Found problems: 85335
Let $ f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ ,$ where $ \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty$ for some $ \alpha > 2$. Let us assume that for some $ \beta > 1/{\alpha}$, we have $ f(x)\equal{}O(e^{\minus{}x^{\beta}})$ as $ x \rightarrow \infty$. Prove that $ a_n$ is identically $ 0$.
[i]G. Halasz[/i]
Point $O$ is the center of circumcircle $\omega$ of the isosceles triangle $ABC$ ($AB = AC$). Bisector of the angle $\angle C$ intersects $\omega$ at the point $W$. Point $Q$ is the center of the circumcircle of the triangle $OWB$. Construct the triangle $ABC$ given the points $Q,W, B$.
(Andrey Mostovy)
The convex quadrilateral $ABCD$ is inscribed in the circle $S_1$. Let $O$ be the intersection of $AC$ and $BD$. Circle $S_2$ passes through $D$ and $ O$, intersecting $AD$ and $CD$ at $ M$ and $ N$, respectively. Lines $OM$ and $AB$ intersect at $R$, lines $ON$ and $BC$ intersect at $T$, and $R$ and $T$ lie on the same side of line $BD$ as $ A$.
Prove that $O$, $R$,$T$, and $B$ are concyclic.
Define a function $f$ as follows. For any positive integer $i$, let $f(i)$ be the smallest positive integer $j$ such that there exist pairwise distinct positive integers $a,b,c,$ and $d$ such that $\gcd(a,b)$, $\gcd(a,c)$, $\gcd(a,d)$, $\gcd(b,c)$, $\gcd(b,d)$, and $\gcd(c,d)$ are pairwise distinct and equal to $i, i+1, i+2, i+3, i+4,$ and $j$ in some order, if any such $j$ exists; let $f(i)=0$ if no such $j$ exists. Compute $f(1)+f(2)+\dots +f(2019)$.
[i]Proposed by Edward Wan[/i]
Find all functions $ f : \mathbb{R}\to\mathbb{R}$ satisfying following conditions.
(a) $ f(x) \ge 0 $ for all $ x \in \mathbb{R} $.
(b) For $ a, b, c, d \in \mathbb{R} $ with $ ab + bc + cd = 0 $, equality $ f(a-b) + f(c-d) = f(a) + f(b+c) + f(d) $ holds.
The numbers $1, 2,. . . , 33$ are written on the board . A student performs the following procedure:
choose two numbers from those written on the board so that one of them is a multiple of the other number; after the election he deletes the two numbers and writes on the board their number. The student repeats the procedure so many times until only numbers without multiples remain on the board. Determine how many numbers they remain on the board in the situation where the student can no longer repeat the procedure.
Denote by $s(n)$ the sum of all natural divisors of $n$ which are smaller than $n$. Does there exist a positive integer $a$ such that the equation $$s(n)=a+n$$ has infinitely many solutions in positive integers?
At a mathematical competition $n$ students work on 6 problems each one with three possible answers. After the competition, the Jury found that for every two students the number of the problems, for which these students have the same answers, is 0 or 2. Find the maximum possible value of $n$.
Ten gamblers started playing with the same amount of money. Each turn they cast (threw) five dice. At each stage the gambler who had thrown paid to each of his 9 opponents $\frac{1}{n}$ times the amount which that opponent owned at that moment. They threw and paid one after the other. At the 10th round (i.e. when each gambler has cast the five dice once), the dice showed a total of 12, and after payment it turned out that every player had exactly the same sum as he had at the beginning. Is it possible to determine the total shown by the dice at the nine former rounds ?
For all non-zero real numbers $x$ and $y$ such that $x-y=xy$, $\frac{1}{x}-\frac{1}{y}$ equals
$\textbf{(A) }\frac{1}{xy}\qquad\textbf{(B) }\frac{1}{x-y}\qquad\textbf{(C) }0\qquad\textbf{(D) }-1\qquad\textbf{(E) }y-x$
$(BUL 3)$ One hundred convex polygons are placed on a square with edge of length $38 cm.$ The area of each of the polygons is smaller than $\pi cm^2,$ and the perimeter of each of the polygons is smaller than $2\pi cm.$ Prove that there exists a disk with radius $1$ in the square that does not intersect any of the polygons.
Let $V$ be the set of vertices of a regular $25$ sided polygon with center at point $C.$ How many triangles have vertices in $ V$ and contain the point $C$ in the interior of the triangle?
A game is played on an $m \times n$ chessboard. At the beginning, there is a coin on one of the squares. Two players take turns to move the coin to an adjacent square (horizontally or vertically). The coin may never be moved to a square that has been occupied before. If a player cannot move any more, he loses. Prove:
[list]
[*] If the size (number of squares) of the board is even, then the player to move first has a winning strategy, regardless of the initial position.
[*] If the size of the board is odd, then the player to move first has a winning strategy if and only if the coin is initially placed on a square whose colour is not the same as the colour of the corners.
