Found problems: 85335
Let $G$ be a complete directed graph with $100$ vertices such that for any two vertices $x,y$ one can find a directed path from $x$ to $y$.
a) Show that for any such $G$, one can find a $m$ such that for any two vertices $x,y$ one can find a directed path of length $m$ from $x$ to $y$ (Vertices can be repeated in the path)
b) For any graph $G$ with the properties above, define $m(G)$ to be smallest possible $m$ as defined in part a). Find the minimim value of $m(G)$ over all such possible $G$'s.
Given a cyclic quadrilateral $ABCD$, define $E$ as $AD \cap BC$ and $F$ as $AB \cap CD$.
Let $\Omega_A$ be the circle passing through $A, D$ and tangent to $AB$, and let its center be $O_A$.
Let $\Gamma_B$ be the circle passing through $B, C$ and tangent to $AB$, and let its center be $O_B$.
Let $\Gamma_C$ be the circle passing through $B, C$ and tangent to $CD$, and let its center be $O_C$.
Let $\Omega_D$ be the circle passing through $A, D$ and tangent to $CD$, and let its center be $O_D$.
Prove that $O_AO_BO_CO_D$ is cyclic, and prove that it's center lies on $EF$.
Consider an arbitrary (optional convex) polygon. It's [i]chord [/i] is a segment whose ends lie on the boundary of the polygon, and itself belongs entirely to the polygon. Will there always be a chord of a polygon that divides it into two equal parts? Is it true that any polygon can be divided by some chord into parts, the area of each of which is not less than $\frac13$ the area of the polygon?
$\Delta ABC$ is a right-angled triangle, $\angle C = 90^{\circ}$. Draw a circle centered at $B$ with radius $BC$. Let $D$ be a point on the side $AC$, and $DE$ is tangent to the circle at $E$. The line through $C$ perpendicular to $AB$ meets line $BE$ at $F$. Line $AF$ meets $DE$ at point $G$. The line through $A$ parallel to $BG$ meets $DE$ at $H$. Prove that $GE = GH$.
If, instead, the graph is a graph of VELOCITY vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)?
(A) at B
(B) at C
(C) at D
(D) at both B and D
(E) From C to D
Prove that for each positive integer $K$ there exist infinitely many even positive integers which can be written in more than $K$ ways as the sum of two odd primes.
$\text{Let} S_1,S_2,...S_{2011}$ $\text{be nonempty sets of consecutive integers
such that any}$ $2$ $\text{of them have a common element. Prove that there is a positive
integer that belongs to every}$ $S_i, i=1,...,2011$
(For example, ${2,3,4,5}$ is a set of consecutive integers while ${2,3,5}$ is not.)
The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s = 6 \sqrt{2}$, what is the volume of the solid?
[asy]
import three;
size(170);
pathpen = black+linewidth(0.65);
pointpen = black;
currentprojection = perspective(30,-20,10);
real s = 6 * 2^.5;
triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);
draw(F--B--C--F--E--A--B);
draw(A--D--E, dashed);
draw(D--C, dashed);
label("$2s$", (s/2, s/2, 6), N);
label("$s$", (s/2, 0, 0), SW);
[/asy]
Find the number of ways there are to permute the elements of the set $\{1,2,3,4,5,6,7,8,9\}$ such that no two adjacent numbers are both even or both odd.
[i]Proposed by Ephram Chun[/i]
Let $ABCDEF$ be a convex hexagon. The diagonals $AC$ and $BD$ cross at $P,$ the diagonals $AE{}$ and $DF$ cross at $Q,$ and the line $PQ$ crosses the sides $BC$ and $EF$ at $X$ and $Y,{}$ respectively. Prove that the length of the segment $XY$ does not exceed the sum of the lengths of one of the diagonals through $P{}$ and one of the diagonals through $Q{}$.
[i]The Problem Selection Committee[/i]
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]
Show that it is possible to situate eight parallel planes at equal distances such that each plane contains precisely one vertex of a given cube. How many such configurations of planes are there?
In the country of PUMACsboro, there are $n$ distinct cities labelled $1$ through $n$. There is a rail line going from city $i$ to city $j$ if and only if $i<j$; you can only take this rail line from city $i$ to city $j$. What is the smallest possible value of $n$ such that if each rail line's track is painted orange or black, you can always take the train between $2019$ cities on tracks that are all the same color? (This means there are some cities $c_1,c_2,\dots,c_{2019}$ such that there is a rail line going from city $c_i$ to $c_{i+1}$ for all $1\leq i\leq 2018$ and their rail lines' tracks are either all orange or all black.)
Acute triangle $ABC$ has altitudes $AD$, $BE$, and $CF$. Point $D$ is projected onto $AB$ and $AC$ to points $D_c$ and $D_b$ respectively. Likewise, $E$ is projected to $E_a$ on $BC$ and $E_c$ on $AB$, and $F$ is projected to $F_a$ on $BC$ and $F_b$ on $AC$. Lines $D_bD_c$, $E_cE_a$, $F_aF_b$ bound a triangle of area $T_1$, and lines $E_cF_b$, $D_bE_a$, $F_aD_c$ bound a triangle of area $T_2$. What is the smallest possible value of the ratio $T_2/T_1$?
