Found problems: 85335
A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.
Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.
Let $a_0,a_1,a_2,\dots$ be a periodic sequence of real numbers(that is, there is a fixed positive integer $k$ such that $a_n=a_{n+k}$ for every integer $n\geq 0$). The following equality is true, for all $n\geq 0$:
$a_{n+2}=\frac{1}{n+2} (a_n - \frac{n+1}{a_{n+1}})$
if $a_0=2020$, determine the value of $a_1$.
Find all pairs of integers $(x, y)$ such that ths sum of the fractions $\frac{19}{x}$ and $\frac{96}{y}$ would be equal to their product.
Prove that, for all $n \in \mathbb{N}$ \begin{align*} \frac{1}{1}+\frac{1}{3}+\frac{1}{5}+\ldots+\frac{1}{2n+1} \not\in \mathbb{Z} \end{align*}
The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$.
[i]Proposed by Croatia[/i]
Given an isosceles triangle $ABC$ such that $|AB|=|AC|$ . Let $M$ be the midpoint of the segment $BC$ and let $P$ be a point other than $A$ such that $PA\parallel BC$. The points $X$ and $Y$ are located respectively on rays $PB$ and $PC$, so that the point $B$ is between $P$ and $X$, the point $C$ is between $P$ and $Y$ and $\angle PXM=\angle PYM$. Prove that the points $A,P,X$ and $Y$ are concyclic.
Prove that there are infinitely many positive integers $m$, such that $n^4+m$ is not prime for any positive integer $n$.
Seller reduced price of one shirt for $20\%$,and they raised it by $10\%$. Does he needs to reduce or raise the price and how many, so that price of shirt will be reduced by $10\%$ from the original price
Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.
Set $|A|=n$. $A_1,A_2,\cdots,A_m$ are subsets of $A$, and $A_i\not\subseteq A_j$ for any $1\leq i<j\leq m$. Prove:
[b](a)[/b] $\sum_{i=1}^{m}\frac{1}{\text{C}_n^{|A_i|}}\leq1$.
[b](b)[/b] $\sum_{i=1}^{m}\text{C}_n^{|A_i|}\geq m^2$.
An $n$-dimensional cube is given. Consider all the segments connecting any two different vertices of the cube. How many distinct intersection points do these segments have (excluding the vertices)?
Show that the number of integral-sided right triangles whose ratio of area to semi-perimeter is $p^{m}$, where $p$ is a prime and $m$ is an integer, is $m+1$ if $p=2$ and $2m+1$ if $p \neq 2$.
What is the last digit of ${17^{17^{17^{17}}}}$?
* Given an infinite cone. The measure of its unfolding’s angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the number of the curve’s self-intersections.
Let be a function $ s:\mathbb{N}^2\longrightarrow \mathbb{N} $ that sends $ (m,n) $ to the number of solutions in $ \mathbb{N}^n $ of the equation:
$$ x_1+x_2+\cdots +x_n=m $$
[b]1)[/b] Prove that:
$$ s(m+1,n+1)=s(m,n)+s(m,n+1) =\prod_{r=1}^n\frac{m-r+1}{r} ,\quad\forall m,n\in\mathbb{N} $$
[b]2)[/b] Find $ \max\left\{ a_1a_2\cdots a_{20}\bigg| a_1+a_2+\cdots +a_{20}=2007, a_1,a_2,\ldots a_{20}\in\mathbb{N} \right\} . $
In trapezoid $ ABCD$, $ AB \parallel CD$, $ \angle CAB < 90^\circ$, $ AB \equal{} 5$, $ CD \equal{} 3$, $ AC \equal{} 15$. What are the sum of different integer values of possible $ BD$?
$\textbf{(A)}\ 101 \qquad\textbf{(B)}\ 108 \qquad\textbf{(C)}\ 115 \qquad\textbf{(D)}\ 125 \qquad\textbf{(E)}\ \text{None}$
You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a three-by-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle?
[asy]
import graph; size(5cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((12,0)--(12,1)--(11,1)--(11,0)--cycle);
fill((13.49,0)--(13.49,1)--(12.49,1)--(12.49,0)--cycle, black);
draw((13.49,0)--(13.49,1)--(14.49,1)--(14.49,0)--cycle);
draw((15,0)--(15,1)--(16,1)--(16,0)--cycle);
fill((17,0)--(17,1)--(16,1)--(16,0)--cycle, black);
[/asy]
Can the graphs of a polynomial of degree 20 and the function $\displaystyle y={1\over x^{40}}$ have exactly 30 points of intersection?
[i]Proposed by K. Kokhas[/i]
For how many integers $n$ between $1$ and $50$, inclusive, is
\[
\frac{(n^2-1)!}{(n!)^n}
\]an integer? (Recall that $0! = 1$.)
$\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$
Let $[ABCD]$ be a convex quadrilateral with area $2014$, and let $P$ be a point on $[AB]$ and $Q$ a point on $[AD]$ such that triangles $[ABQ]$ and $[ADP]$ have area $1$. Let $R$ be the intersection of $[AC]$ and $[PQ]$. Determine $\frac{\overline{RC}}{\overline{RA}}$.
Let $ m,n > 1$ are integers which satisfy $ n|4^m \minus{} 1$ and $ 2^m|n \minus{} 1$. Is it a must that $ n \equal{} 2^{m} \plus{} 1$?
Consider points $A, B, C$. Draw a line through $A$ so that the sum of distances from $B$ and $C$ to this line is equal to the length of a given segment.
Given positive integers $d \ge 3$, $r>2$ and $l$, with $2d \le l <rd$. Every vertice of the graph $G(V,E)$ is assigned to a positive integer in $\{1,2,\cdots,l\}$, such that for any two consecutive vertices in the graph, the integers they are assigned to, respectively, have difference no less than $d$, and no more than $l-d$.
A proper coloring of the graph is a coloring of the vertices, such that any two consecutive vertices are not the same color. It's given that there exist a proper subset $A$ of $V$, such that for $G$'s any proper coloring with $r-1$ colors, and for an arbitrary color $C$, either all numbers in color $C$ appear in $A$, or none of the numbers in color $C$ appear in $A$.
Show that $G$ has a proper coloring within $r-1$ colors.
Suppose $a_1, a_2, ... , a_{100}$ are positive real numbers such that $$a_k =\frac{ka_{k-1}}{a_{k-1} - (k - 1)}$$ for $k = 2, 3, ... , 100$. Given that $a_{20} = a_{23}$, compute $a_{100}$.