[/list]
The axis of a parabola is its axis of symmetry and its vertex is its point of intersection with its axis. Find: the equation of the parabola which touches $y = 0$ at $(1,0)$ and $x = 0$ at $(0,2);$ the equation of its axis; and its vertex.
Given a trapezoid $ABCD$, in which $AD \parallel BC$, and rays $AB$ and $DC$ intersect at point $G$. The common external tangents to the circles $(ABC), (ACD)$ intersect at point $E$. The common external tangents to circles $(ABD), (CBD)$ meet at $F$. Prove that the points $E, F$ and $G$ are collinear.
(a) Prove that \[ [5x]\plus{}[5y] \ge [3x\plus{}y] \plus{} [3y\plus{}x],\] where $ x,y \ge 0$ and $ [u]$ denotes the greatest integer $ \le u$ (e.g., $ [\sqrt{2}]\equal{}1$).
(b) Using (a) or otherwise, prove that \[ \frac{(5m)!(5n)!}{m!n!(3m\plus{}n)!(3n\plus{}m)!}\] is integral for all positive integral $ m$ and $ n$.
You are initially given the number $n=1$. Each turn, you may choose any positive divisor $d\mid n$, and multiply $n$ by $d+1$. For instance, on the first turn, you must select $d=1$, giving $n=1\cdot(1+1)=2$ as your new value of $n$. On the next turn, you can select either $d=1$ or $2$, giving $n=2\cdot(1+1)=4$ or $n=2\cdot(2+1)=6$, respectively, and so on.
Find an algorithm that, in at most $k$ steps, results in $n$ being divisible by the number $2021^{2021^{2021}} - 1$.
An algorithm that completes in at most $k$ steps will be awarded:
1 pt for $k>2021^{2021^{2021}}$
20 pts for $k=2021^{2021^{2021}}$
50 pts for $k=10^{10^4}$
75 pts for $k=10^{10}$
90 pts for $k=10^5$
95 pts for $k=6\cdot10^4$
100 pts for $k=5\cdot10^4$
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that \[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\] What is $k?$
$\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$
Let $n$ be a positive integer. Find all $n$- rows $( a_1 , a_2 ,..., a_n )$ of different positive integers such that $$ \frac{(a_1 + d ) (a_2 + d ) \cdot\cdot\cdot ( a_n + d )}{a_1a_2\cdot \cdot \cdot a_n }$$
is integer for every integer $d\ge 0$
$\mathbb{N}_{10}$ is generalization of $\mathbb{N}$ that every hypernumber in $\mathbb{N}_{10}$ is something like: $\overline{...a_2a_1a_0}$ with $a_i \in {0,1..9}$
(Notice that $\overline {...000} \in \mathbb{N}_{10}$)
Also we easily have $+,*$ in $\mathbb{N}_{10}$.
first $k$ number of $a*b$= first $k$ nubmer of (first $k$ number of a * first $k$ number of b)
first $k$ number of $a+b$= first $k$ nubmer of (first $k$ number of a + first $k$ number of b)
Fore example $\overline {...999}+ \overline {...0001}= \overline {...000}$
Prove that every monic polynomial in $\mathbb{N}_{10}[x]$ with degree $d$ has at most $d^2$ roots.
Let $k$ be a positive integer. Find all collection of integers $(a_1, a_2,\cdots, a_k)$ such that there exist a non-linear polynomial $P$ with integer coefficients, so that for all positive integers $n$ there exist a positive integer $m$ satisfying: $$P(n+a_1)+P(n+a_2)+...+P(n+a_k)=P(m)$$
[i]Proposed by Ivan Chan Kai Chin[/i]
Consider five segments $AB_1, AB_2, AB_3, AB_4, AB_5$. From each point $B_i$ there can exit either $5$ segments or no segments at all, so that the endpoints of any two segments of the resulting graph (system of segments) do not coincide. Can the number of free endpoints of the segments thus constructed be equal to $1001$? (A free endpoint is an endpoint from which no segment begins.)
On a rectangular board with $m \times n$ squares ($m, n \ge 3$) there are dominoes ($2 \times 1$ or $1\times 2$ tiles), which do not overlap and do not extend beyond the board. Every domino covers exactly two squares of the board. Assume that the dominos cover the has the property that no more dominos can be added to the board and that the four corner spaces of the board are not all empty. Prove that at least $2/3$ of the squares of the board are covered with dominos.
Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$
[list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good.
[*][b](b)[/b] Show that all 2010-good pairs are very good.[/list]
[i]Proposed by Okan Tekman, Turkey[/i]
Let $(a_n)_{n\ge 1}$ be a sequence such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n + 2}$, for any $n\ge 1$. Show that the sequence $(x_n)_{n\ge 1}$ given by $x_n = \log_{a_n} a_{n + 1}$ for $n\ge 1$ is convergent and compute its limit.