Found problems: 85335
Consider all regular convex and star polygons inscribed in a given circle and having $n$ [i]sides[/i]. We call two such polygons to be equivalent if it is possible to obtain one from the other using a rotation about the center of the circle. How many classes of such polygons exist?
[i]Mircea Becheanu[/i]
Let $ABCDEF$ be a cyclic hexagon satisfying $AB\perp BD$ and $BC=EF$.Let $P$ be the intersection of lines $BC$ and $AD$ and let $Q$ be the intersection of lines $EF$ and $AD$.Assume that $P$ and $Q$ are on the same side of $D$ and $A$ is on the opposite side.Let $S$ be the midpoint of $AD$.Let $K$ and $L$ be the incentres of $\triangle BPS$ and $\triangle EQS$ respectively.Prove that $\angle KDL=90^0$.
Amir and Bella play a game on a gameboard with $6$ spaces, labeled $0, 1, 2, 3, 4,$ and $5.$ Each turn, each player flips a coin. If it is heads, their character moves forward one space, and if it is tails, their character moves back one space, unless it was already at space $0,$ in which case it moves forward one space instead. If Amir and Bella each have a character that starts at space $0,$ the probability that they end turn $5$ on the same space can be expressed as a common fraction $\tfrac{a}{b}.$ Find $a+b.$
The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$.
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{3}{4}\qquad\textbf{(C) }\frac{\sqrt{2}}{2}\qquad\textbf{(D) }\frac{\sqrt{3}}{2}\qquad\textbf{(E) }\frac{\sqrt{3}}{3}$
[asy]
size(150);
import patterns;
pair D=(0,0),C=(1,-1),B=(2.5,-0.2),A=(1,2),AA,BB,CC,DD,P,Q,aux;
add("hatch",hatch());
//AA=new A and etc.
draw(rotate(100,D)*(A--B--C--D--cycle));
AA=rotate(100,D)*A;
BB=rotate(100,D)*D;
CC=rotate(100,D)*C;
DD=rotate(100,D)*B;
aux=midpoint(AA--BB);
draw(BB--DD);
P=midpoint(AA--aux);
aux=midpoint(CC--DD);
Q=midpoint(CC--aux);
draw(AA--CC,dashed);
dot(P);
dot(Q);
fill(DD--BB--CC--cycle,pattern("hatch"));
label("$A$",AA,W);
label("$B$",BB,S);
label("$C$",CC,E);
label("$D$",DD,N);
label("$P$",P,S);
label("$Q$",Q,E);
//Credit to TheMaskedMagician for the diagram
[/asy]
Find a continuous function $f(x)$ satisfying the identity $f(x)-f(ax)=x^n-x^m$, where $n,m\in N , 0<a<1$
We are given $101$ rectangles with sides of integer lengths not exceeding $100$ . Prove that among these $101$ rectangles there are $3$ rectangles, say $A , B$ and $C$ such that $A$ will fit inside $B$ and $B$ inside $C$.
( N . Sedrakyan, Yerevan)
Let $a, b$ be positive integers. Prove that if $a^b + b^a \equiv 3 \pmod{4}$, then either $a+1$ or $b+1$ can't be written as the sum of two integer squares.
[i](Proposed by Orestis Lignos, Greece)[/i]
Two decompositions of a square into three rectangles are called substantially different, if reordering the rectangles does not change one into the other.
How many substantially different decompositions of a $2010 \times 2010$ square in three rectangles with integer side lengths are there such that the area of one rectangle is equal to the arithmetic mean of the areas of the other rectangles?
Let $d(n)=\sum_{0<d|n}{1}$. Show that, for any natural $n>1$,
\[ \sum_{2 \leq i \leq n}{\frac{1}{i}} \leq \sum{\frac{d(i)}{n}} \leq \sum_{1 \leq i \leq n}{\frac{1}{i}} \]
For all reals $x,y,z$ prove that $$\sqrt {x^2+\frac {1}{y^2}}+ \sqrt {y^2+\frac {1}{z^2}}+ \sqrt {z^2+\frac {1}{x^2}}\geq 3\sqrt {2}. $$
Let \(a\), \(b\), and \(n\) be positive integers. A lemonade stand owns \(n\) cups, all of which are initially empty. The lemonade stand has a [i]filling machine[/i] and an [i]emptying machine[/i], which operate according to the following rules: [list] [*]If at any moment, \(a\) completely empty cups are available, the filling machine spends the next \(a\) minutes filling those \(a\) cups simultaneously and doing nothing else. [*]If at any moment, \(b\) completely full cups are available, the emptying machine spends the next \(b\) minutes emptying those \(b\) cups simultaneously and doing nothing else. [/list] Suppose that after a sufficiently long time has passed, both the filling machine and emptying machine work without pausing. Find, in terms of \(a\) and \(b\), the least possible value of \(n\).
