Found problems: 85335
Prove that $1^{1987} + 2^{1987} + ... + n^{1987}$ is divisible by $n+2$.
This problem is generalization of [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=5918]this one[/url].
Suppose $G$ is a graph and $S\subset V(G)$. Suppose we have arbitrarily assign real numbers to each element of $S$. Prove that we can assign numbers to each vertex in $G\backslash S$ that for each $v\in G\backslash S$ number assigned to $v$ is average of its neighbors.
A house worth $ \$9000$ is sold by Mr. A to Mr. B at a $ 10\%$ loss. Mr. B sells the house back to Mr. A at a $ 10\%$ gain. The result of the two transactions is:
$ \textbf{(A)}\ \text{Mr. A breaks even} \qquad\textbf{(B)}\ \text{Mr. B gains }\$900 \qquad\textbf{(C)}\ \text{Mr. A loses }\$900\\
\textbf{(D)}\ \text{Mr. A loses }\$810 \qquad\textbf{(E)}\ \text{Mr. B gains }\$1710$
Let $ABC$ be a triangle and $D$ a point inside the triangle, located on the median of $A$. Prove that if $\angle BDC = 180^o - \angle BAC$, then $AB \cdot CD = AC \cdot BD$.
Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$?
If $f(x)=\cos(x)-1+\frac{x^2}2$, then
$\textbf{(A)}~f(x)\text{ is an increasing function on the real line}$
$\textbf{(B)}~f(x)\text{ is a decreasing function on the real line}$
$\textbf{(C)}~f(x)\text{ is increasing on }-\infty<x\le0\text{ and decreasing on }0\le x<\infty$
$\textbf{(D)}~f(x)\text{ is decreasing on }-\infty<x\le0\text{ and increasing on }0\le x<\infty$
Let $a_{n}$ be the last nonzero digit in the decimal representation of the number $n!$. Does the sequence $a_{1}$, $a_{2}$, $a_{3}$, $\cdots$ become periodic after a finite number of terms?
On a line, $44$ points are marked and numbered $1, 2, 3,…,44$ from left to right. Various crickets jump around the line. Each starts at point $1$, jumping on the marked points and ending up at point $44$. In addition, each cricket jumps from a marked point to another marked point with a greater number.
When all the crickets have finished jumping, it turns out that for pair $i, j$ with ${1}\leq{i}<{j}\leq{44}$, there was a cricket that jumped directly from point $i$ to point $j$, without visiting any of the points in between the two.
Determine the smallest number of crickets such that this is possible.
Bobby Fisherman played a tournament in which he played $2009$ players. He either won or lost every game. He lost his first two games, but won $2002$ total games. At the conclusion of each game, he computed his exact winning percentage at that moment. Let $w_1,w_2,\ldots, w_{2009}$ be his winning percentages after games $1$, $2$, $\ldots$, $2009$ respectively. There are some real numbers, such as $0$, which are necessarily members of the set $W = \{w_1,w_2,\ldots, w_{2009}\}$. How many positive real numbers are necessarily elements of set $W$, regardless of the order in which he won or lost his games?
In an acute-angled triangle, each angle is an integral number of degrees, and the smallest angle is one-fifth of the largest one. Find these angles.
(G Galperin)
$f(x) = [x] + [2x] + [3x] + [4x] + [5x] + [6x]$. What values does $f$ take?
Shenelle has some square tiles. Some of the tiles have side length $5\text{ cm}$ while the others have side length $3\text{ cm}$. The total area that can be covered by the tiles is exactly $2014\text{ cm}^2$. Find the least number of tiles that Shenelle can have.
$$\int_{-\infty}^\infty \frac{\sin(\pi x)}{x(1+x^2)}\,dx$$
[i]Proposed by Vlad Oleksenko[/i]
The pages of a notebook are numbered consecutively so that the numbers $1$ and $2$ are on the second sheet, numbers $3$ and $4$, and so on. A sheet is torn out of this notebook. All of the remaining page numbers are addedand have sum $2021$.
(a) How many pages could the notebook originally have been?
(b) What page numbers can be on the torn sheet?
(Walther Janous)
Prove that there are no positive integers $x$ and $y$ such that $x^2+y+2$ and $y^2+4x$ are perfect squares
[color=darkred]Let $M=\{x\in\mathbb{C}\, |\, |z|=1,\ \text{Re}\, z\in\mathbb{Q}\}\, .$ Prove that there exist infinitely many equilateral triangles in the complex plane having all affixes of their vertices in the set $M$ .[/color]
For how many integers $n$ between $1$ and $50$, inclusive, is
\[
\frac{(n^2-1)!}{(n!)^n}
\]an integer? (Recall that $0! = 1$.)
$\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$
Students have taken a test paper in each of $n \ge 3$ subjects. It is known that in any subject exactly three students got the best score, and for any two subjects exactly one student got the best scores in both subjects. Find the smallest $n$ for which the above conditions imply that exactly one student got the best score in each of the $n$ subjects.
In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three
5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as
necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile.
Find the number of different positive weights of chemicals that Gerome could measure.
$\vartriangle ABC$ is an equilateral triangle. $L, M$ and $N$ are points on $BC, CA$ and $AB$ respectively. Prove that $MA \cdot AN + NB \cdot BL + LC \cdot CM < BC^2$.
From point $A$ to circle $\omega$ tangent $AD$ and arbitrary a secant intersecting a circle at points $B$ and $C$ (B lies between points $A$ and $C$). Prove that the circle passing through points $C$ and $D$ and touching the straight line $BD$, passes through a fixed point (other than $D$).
Find all nonnegative integers $(x,y,z,u)$ with satisfy the following equation:
$2^x + 3^y + 5^z = 7^u.$
Write the following polynomial as a product of irreducible polynomials in $\mathbb{Z}[X]$
\[ f(X) = X^{2005} - 2005 X + 2004 . \]Justify your answer.
Let $ABC$ to be an acute triangle. Also, let $K$ and $L$ to be the two intersections of the perpendicular from $B$ with respect to side $AC$ with the circle of diameter $AC$, with $K$ closer to $B$ than $L$. Analogously, $X$ and $Y$ are the two intersections of the perpendicular from $C$ with respect to side $AB$ with the circle of diamter $AB$, with $X$ closer to $C$ than $Y$. Prove that the intersection of $XL$ and $KY$ lies on $BC$.
A positive integer $n$ is called almost-square if $n$ can be represented as $n=ab$ where $a,b$ are positive integers that $a\leq b\leq 1.01a$. Prove that there exists infinitely many positive integers $m$ that there’re no almost-square positive integer among $m,m+1,…,m+198$.