Found problems: 5802
A positive integer $n$ is given. A cube $3\times3\times3$ is built from $26$ white and $1$ black cubes $1\times1\times1$ such that the black cube is in the center of $3\times3\times3$-cube. A cube $3n\times 3n\times 3n$ is formed by $n^3$ such $3\times3\times3$-cubes. What is the smallest number of white cubes which should be colored in red in such a way that every white cube will have at least one common vertex with a red one.
[hide=thanks] Thanks to the user Vlados021 for translating the problem.[/hide]
The terms of a sequence $(T_n)$ satisfy $T_n T_{n+1} =n$ for all positive integers $n$ and
$$\lim_{n\to \infty} \frac{ T_{n} }{ T_{n+1}}=1.$$
Show that $ \pi T_{1}^{2}=2.$
Let $a_1, a_2, \ldots$ be a sequence of real numbers such that $a_1=4$ and $a_2=7$ such that for all integers $n$, $\frac{1}{a_{2n-1}}, \frac{1}{a_{2n}}, \frac{1}{a_{2n+1}}$ forms an arithmetic progression, and $a_{2n}, a_{2n+1}, a_{2n+2}$ forms an arithmetic progression. Find, with proof, the prime factorization of $a_{2019}$.
We are given more than $2^k$ integers, where $k\in\mathbb{N}$. Prove that we can choose $k+2$ of them such that if some of our selected numbers satisfy
\[x_1+x_2+\dots+x_m=y_1+y_2+\dots+y_m\]
where $x_1<\dots<x_m$ and $y_1<\dots<y_m$, then $x_i=y_i$ for any $1\le i\le m$.
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a monotonically increasing function over the natural numbers, such that $f(f(n))=n^{2}$. What is the smallest, and what is the largest value that $f(2023)$ can take?
[i]Proposed by Ilija Jovcheski[/i]
Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$
[i]Proposed by Carl Schildkraut[/i]
Let $A=\{a_1,a_2,\cdots,a_{2010}\}$ and $B=\{b_1,b_2,\cdots,b_{2010}\}$ be two sets of complex numbers. Suppose
\[\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k\]
holds for every $k=1,2,\cdots, 2010$. Prove that $A=B$.
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$.
Prove that $S$ can be covered by a strip of width $2$.
Suppose $P(x)$ is a polynomial with real coefficients, satisfying the condition $P(\cos \theta+\sin \theta)=P(\cos \theta-\sin \theta)$, for every real $\theta$. Prove that $P(x)$ can be expressed in the form$$P(x)=a_0+a_1(1-x^2)^2+a_2(1-x^2)^4+\dots+a_n(1-x^2)^{2n}$$for some real numbers $a_0, a_1, \dots, a_n$ and non-negative integer $n$.
[i]Proposed by C.R. Pranesacher[/i]
A positive integer $N$ is said to be $n-$ good if
(i) $N$ has at least $n$ distinct prime divisors, and
(ii) there exist distinct positive divisors $1, x_{2}, . . . , x_{n}$ whose sum is $N$ .
Show that there exists an $n-$ good number for each $n\geq 6$.
Assume $ a_{1} \ge a_{2} \ge \dots \ge a_{107} > 0 $ satisfy $ \sum\limits_{k=1}^{107}{a_{k}} \ge M $ and $ b_{107} \ge b_{106} \ge \dots \ge b_{1} > 0 $ satisfy $ \sum\limits_{k=1}^{107}{b_{k}} \le M $. Prove that for any $ m \in \{1,2, \dots, 107\} $, the arithmetic mean of the following numbers $$ \frac{a_{1}}{b_{1}}, \frac{a_{2}}{b_{2}}, \dots, \frac{a_{m}}{b_{m}} $$ is greater than or equal to $ \frac{M}{N} $
A sequence of integers is defined as follows: $a_1=1,a_2=2,a_3=3$ and for $n>3$, $$a_n=\textsf{The smallest integer not occurring earlier, which is relatively prime to }a_{n-1}\textsf{ but not relatively prime to }a_{n-2}.$$Prove that every natural number occurs exactly once in this sequence.
[i]M. Ivanov[/i]
Let $n>1$ be an integer and $a,b,c$ be three complex numbers such that $a+b+c=0$ and $a^n+b^n+c^n=0$. Prove that two of $a,b,c$ have the same magnitude.
[i]Evan O'Dorney.[/i]
a tournament is playing between n persons. Everybody plays with everybody one time. There is no draw here. A number $k$ is called $n$ good if there is any tournament such that in that tournament they have any player in the tournament that has lost all of $k$'s.
prove that
1. $n$ is greater than or equal to $2^{k+1}-1$
2.Find all $n$ such that $2$ is a n-good
Let $(a_n)_{n\in \mathbb N}$ be the sequence of integers defined recursively by $a_1 = a_2 = 1, a_{n+2} = 7a_{n+1} - a_n - 2$ for $n \geq 1$. Prove that $a_n$ is a perfect square for every $n.$
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.
Prove: \[|\sin a_1|+|\sin a_2|+|\sin a_3|+\ldots+|\sin a_n|+|\cos(a_1+a_2+a_3+\ldots+a_n)|\geq 1.\]
Find all integer $c\in\{0,1,...,2016\}$ such that the number of $f:\mathbb{Z}\rightarrow\{0,1,...,2016\}$ which satisfy the following condition is minimal:\\
(1) $f$ has periodic $2017$\\
(2) $f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017}$\\
Proposed by William Chao
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 $x$. )
10.4, 11.3 Given $N\geq 3$ points enumerated with 1, 2, ..., $N$. Each two numbers are connected by mean of arrow from a lesser number to a greater one. A coloring of all arrows into red and blue is called [i]monochromatic[/i] iff for any numbers $A$ and $B$ there are [color=red]no[/color] two monochromatic paths from $A$ to $B$ of different colors. Find the number of monochromatic colorings.
([i]I. Bogdanov, G. Chelnokov[/i])
Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.)
[i]Proposed by Milan Haiman[/i]
Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$
[i]Andrew Carratu[/i]
Let $n$ be a positive integer. Prove that
\[ \binom{n}{0}^{-1} + \binom{n}{1}^{-1} + \cdots + \binom{n}{n}^{-1} = \frac{n+1}{2^{n+1}} \left( \frac{2}{1} + \frac{2^2}{2} + \cdots + \frac{2^{n+1}}{n+1} \right). \]