[asy]
size(120);
real t = 2/sqrt(3);
real x = 1 + sqrt(3);
pair A = t*dir(90), D = x*A;
pair B = t*dir(210), E = x*B;
pair C = t*dir(330), F = x*C;
draw(D--E--F--cycle);
draw(Circle(A, 1));
draw(Circle(B, 1));
draw(Circle(C, 1));
//Credit to MSTang for the diagram[/asy]
Each of the three circles in the adjoining figure is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius three, then the perimeter of the triangle is
$\textbf{(A) }36+9\sqrt{2}\qquad\textbf{(B) }36+6\sqrt{3}\qquad\textbf{(C) }36+9\sqrt{3}\qquad\textbf{(D) }18+18\sqrt{3}\qquad \textbf{(E) }45$
Let $ABC$ be an acute triangle with altitude $\overline{AH}$, and let $P$ be a variable point such that the angle bisectors $k$ and $\ell$ of $\angle PBC$ and $\angle PCB$, respectively, meet on $\overline{AH}$. Let $k$ meet $\overline{AC}$ at $E$, $\ell$ meet $\overline{AB}$ at $F$, and $\overline{EF}$ meet $\overline{AH}$ at $Q$. Prove that as $P$ varies, line $PQ$ passes through a fixed point.
Derek is bored in math class and is drawing a flower. He first draws $8$ points $A_1, A_2, \ldots, A_8$ equally spaced around an enormous circle. He then draws $8$ arcs outside the circle where the $i$th arc for $i = 1, 2, \ldots, 8$ has endpoints $A_i, A_{i+1}$ with $A_9 = A_1,$ such that all of the arcs have radius $1$ and any two consecutive arcs are tangent. Compute the perimeter of Derek’s $8$-petaled flower (not including the central circle).
[center]
[img]
https://cdn.artofproblemsolving.com/attachments/8/4/e8b23c587762c089adb77b29cae155209f5db5.png
[/img]
[/center]
The slope of the line $\frac{x}{3} + \frac{y}{2} = 1$ is
$ \textbf{(A)}\ -\frac{3}{2}\qquad\textbf{(B)}\ -\frac{2}{3}\qquad\textbf{(C)}\ \frac{1}{3}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{3}{2} $
Let $a,b$, and $x$ be positive integers such that $x^{a+b}=a^b{b}$. Prove that $a=x$ and $b=x^{x}$.
An acute triangle \(ABC\) has circumcircle \(\Gamma\) and circumcentre \(O\). The incentres of \(AOB\) and \(AOC\) are \(I_b\) and \(I_c\) respectively. Let \(M\) be the the point on \(\Gamma\) such that \(MB = MC\) and \(M\) lies on the same side of \(BC\) as \(A\). Prove that the points \(M\), \(A\), \(I_b\), and \(I_c\) are concyclic.
Let $A,B,C$ be variable points on edges $OX,OY,OZ$ of a trihedral angle $OXYZ$, respectively.
Let $OA = a, OB = b, OC = c$ and $R$ be the radius of the circumsphere $S$ of $OABC$.
Prove that if points $A,B,C$ vary so that $a+b+c = R+l$, then the sphere $S$ remains tangent to a fixed sphere.
Does there exist natural numbers $a,b,c$ all greater than $10^{10}$ such that their product is divisible by each of these numbers increased by $2012$?
A cone with semivertical angle $30^{\circ}$ is half filled with water. What is the angle it must be tilted by so that water starts spilling?
Let $c_1, \ldots, c_n \in \mathbb{R}$ with $n \geq 2$ such that \[ 0 \leq \sum^n_{i=1} c_i \leq n. \] Show that we can find integers $k_1, \ldots, k_n$ such that \[ \sum^n_{i=1} k_i = 0 \] and \[ 1-n \leq c_i + n \cdot k_i \leq n \] for every $i = 1, \ldots, n.$
[hide="Another formulation:"]
Let $x_1, \ldots, x_n,$ with $n \geq 2$ be real numbers such that \[ |x_1 + \ldots + x_n| \leq n. \] Show that there exist integers $k_1, \ldots, k_n$ such that \[ |k_1 + \ldots + k_n| = 0. \] and \[ |x_i + 2 \cdot n \cdot k_i| \leq 2 \cdot n -1 \] for every $i = 1, \ldots, n.$ In order to prove this, denote $c_i = \frac{1+x_i}{2}$ for $i = 1, \ldots, n,$ etc.
[/hide]
Find all nonnegative integer solutions to $2^a + 3^b + 5^c = n!$.
[i]Proposed by Mark Sellke[/i]