Found problems: 85335
For a positive integer $n$, define $f(n)=\sum_{i=0}^{\infty}\frac{\gcd(i,n)}{2^i}$ and let $g:\mathbb N\rightarrow \mathbb Q$ be a function such that $\sum_{d\mid n}g(d)=f(n)$ for all positive integers $n$. Given that $g(12321)=\frac{p}{q}$ for relatively prime integers $p$ and $q$, find $v_2(p)$.
[i]Proposed by Michael Ren[/i]
Let $M$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$. Let $P$ and $Q$ be the centroids of triangles $AMD$ and $BMC$ respectively. Let $R$ and $S$ are the orthocenters of triangles $AMB$ and $CMD$. Prove that the lines $P Q$ and $RS$ are perpendicular to each other.
The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular.
The following diagram shows a square where each side has seven dots that divide the side into six equal segments. All the line segments that connect these dots that form a $45^{\circ}$ angle with a side of the square are drawn as shown. The area of the shaded region is 75. Find the area of the original square.
[center][img]https://i.snag.gy/Jzx9Fn.jpg[/img][/center]
Let $a_1, a_2, \dots, a_{2019}$ be positive integers and $P$ a polynomial with integer coefficients such that, for every positive integer $n$,
$$P(n) \text{ divides } a_1^n+a_2^n+\dots+a_{2019}^n.$$
Prove that $P$ is a constant polynomial.
Let $ n$ be a positive integer. Show that \[ \left(\sqrt{2} \plus{} 1 \right)^n \equal{} \sqrt{m} \plus{} \sqrt{m\minus{}1}\] for some positive integer $ m.$
Let $m, n, a, k$ be positive integers and $k>1$ such that the equality $$5^m+63n+49=a^k$$
holds. Find the minimum value of $k$.
Let $f(x)=\frac{1+\cos(2 \pi x)}{2}$, for $x \in \mathbb{R}$, and $f^n=\underbrace{ f \circ \cdots \circ f}_{n}$. Is it true that for Lebesgue almost every $x$, $\lim_{n \to \infty} f^n(x)=1$?
Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$.
Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?
An $ (n, k) \minus{}$ tournament is a contest with $ n$ players held in $ k$ rounds such that:
$ (i)$ Each player plays in each round, and every two players meet at most once.
$ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$.
Determine all pairs $ (n, k)$ for which there exists an $ (n, k) \minus{}$ tournament.
[i]Proposed by Carlos di Fiore, Argentina[/i]
Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$.
a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$.
b) Find the sum of the other components of all such pairs of numbers.
If $x$ men working $x$ hours a day for $x$ days produce $x$ articles, then the number of articles (not necessarily an integer) produced by $y$ men working $y$ hours a day for $y$ days is:
${{ \textbf{(A)}\ \frac{x^3}{y^2} \qquad\textbf{(B)}\ \frac{y^3}{x^2} \qquad\textbf{(C)}\ \frac{x^2}{y^3} \qquad\textbf{(D)}\ \frac{y^2}{x^3} }\qquad\textbf{(E)}\ y} $
$ (a_n)_{n \equal{} 0}^\infty$ is a sequence on integers. For every $ n \ge 0$, $ a_{n \plus{} 1} \equal{} a_n^3 \plus{} a_n^2$. The number of distinct residues of $ a_i$ in $ \pmod {11}$ can be at most?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$
Circle $\omega_1$ with radius 3 is inscribed in a strip $S$ having border lines $a$ and $b$. Circle $\omega_2$ within $S$ with radius 2 is tangent externally to circle $\omega_1$ and is also tangent to line $a$. Circle $\omega_3$ within $S$ is tangent externally to both circles $\omega_1$ and $\omega_2$, and is also tangent to line $b$. Compute the radius of circle $\omega_3$.
Assume we are given a set of weights, $x_1$ of which have mass $d_1$, $x_2$ have mass $d_2$, etc, $x_k$ have mass $d_k$, where $x_i,d_i$ are positive integers and $1\le d_1<d_2<\ldots<d_k$. Let us denote their total sum by $n=x_1d_1+\ldots+x_kd_k$. We call such a set of weights [i]perfect[/i] if each mass $0,1,\ldots,n$ can be uniquely obtained using these weights.
(a) Write down all sets of weights of total mass $5$. Which of them are perfect?
(b) Show that a perfect set of weights satisfies $$(1+x_1)(1+x_2)\cdots(1+x_k)=n+1.$$
(c) Conversely, if $(1+x_1)(1+x_2)\cdots(1+x_k)=n+1$, prove that one can uniquely choose the corresponding masses $d_1,d_2,\ldots,d_k$ with $1\le d_1<\ldots<d_k$ in order for the obtained set of weights is perfect.
(d) Determine all perfect sets of weights of total mass $1993$.
Find all functions $f:\mathbb{R}\longrightarrow\mathbb{R}$ such that $f(x+y)+xy=f(x)f(y)$ for all reals $x, y$
Let a be a real number such that $0 < a < 1$, and let $n$ be a positive integer. Define the sequence $a_0, a_1, a_2, \ldots, a_n$ an recursively by
\[a_0 = a, \quad a_{k+1} = a_k +\frac 1n a_k^2 \quad \text{ for } k = 0, 1, \ldots, n - 1.\]
Prove that there exists a real number $A$, depending on $a$ but independent of $n$, such that
\[0 < n(A - a_n) < A^3.\]
In Thailand there are $n$ cities. Each pair of cities is connected by a one-way street which can be borrowed, depending on its type, only by bike or by car. Show that there is a city from which you can reach any other city, either by bike or by car.
[i]Remark : It is not necessary to use the same means of transport for each city[/i]
Find the smallest integer $n$ such that \[(x^2+y^2+z^2)^2\le n(x^4+y^4+z^4)\] for all real numbers $x,y,$ and $z$.
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad \textbf{(E) }\text{There is no such integer }n.$
Given a positive integer $n$, show that the set $\{1,2,...,n\}$ can be partitioned into $m$ sets, each with the same sum, if and only if m is a divisor of $\frac{n(n + 1)}{2}$ which does not exceed $\frac{n + 1}{2}$.
Prove the identity \[\sum\limits_{k=0}^n\binom{n}{k}\left(\tan\frac{x}{2}\right)^{2k}\left(1+\frac{2^k}{\left(1-\tan^2\frac{x}{2}\right)^k}\right)=\sec^{2n}\frac{x}{2}+\sec^n x\] for any natural number $n$ and any angle $x.$
Let ABC be an acute triangle with an incenter $I$.The Incircle touches sides $AC$ and $AB$ in $E$ and $F$ ,respectively. Lines CI and EF intersect at $S$. The point $T$≠$I$ is on the line AI so that $EI$=$ET$.If $K$ is the foot of the altitude from $C$ in triangle $ABC$,prove that points $K$,$S$ and $T$ are colinear.
$25$ chess players are going to participate in a chess tournament. All are on distinct skill levels, and of the two players the one who plays better always wins. What is the least number of games needed to select the two best players?
Let $p$ be a positive integer and $q, z$ be real numbers with $0\le q\le 1$ and $q^{p+1}\le z\le 1$. Prove that
\[\prod_{k=1}^p \left|\frac{z - q^k}{z + q^k}\right| \le\prod_{k=1}^p \left|\frac{1 - q^k}{1 + q^k}\right|.\]