Found problems: 5802
Let $ x,y$ be real numbers such that the numbers $ x\plus{}y, x^2\plus{}y^2, x^3\plus{}y^3$ and $ x^4\plus{}y^4$ are integers. Prove that for all positive integers $ n$, the number $ x^n \plus{} y^n$ is an integer.
Let $P(x,y)$ be a polynomial such that $\deg_x(P), \deg_y(P)\le 2020$ and \[P(i,j)=\binom{i+j}{i}\] over all $2021^2$ ordered pairs $(i,j)$ with $0\leq i,j\leq 2020$. Find the remainder when $P(4040, 4040)$ is divided by $2017$.
Note: $\deg_x (P)$ is the highest exponent of $x$ in a nonzero term of $P(x,y)$. $\deg_y (P)$ is defined similarly.
[i]Proposed by Michael Ren[/i]
It is known that for every integer $n > 1$ there is a prime number among the numbers $n+1,n+2,...,2n-1.$ Determine all positive integers $n$ with the following property: Every integer $m > 1$ less than $n$ and coprime to $n$ is prime.
Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$
[b]a)[/b] Let be a nonnegative integer $ n. $ Solve in the complex numbers the equation $ z^n\cdot\Re z=\bar z^n\cdot\Im z. $
[b]b)[/b] Let be two complex numbers $ v,d $ satisfying $ v+1/v=d/\bar d +\bar d/d. $ Show that
$$ v^n+1/v^n=d^n/\bar d^n + \bar d^n/d^n, $$
for any nonnegative integer $ n. $
Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.
Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by
\[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \]
is a decreasing function.
[i]Dan Marinescu et al.[/i]
Given $a_0 > 0$ and $c > 0$, the sequence $(a_n)$ is defined by
\[a_{n+1}=\frac{a_n+c}{1-ca_n}\quad\text{for }n=1,2,\dots\]
Is it possible that $a_0, a_1, \dots , a_{1989}$ are all positive but $a_{1990}$ is negative?
Show that there is a unique positive integer which consists of the digits $2$ and $5$, having $2005$ digits and divisible by $2^{2005}$.
The sides of an equilateral triangle with sides of length $n$ have been divided into equal parts, each of length $1$, and lines have been drawn through the points of division parallel to the sides of the triangle, thus dividing the large triangle into many small triangles. Nils has a pile of rhombic tiles, each of side $1$ and angles $60^\circ$ and $120^\circ$, and wants to tile most of the triangle using these, so that each tile covers two small triangles with no overlap. In the picture, three tiles are placed somewhat arbitrarily as an illustration. How many tiles can Nils fit inside the triangle?
[asy]
/* original code by fedja: https://artofproblemsolving.com/community/c68h207503p1220868
modified by Klaus-Anton: https://artofproblemsolving.com/community/c2083h3267391_draw_me_a_grid_of_regular_triangles
*/
size(5cm);
int n=6;
pair A=(1,0), B=dir(60);
path P=A--B--(0,0)--cycle;
path Pp=A--shift(A)*B--B--cycle;
/*
label("$A$",A,S);
label("$B$",B,dir(120));
label("$(0,0)$",(0,0),dir(210));
fill(shift(2*A-1+2*B-1)*P,yellow+white);
fill(shift(2*A-1+2*B-0)*P,yellow+white);
fill(shift(2*A-1+2*B+1)*P,yellow+white);
fill(shift(2*A-1+2*B+2)*P,yellow+white);
fill(shift(1*A-1+1*B)*P,blue+white);
fill(shift(2*A-1+1*B)*P,blue+white);
fill(shift(3*A-1+1*B)*P,blue+white);
fill(shift(4*A-1+1*B)*P,blue+white);
fill(shift(5*A-1+1*B)*P,blue+white);
fill(shift(0*A+0*B)*P,green+white);
fill(shift(0*A+1+0*B)*P,green+white);
fill(shift(0*A+2+0*B)*P,green+white);
fill(shift(0*A+3+0*B)*P,green+white);
fill(shift(0*A+4+0*B)*P,green+white);
fill(shift(0*A+5+0*B)*P,green+white);
fill(shift(2*A-1+3*B-1)*P,magenta+white);
fill(shift(3*A-1+3*B-1)*P,magenta+white);
fill(shift(4*A-1+3*B-1)*P,magenta+white);
fill(shift(5*A+5*B-5)*P,heavyred+white);
fill(shift(4*A+4*B-4)*P,palered+white);
fill(shift(4*A+4*B-3)*P,palered+white);
fill(shift(0*A+0*B)*Pp,gray);
fill(shift(0*A+1+0*B)*Pp,gray);
fill(shift(0*A+2+0*B)*Pp,gray);
fill(shift(0*A+3+0*B)*Pp,gray);
fill(shift(0*A+4+0*B)*Pp,gray);
fill(shift(1*A+1*B-1)*Pp,lightgray);
fill(shift(1*A+1*B-0)*Pp,lightgray);
fill(shift(1*A+1*B+1)*Pp,lightgray);
fill(shift(1*A+1*B+2)*Pp,lightgray);
fill(shift(2*A+2*B-2)*Pp,red);
fill(shift(2*A+2*B-1)*Pp,red);
fill(shift(2*A+2*B-0)*Pp,red);
fill(shift(3*A+3*B-2)*Pp,blue);
fill(shift(3*A+3*B-3)*Pp,blue);
fill(shift(4*A+4*B-4)*Pp,cyan);
fill(shift(0*A+1+0*B)*Pp,gray);
fill(shift(0*A+2+0*B)*Pp,gray);
fill(shift(0*A+3+0*B)*Pp,gray);
fill(shift(0*A+4+0*B)*Pp,gray);
*/
fill(Pp, rgb(244, 215, 158));
fill(shift(dir(60))*P, rgb(244, 215, 158));
fill(shift(1.5,(-sqrt(3)/2))*shift(2*dir(60))*Pp, rgb(244, 215, 158));
fill(shift(1.5,(-sqrt(3)/2))*shift(2*dir(60))*P, rgb(244, 215, 158));
fill(shift(-.5,(-sqrt(3)/2))*shift(4*dir(60))*Pp, rgb(244, 215, 158));
fill(shift(.5,(-sqrt(3)/2))*shift(4*dir(60))*P, rgb(244, 215, 158));
for(int i=0;i<n;++i){
for(int j;j<n-i;++j)
{draw(shift(i*A+j*B)*P);}}
shipout(bbox(2mm,Fill(white)));
[/asy]
Let $ \,n > 6\,$ be an integer and $ \,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $ n$ and relatively prime to $ n$. If
\[ a_{2} \minus{} a_{1} \equal{} a_{3} \minus{} a_{2} \equal{} \cdots \equal{} a_{k} \minus{} a_{k \minus{} 1} > 0,
\]
prove that $ \,n\,$ must be either a prime number or a power of $ \,2$.
