Found problems: 5802
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Two ants are moving along the edges of a convex polyhedron. The route of every ant ends in its starting point, so that one ant does not pass through the same point twice along its way. On every face $F$ of the polyhedron are written the number of edges of $F$ belonging to the route of the first ant and the number of edges of $F$ belonging to the route of the second ant. Is there a polyhedron and a pair of routes described as above, such that only one face contains a pair of distinct numbers?
[i]Proposed by Nikolai Beluhov[/i]
Find all positive integers \( m \) for which there exists an infinite subset \( A \) of the positive integers such that: for any pairwise distinct positive integers \( a_1, a_2, \cdots, a_m \in A \), the sum \( a_1 + a_2 + \cdots + a_m \) and the product \( a_1a_2 \cdots a_m \) are both square-free.
Given coprime positive integers $p,q>1$, call all positive integers that cannot be written as $px+qy$(where $x,y$ are non-negative integers) [i]bad[/i], and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$. Prove that there exists a positive integer $n$, such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$.
Let $r_n$ be the $n$th smallest positive solution to $\tan x=x$, where the argument of tangent is in radians. Prove that
\[
0<r_{n+1}-r_n-\pi<\frac{1}{(n^2+n)\pi}
\]
for $n\geq 1$.
The sequence $ \{a_n\}$ of integers is defined by
\[ a_1 \equal{} 2, a_2 \equal{} 7
\]
and
\[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2.
\]
Prove that $ a_n$ is odd for all $ n > 1.$
We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell. Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?
The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$.
We wish to find the sum of $40$ given numbers utilizing $40$ processors. Initially, we have the number $0$ on the screen of each processor. Each processor adds the number on its screen with a number entered directly (only the given numbers could be entered directly to the processors) or transferred from another processor in a unit time. Whenever a number is transferred from a processor to another, the former processor resets. Find the least time needed to find the desired sum.
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y, \in \mathbb{R}$,
\[xf(x+xy)=xf(x)+f(x^{2})\cdot f(y).\]
Let $n$ be a positive integer not divisible by $2$ or $3$. Prove that for all integers $k$, the number $(k+1)^n-k^n-1$ is divisible by $k^2+k+1$.
Let $n$ be a positive integer, and $a_j$, for $j=1,2,\ldots,n$ are complex numbers. Suppose $I$ is an arbitrary nonempty subset of $\{1,2,\ldots,n\}$, the inequality $\left|-1+ \prod_{j\in I} (1+a_j) \right| \leq \frac 12$ always holds.
Prove that $\sum_{j=1}^n |a_j| \leq 3$.
Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.
The audience arranges $n$ coins in a row. The sequence of heads and tails is chosen arbitrarily. The audience also chooses a number between $1$ and $n$ inclusive. Then the assistant turns one of the coins over, and the magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by the audience.
[list][b](a)[/b] Prove that if this is possible for some $n$, then it is also possible for $2n$.
[b](b)[/b] Determine all $n$ for which this is possible.[/list]
Let be a sequence $ \left( x_n \right)_{n\ge 0} $ with $ x_0\in (0,1) $ and defined as
$$ 2x_n=x_{n-1}+\sqrt{3-3x_{n-1}^2} . $$
Prove that this sequence is bounded and periodic. Moreover, find $ x_0 $ for which this sequence is convergent.
[i]Ovidiu Țâțan[/i]
Calculate the exact value of the series $\sum _{n=2} ^\infty \log (n^3 +1) - \log (n^3 - 1)$ and provide justification.
Let $F$ be a family of subsets of $S = \left \{ 1,2,...,n \right \}$ ($n \geq 2$). A valid play is to choose two disjoint sets $A$ and $B$ from $F$ and add $A \cup B$ to $F$ (without removing $A$ and $B$).
Initially, $F$ has all the subsets that contain only one element of $S$. The goal is to have all subsets of $n - 1$ elements of $S$ in $F$ using valid plays.
Determine the lowest number of plays required in order to achieve the goal.
Vincent has a fair die with sides labeled $1$ to $6$. He first rolls the die and records it on a piece of paper. Then, every second thereafter, he re-rolls the die. If Vincent rolls a different value than his previous roll, he records the value and continues rolling. If Vincent rolls the same value, he stops, does \emph{not} record his final roll, and computes the average of his previously recorded rolls. Given that Vincent first rolled a $1$, let $E$ be the expected value of his result. There exist rational numbers $r,s,t > 0$ such that $E = r-s\ln t$ and $t$ is not a perfect power. If $r+s+t = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Sean Li[/i]
A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation
$$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$
Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.
Let $m$ be a positive integer, and $A_1, A_2, \ldots, A_m$ (not necessarily different) be $m$ subsets of a finite set $A$. It is known that for any nonempty subset $I$ of $\{1, 2 \ldots, m \}$,
\[ \Big| \bigcup_{i \in I} A_i \Big| \ge |I|+1. \]
Show that the elements of $A$ can be colored black and white, so that each of $A_1,A_2,\ldots,A_m$ contains both black and white elements.
Given a positive integer $ n\geq 2$, consider a set of $ n$ islands arranged in a circle. Between every two neigboring islands two bridges are built as shown in the figure.
Starting at the island $ X_1$, in how many ways one can one can cross the $ 2n$ bridges so that no bridge is used more than once?
Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.
A function $ f:\mathbb{Q}_{>0}\longrightarrow\mathbb{Q} $ has the following property:
$$ f(xy)=f(x)+f(y),\quad x,y\in\mathbb{Q}_{>0} $$
[b]a)[/b] Demonstrate that there are no injective functions with this property.
[b]b)[/b] Do exist surjective functions having this property?
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
A family $L$ of 2006 lines on the plane is given in such a way that it doesn't contain
parallel lines and it doesn't contain three lines with a common point.We say that
the line $l_1\in L$ is [i]bounding[/i] the line $l_2\in L$,if all intersection points
of the line $l_2$ with other lines from $L$ lie on the one side of the line $l_1$.
Prove that in the family $L$ there are two lines $l$ and $l'$ such that the following
2 conditions are satisfied simultaneously:
[b]1)[/b] The line $l$ is bounding the line $l'$;
[b]2)[/b] the line $l'$ is not bounding the line $l$.