Found problems: 5802
In a row, $1000$ numbers \(2\) and $2000$ numbers \(-1\) are written in some order.
Mykhailo counted the number of groups of adjacent numbers, consisting of at least two numbers, whose sum equals \(0\).
(a) Find the smallest possible value of this number.
(b) Find the largest possible value of this number.
[i]Proposed by Anton Trygub[/i]
Let $x_{0}$, $x_{1}$, $x_{2}$, $\cdots$ be a sequence of numbers, where each $x_{k}$ is either $0$ or $1$. For each positive integer $n$, define
\[S_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}\]
Suppose $7S_{n} \equiv 1\pmod {2^{n}}$ for all $n\geq 1$. What is the value of the sum
\[x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?\]
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 15$
Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$
[i]Proposed by Gabriel Carroll, USA[/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]
Let ${\bf R}$ denote the set of all real numbers. Find all functions $f$ from ${\bf R}$ to ${\bf R}$ satisfying:
(i) there are only finitely many $s$ in ${\bf R}$ such that $f(s)=0$,
and
(ii) $f(x^4+y)=x^3f(x)+f(f(y))$ for all $x,y$ in ${\bf R}$.
Let $f(x)$ and $g(x)$ be degree $n$ polynomials, and $x_0,x_1,\ldots,x_n$ be real numbers such that
$$f(x_0)=g(x_0),f'(x_1)=g'(x_1),f''(x_2)=g''(x_2),\ldots,f^{(n)}(x_n)=g^{(n)}(x_n).$$Prove that $f(x)=g(x)$ for all $x$.
A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs.
Juniors) How many vertical sides can there be in this set?
Seniors) How many ways are there to do that?
[asy]
size(120);
defaultpen(linewidth(0.8));
path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle;
for(int i=0;i<=3;i=i+1)
{
for(int j=0;j<=2;j=j+1)
{
real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2;
draw(shift(shiftx,shifty)*hex);
}
}
[/asy]
[i](T. Doslic)[/i]
2011 storage buildings are connected by roads so that it is possible to reach any building from any other building, possibly using multiple roads. The buildings contain $x_1,\dots,x_{2011}$ kilogram of cement. In one move, it is possible to relocate any quantity of cement from one building to any other building that is connected to it.
The target is to have $y_1,\dots,y_{2011}$ redistributed across storage buildings and
\[x_1+x_2+\dots+x_{2011}=y_1+y_2+\dots+y_{2011}.\] What is the minimal number of moves that the redistribution can take regardless of values of $x_i$ and $y_i$ and of the road plan?
(Author: P. Karasev)
Given an integer $k\ge 2$. Prove that there exist $k$ pairwise distinct positive integers $a_1,a_2,\ldots,a_k$ such that for any non-negative integers $b_1,b_2,\ldots,b_k,c_1,c_2,\ldots,c_k$ satisfying $a_1\le b_i\le 2a_i, i=1,2,\ldots,k$ and $\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i$, we have
\[k\prod_{i=1}^{k}b_i^{c_i}<\prod_{i=1}^{k}b_i.\]
Find all $k>0$ for which a strictly decreasing function $g:(0;+\infty)\to(0;+\infty)$ exists such that $g(x)\geq kg(x+g(x))$ for all positive $x$.
Numbers $1$ to $22$ are written on a board. A "move" is a procedure of picking two numbers $a,b$ on the board such that $b \geq a+2$, then erasing $a$ and $b$ to be replaced with $a+1$ and $b-1$. Determine the maximum possible number of moves that can be done on the board.
Determine all functions $f: \mathbb{Q} \rightarrow \mathbb{Z} $ satisfying
\[ f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right) \]
for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)
Let $n$ be a positive integer such that the number
\[\frac{1^k + 2^k + \dots + n^k}{n}\]
is an integer for any $k \in \{1, 2, \dots, 99\}$. Prove that $n$ has no divisors between 2 and 100, inclusive.
Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$.
Find $2^{2006}$ positive integers satisfying the following conditions.
(i) Each positive integer has $2^{2005}$ digits.
(ii) Each positive integer only has 7 or 8 in its digits.
(iii) Among any two chosen integers, at most half of their corresponding digits are the same.
Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions
\[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\]
for all $x \in \mathbb{Q^+}.$
De fine the [i]quasi-primes[/i] as follows.
$\bullet$ The first quasi-prime is $q_1 = 2$
$\bullet$ For $n \ge 2$, the $n^{th}$ quasi-prime $q_n$ is the smallest integer greater than $q_{n_1}$ and not of the form $q_iq_j$ for some $1 \le i \le j \le n - 1$.
Determine, with proof, whether or not $1000$ is a quasi-prime.
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that for every pair of real numbers $x$ and $y$,
\[f(x+y^2)=f(x)+|yf(y)|.\]
Raashan, Sylvia, and Ted play the following game. Each starts with $\$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $\$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $\$1$? (For example, Raashan and Ted may each decide to give $\$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $\$0$, Sylvia would have $\$2$, and Ted would have $\$1$, and and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their $\$1$ to, and and the holdings will be the same as the end of the second [sic] round.
$\textbf{(A) } \frac{1}{7} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{2}{3}$
An infinite sequence of real numbers $a_1,a_2,a_3,\dots$ is called $\emph{spooky}$ if $a_1=1$ and for all integers $n>1$,
\[\begin{array}{c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c}
na_1&+&(n-1)a_2&+&(n-2)a_3&+&\dots&+&2a_{n-1}&+&a_n&<&0,\\
n^2a_1&+&(n-1)^2a_2&+&(n-2)^2a_3&+&\dots&+&2^2a_{n-1}&+&a_n&>&0.
\end{array}\]Given any spooky sequence $a_1,a_2,a_3,\dots$, prove that
\[2013^3a_1+2012^3a_2+2011^3a_3+\cdots+2^3a_{2012}+a_{2013}<12345.\]
For any permutation $ f : \{ 1, 2, \cdots , n \} \to \{1, 2, \cdots , n \} $, and define
\[ A = \{ i | i > f(i) \} \]
\[ B = \{ (i, j) | i<j \le f(j) < f(i) \ or \ f(j) < f(i) < i < j \} \]
\[ C = \{ (i, j) | i<j \le f(i) < f(j) \ or \ f(i) < f(j) < i < j \} \]
\[ D = \{ (i, j) | i< j \ and \ f(i) > f(j)\} \]
Prove that $ |A| + 2|B| + |C| = |D| $.
A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.
(a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced.
(b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.
[i]Proposed by Jorge Tipe, Peru[/i]
Find all functions $f\colon R \to R$ such that
\[f\left(x^{2}+yf(x)\right) = f(x)^{2}+xf(y)\]
for all reals $x,y$.
Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.
Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$
[i]Michael Ren and Ankan Bhattacharya, USA[/i]