Let $ \{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \} $ . Find the maximum value of
\[ \sum_{n=1}^{10}(na_n ^2 - n^2 a_n ) \]
Let $M(n)$ and $m(n)$ are maximal and minimal proper divisors of $n$
Natural number $n>1000$ is on the board. Every minute we replace our number with $n+M(n)-m(n)$. If we get prime, then process is stopped.
Prove that after some moves we will get number, that is not divisible by $17$
In a 2025 by 2025 grid, every cell initially contains a `1'. Every minute, we simultaneously replace the number in each cell with the sum of numbers in the cells that share an edge with it. (For example, after the first minute, the number 2 is written in each of the four
corner cells.)
After 2025 minutes, we colour the board in checkerboard fashion, such that the top left corner is black. Find the difference between the sum of numbers in black cells and the sum of numbers in white cells.
[i]Proposed by chorn[/i]
A table tennis club hosts a series of doubles matches following several rules:
(i) each player belongs to two pairs at most;
(ii) every two distinct pairs play one game against each other at most;
(iii) players in the same pair do not play against each other when they pair with others respectively.
Every player plays a certain number of games in this series. All these distinct numbers make up a set called the “[i]set of games[/i]”. Consider a set $A=\{a_1,a_2,\ldots ,a_k\}$ of positive integers such that every element in $A$ is divisible by $6$. Determine the minimum number of players needed to participate in this series so that a schedule for which the corresponding [i]set of games [/i] is equal to set $A$ exists.
Does there exist a function $f:\mathbb{Z}\to\mathbb{Z}$ such that $f(x+f(y))=f(x)-y$ for all integers $x$ and $y$?
Let $a_1, a_2, \dots, a_n$ be a sequence of real numbers, and let $m$ be a fixed positive integer less than $n$. We say an index $k$ with $1\le k\le n$ is good if there exists some $\ell$ with $1\le \ell \le m$ such that $a_k+a_{k+1}+...+a_{k+\ell-1}\ge0$, where the indices are taken modulo $n$. Let $T$ be the set of all good indices. Prove that $\sum\limits_{k \in T}a_k \ge 0$.
[i]Proposed by Mark Sellke[/i]
Find all sequences $0\le a_0\le a_1\le a_2\le \ldots$ of real numbers such that
\[a_{m^2+n^2}=a_m^2+a_n^2 \]
for all integers $m,n\ge 0$.
Given a sequence $ (c_n) $ of natural numbers defined recursively: $ c_1 = 2 $, $ c_{n+1} = \left[ \frac{3}{2}c_n\right] $. Prove that there are infinitely many even numbers and infinitely many odd numbers among the terms of this sequence.
Find all polynomials $P (x)$ with complex coefficients such that $$P (x^2) = P (x) · P (x + 2)$$
for any complex number $x.$
Let $ p \ge 3$ be a prime number. The sequence $ \{a_{n}\}_{n \ge 0}$ is defined by $ a_{n}=n$ for all $ 0 \le n \le p-1$, and $ a_{n}=a_{n-1}+a_{n-p}$ for all $ n \ge p$. Compute $ a_{p^{3}}\; \pmod{p}$.
Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$.
[i]Proposed by Warut Suksompong, Thailand[/i]
In a company of people some pairs are enemies. A group of people is called [i]unsociable[/i] if the number of members in the group is odd and at least $3$, and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most $2015$ unsociable groups, prove that it is possible to partition the company into $11$ parts so that no two enemies are in the same part.
[i]Proposed by Russia[/i]
Determine all sets of real numbers $S$ such that:
[list]
[*] $1$ is the smallest element of $S$,
[*] for all $x,y\in S$ such that $x>y$, $\sqrt{x^2-y^2}\in S$
[/list]
[i]Adian Anibal Santos Sepcic[/i]
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]