Found problems: 5802
For an $A=\{ a_i\}^{\infty}_{i=0}$ sequence let $SA=\{ a_0, a_0+a_1, a_0+a_1+a_2, \ldots\}$ be the sequence of partial sums of the $a_0+a_1+\ldots$ series. Does there exist a non-identically zero sequence $A$ such that all of the sequences $A, SA, SSA, SSSA, \ldots$ are convergent?
(translated by Miklós Maróti)
Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$.
[i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]
Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$?
[i]Proposed by David Yang[/i]
Suppose $p$ is an odd prime number. We call the polynomial $f(x)=\sum_{j=0}^n a_jx^j$ with integer coefficients $i$-remainder if $ \sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}$. Prove that the set $\{f(0),f(1),...,f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), (f(x))^2,...,(f(x))^{p-2}$ are $0$-remainder and the polynomial $(f(x))^{p-1}$ is $1$-remainder.
[i]Proposed by Yahya Motevassel[/i]
Define the sequence of real numbers $\{x_n\}_{n \geq 1}$, where $x_1$ is any real number and \[x_n = 1 - x_1x_2\ldots x_{n-1} \text{ for all } n > 1.\] Show that $x_{2011} > \frac{2011}{2012}$.
Let $n \neq 0$. For every sequence of integers \[ A = a_0,a_1,a_2,\dots, a_n \] satisfying $0 \le a_i \le i$, for $i=0,\dots,n$, define another sequence \[ t(A)= t(a_0), t(a_1), t(a_2), \dots, t(a_n) \] by setting $t(a_i)$ to be the number of terms in the sequence $A$ that precede the term $a_i$ and are different from $a_i$. Show that, starting from any sequence $A$ as above, fewer than $n$ applications of the transformation $t$ lead to a sequence $B$ such that $t(B) = B$.
A biologist watches a chameleon. The chameleon catches flies and rests after each catch. The biologist notices that:
[list=1][*]the first fly is caught after a resting period of one minute;
[*]the resting period before catching the $2m^\text{th}$ fly is the same as the resting period before catching the $m^\text{th}$ fly and one minute shorter than the resting period before catching the $(2m+1)^\text{th}$ fly;
[*]when the chameleon stops resting, he catches a fly instantly.[/list]
[list=a][*]How many flies were caught by the chameleon before his first resting period of $9$ minutes in a row?
[*]After how many minutes will the chameleon catch his $98^\text{th}$ fly?
[*]How many flies were caught by the chameleon after 1999 minutes have passed?[/list]
For any positive integer $ x$ define $ g(x)$ as greatest odd divisor of $ x,$ and
\[ f(x) \equal{} \begin{cases} \frac {x}{2} \plus{} \frac {x}{g(x)} & \text{if \ \(x\) is even}, \\
2^{\frac {x \plus{} 1}{2}} & \text{if \ \(x\) is odd}. \end{cases}
\]
Construct the sequence $ x_1 \equal{} 1, x_{n \plus{} 1} \equal{} f(x_n).$ Show that the number 1992 appears in this sequence, determine the least $ n$ such that $ x_n \equal{} 1992,$ and determine whether $ n$ is unique.
Carl chooses a [i]functional expression[/i]* $E$ which is a finite nonempty string formed from a set $x_1, x_2, \dots$ of variables and applications of a function $f$, together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation $E = 0$, and lets $S$ denote the set of functions $f \colon \mathbb R \to \mathbb R$ such that the equation holds for any choices of real numbers $x_1, x_2, \dots$. (For example, if Carl chooses the functional equation
$$ f(2f(x_1)+x_2) - 2f(x_1)-x_2 = 0, $$
then $S$ consists of one function, the identity function.
(a) Let $X$ denote the set of functions with domain $\mathbb R$ and image exactly $\mathbb Z$. Show that Carl can choose his functional equation such that $S$ is nonempty but $S \subseteq X$.
(b) Can Carl choose his functional equation such that $|S|=1$ and $S \subseteq X$?
*These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer $i$, the variable $x_i$ is a functional expression, and (iii) if $V$ and $W$ are functional expressions, then so are $f(V)$, $V+W$, $V-W$, and $V \cdot W$.
[i]Proposed by Carl Schildkraut[/i]
Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.
Find all functions $f$ from the set of real numbers to itself satisfying
\[ f(x(1+y)) = f(x)(1 + f(y)) \]
for all real numbers $x, y$.
Let $K$ be the set of all positive integers that do not contain the digit $7$ in their base-$10$ representation. Find all polynomials $f$ with nonnegative integer coefficients such that $f(n)\in K$ whenever $n\in K$.