[i]Proposed by Raymond Feng[/i]
The summation $\sum_{k=1}^{360} \frac{1}{k \sqrt{k+1} + (k+1)\sqrt{k}}$ is the ratio of two relatively prime positive integers $m$ and $n$. Find $m + n$.
Evaluate: $\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}$
A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression.
Let $S_n$ represent the sum of the first $n$ terms of the harmonic progression; for example $S_3$ represents the sum of the first three terms. If the first three terms of a harmonic progression are $3,4,6$, then:
$ \textbf{(A)}\ S_4=20 \qquad\textbf{(B)}\ S_4=25\qquad\textbf{(C)}\ S_5=49\qquad\textbf{(D)}\ S_6=49\qquad\textbf{(E)}\ S_2=\frac12 S_4 $
Find the set of $x$-values satisfying the inequality $|\frac{5-x}{3}|<2$. [The symbol $|a|$ means $+a$ if $a$ is positive, $-a$ if $a$ is negative, 0 if $a$ is zero. The notation $1<a<2$ means that $a$ can have any value between $1$ and $2$, excluding $1$ and $2$. ]
$ \textbf{(A)}\ 1 < x < 11\qquad\textbf{(B)}\ -1 < x < 11\qquad\textbf{(C)}\ x< 11\qquad$
$\textbf{(D)}\ x>11\qquad\textbf{(E)}\ |x| < 6 $
Given two positive integers $n$ and $m$ and a function $f : \mathbb{Z} \times \mathbb{Z} \to \left\{0,1\right\}$ with the property that
\begin{align*}
f\left(i, j\right) = f\left(i+n, j\right) = f\left(i, j+m\right) \qquad \text{for all } \left(i, j\right) \in \mathbb{Z} \times \mathbb{Z} .
\end{align*}
Let $\left[k\right] = \left\{1,2,\ldots,k\right\}$ for each positive integer $k$.
Let $a$ be the number of all $\left(i, j\right) \in \left[n\right] \times \left[m\right]$ satisfying
\begin{align*}
f\left(i, j\right) = f\left(i+1, j\right) = f\left(i, j+1\right) .
\end{align*}
Let $b$ be the number of all $\left(i, j\right) \in \left[n\right] \times \left[m\right]$ satisfying
\begin{align*}
f\left(i, j\right) = f\left(i-1, j\right) = f\left(i, j-1\right) .
\end{align*}
Prove that $a = b$.
All decimal digits of some natural number are $1,3,7$, and $9$. Prove that one can rearrange its digits so as to obtain a number divisible by $7$.
Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints:
[list]
[*]each cell contains a distinct divisor;
[*]the sums of all rows are equal; and
[*]the sums of all columns are equal.
[/list]
Prove that $\left\{ \frac{m}{n}\right\}+\left\{ \frac{n}{m}\right\} \ne 1$ , for any positive integers $m, n$.
Find the smallest positive integer that is a multiple of $18$ and whose digits can only be $4$ or $7$.
Consider all ordered pairs of integers $(a,b)$ such that $1\le a\le b\le 100$ and $$\frac{(a+b)(a+b+1)}{ab}$$ is an integer.
Among these pairs, find the one with largest value of $b$. If multiple pairs have this maximal value of $b$, choose the one with largest $a$. For example choose $(3,85)$ over $(2,85)$ over $(4,84)$. Note that your answer should be an ordered pair.
Let $2n + 3$ points be given in the plane in such a way that no three lie on a line and no four lie on a circle. Prove that the number of circles that pass through three of these points and contain exactly $n$ interior points is not less than $\frac 13 \binom{2n+3}{2}.$
Fine all positive integers $m,n\geq 2$, such that
(1) $m+1$ is a prime number of type $4k-1$;
(2) there is a (positive) prime number $p$ and nonnegative integer $a$, such that
\[\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.\]
Given a positive real number $a_1$, we recursively define $a_{n+1} = 1+a_1 a_2 \cdots \cdot a_n.$ Furthermore, let
$$b_n = \frac{1}{a_1 } + \frac{1}{a_2 } +\cdots + \frac{1}{a_n }.$$
Prove that $b_n < \frac{2}{a_1}$ for all positive integers $n$ and that this is the smallest possible bound.
Let $ABC$ be a triangle, and let $P$ be a point inside it such that $\angle PAC = \angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$, respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM.$