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.
[i]Proposed by Mohsen Jamaali, Iran[/i]
An infinite sequence of positive real numbers $x_0,x_1,x_2,...$ is called $vasco$ if it satisfies the following properties:
(a) $x_0=1,x_1=3$; and
(b) $x_0+x_1+...+x_{n-1}\ge3x_{n}-x_{n+1}$, for every $n\ge1$.
Find the greatest real number $M$ such that, for every $vasco$ sequence, the inequality $\frac{x_{n+1}}{x_{n}}>M$ is true for every $n\ge0$.
Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board.
[list=i]
[*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$.
[*] If no such pair exists, we write two times the number $0$.
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Prove that, no matter how we make the choices in $(i)$, operation $(ii)$ will be performed only finitely many times.
Proposed by [I]Serbia[/I].
Find all sequences $(a_n)_{n\geq 1}$ of positive integers such that for all integers $n\geq 3$ we have
$$
\dfrac{1}{a_1 a_3} + \dfrac{1}{a_2a_4} + \cdots
+ \dfrac{1}{a_{n-2}a_n}= 1 - \dfrac{1}{a_1^2+a_2^2+\cdots +a_{n-1}^2}.
$$
Show that
$$ \int_{0}^{1} x^{x} \, dx = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^n }.$$
Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.
There are $n$ people in a city, and each of them has exactly $1000$ friends (friendship is always symmetric). Prove that it is possible to select a group $S$ of people such that at least $\frac{n}{2017}$ persons in $S$ have exactly two friends in $S$.