[i]Proposed by Titu Andreescu, Cosmin Pohoata, and Vlad Matei[/i]
Elmo calls a monic polynomial with real coefficients [i]tasty[/i] if all of its coefficients are in the range $[-1,1]$. A monic polynomial $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$.
[i]Proposed by Carl Schildkraut[/i]
We are given a combination lock consisting of $6$ rotating discs. Each disc consists of digits $0, 1, 2,\ldots , 9$ in that order (after digit $9$ comes $0$). Lock is opened by exactly one combination. A move consists of turning one of the discs one digit in any direction and the lock opens instantly if the current combination is correct. Discs are initially put in the position $000000$, and we know that this combination is not correct.
[list]
a) What is the least number of moves necessary to ensure that we have found the correct combination?
b) What is the least number of moves necessary to ensure that we have found the correct combination, if we
know that none of the combinations $000000, 111111, 222222, \ldots , 999999$ is correct?[/list]
[i]Proposed by Ognjen Stipetić and Grgur Valentić[/i]
There are $n$ cities in a country, where $n>1$. There are railroads connecting some of the cities so that you can travel between any two cities through a series of railroads (railroads run in both direction.) In addition, in this country, it is impossible to travel from a city, through a series of distinct cities, and return back to the original city. We define the [b]degree[/b] of a city as the number of cities directly connected to it by a single segment of railroad. For a city $A$ that is directly connected to $x$ cities, with $y$ of those cities having a smaller degree than city $A$, the [b]significance[/b] of city $A$ is defined as $\frac{y}{x}$.
Find the smallest positive real number $t$ so that, for any $n>1$, the sum of the significance of all cities is less than $tn$, no matter how the railroads are paved.
[i]Proposed by houkai[/i]
In a country there are $n\geq 2$ cities. Any two cities has exactly one two-way airway. The government wants to license several airlines to take charge of these airways with such following conditions:
i) Every airway can be licensed to exactly one airline.
ii) By choosing one arbitrary airline, we can move from a city to any other cities, using only flights from this airline.
What is the maximum number of airlines that the government can license to satisfy all of these conditions?
Prove that there is a multiple of $2^{2022}$ that has $2022$ digits, and can be written using digits $1$ and $2$ only.
Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$.
Prove that Sisyphus cannot reach the aim in less than
\[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \]
turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$.
Let $\mathbb{Q}$ be the set of rational numbers. Determine all functions $f : \mathbb{Q}\to\mathbb{Q}$ satisfying both of the following conditions.
[list=disc]
[*] $f(a)$ is not an integer for some rational number $a$.
[*] For any rational numbers $x$ and $y$, both $f(x + y) - f(x) - f(y)$ and $f(xy) - f(x)f(y)$ are integers.
[/list]
Let $\mathcal{P}$ be a finite, non-self-intersecting loop in the unit hexagonal grid.¹ All edges enclosed by, but not on $\mathcal P$, are colored green. Let $A$ be the number of green edges, $B$ be the number of connected components of green edges,² and $C$ be the number of unit hexagons enclosed by but not touching $\mathcal{P}$. Prove that $A + B + C$ is even.
For example, for the following closed loop highlighted in black, $A=34$, $B = 2$, and $C = 2$.
[center][img width=40]https://i.imgur.com/PDJdoS5.png[/img][/center]
[size=75]¹The [i]unit hexagonal grid[/i] is the unique arrangement, up to rotation and translation, of unit segments in the plane which partitions the plane into regular hexagons.[/size]
[size=75]²Two green edges are said to be in the same connected component if it is possible to walk from one to the other by only walking along green edges.[/size]
[i]Ethan Liu[/i]
Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$
Determine whether or not there exist positive integers $ a$ and $ b$ such that $ a$ does not divide $ b^n \minus{} n$ for all positive integers $ n$.
Determine all polynomials $P{}$ with integer coefficients, satisfying $0 \leqslant P (n) \leqslant n!$ for all non-negative integers $n$.
[i]Andrei Chirita[/i]
Magicman and his helper want to do some magic trick. They have special card desk. Back of all cards is common color and face is one of $2017$ colors.
Magic trick: magicman go away from scene. Then viewers should put on the table $n>1$ cards in the row face up. Helper looks at these cards, then he turn all cards face down, except one, without changing order in row. Then magicman returns on the scene, looks at cards, then show on the one card, that lays face down and names it face color.
What is minimal $n$ such that magicman and his helper can has strategy to make magic trick successfully?