A factorian is defined to be a number such that it is equal to the sum of it's digits' factorials. What is the smallest three digit factorian?

Two rectangles on a plane intersect at eight points. Consider every other intersection point, they are connected with line segments, these segments form a quadrilateral. Prove that the area of this quadrilateral does not vary under translations of one of the rectangles.

The convex pentagon $ABCDE$ was drawn in the plane. $A_1$ was symmetric to $A$ with respect to $B$. $B_1$ was symmetric to $B$ with respect to $C$. $C_1$ was symmetric to $C$ with respect to $D$. $D_1$ was symmetric to $D$ with respect to $E$. $E_1$ was symmetric to $E$ with respect to $A$. How is it possible to restore the initial pentagon with the compasses and ruler, knowing $A_1,B_1,C_1,D_1,E_1$ points?

Michael walks at the rate of $ 5$ feet per second on a long straight path. Trash pails are located every $ 200$ feet along the path. A garbage truck travels at $ 10$ feet per second in the same direction as Michael and stops for $ 30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

Find the number of the subsets $B$ of the set $\{1,2,\cdots, 2005 \}$ such that the sum of the elements of $B$ is congruent to $2006$ modulo $2048$

Al has three bags, each with three marbles each. Bag $1$ has two blue marbles and one red marble, Bag $2$ has one blue marble and two red marbles, and Bag $3$ has three red marbles. He chooses two distinct bags at random, then one marble at random from each of the chosen bags. What is the probability that he chooses two blue marbles?

Prove that for $\forall n>1$, $n\in \mathbb{N}$ , there exist infinitely many pairs of positive irrational numbers $a$ and $b$, such that $a^n=b$.

On the sides $ AB,AC $ of a triangle $ ABC, $ consider the points $ M, $ respectively, $ N $ such that $ M\neq A\neq N $ and $ \frac{MB}{MA}\neq\frac{NC}{NA}. $ Show that the line $ MN $ passes through a point not dependent on $ M $ and $ N. $

For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$, each to some positive power. What is $N$? $\textbf{(A) }9 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$

$O$ is a circle with a radius of $1$, with strings $CD$ and $EF$. $CD//EF$, and diameter $AB$ intersects $CD,EF$ at $P,Q$. If $\angle BPD=\frac{\pi}{4}$, prove that $$PC\cdot QE+PD \cdot QF<2.$$

[asy] label("$1$",(0,0),S); label("$1$",(-1,-1),S); label("$1$",(-2,-2),S); label("$1$",(-3,-3),S); label("$1$",(-4,-4),S); label("$1$",(1,-1),S); label("$1$",(2,-2),S); label("$1$",(3,-3),S); label("$1$",(4,-4),S); label("$2$",(0,-2),S); label("$3$",(-1,-3),S); label("$3$",(1,-3),S); label("$4$",(-2,-4),S); label("$4$",(2,-4),S); label("$6$",(0,-4),S); label("etc.",(0,-5),S); //Credit to chezbgone2 for the diagram[/asy] Pascal's triangle is an array of positive integers(See figure), in which the first row is $1$, the second row is two $1$'s, each row begins and ends with $1$, and the $k^\text{th}$ number in any row when it is not $1$, is the sum of the $k^\text{th}$ and $(k-1)^\text{th}$ numbers in the immediately preceding row. The quotient of the number of numbers in the first $n$ rows which are not $1$'s and the number of $1$'s is $\textbf{(A) }\dfrac{n^2-n}{2n-1}\qquad\textbf{(B) }\dfrac{n^2-n}{4n-2}\qquad\textbf{(C) }\dfrac{n^2-2n}{2n-1}\qquad\textbf{(D) }\dfrac{n^2-3n+2}{4n-2}\qquad \textbf{(E) }\text{None of these}$

Find all positive integers $k$ such that there exists positive integers $a, b$ such that \[a^2+4=(k^2-4)b^2.\]

In the adjoining figure triangle $ ABC$ is such that $ AB \equal{} 4$ and $ AC \equal{} 8$. If $ M$ is the midpoint of $ BC$ and $ AM \equal{} 3$, what is the length of $ BC$? $ \textbf{(A)}\ 2\sqrt{26} \qquad \textbf{(B)}\ 2\sqrt{31} \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 4\plus{}2\sqrt{13} \qquad$ $ \textbf{(E)}\ \text{not enough information given to solve the problem}$ [asy]draw((0,0)--(2.8284,2)--(8,0)--cycle); draw((2.8284,2)--(4,0)); label("A",(2.8284,2),N); label("B",(0,0),S); label("C",(8,0),S); label("M",(4,0),S);[/asy]

Define the sequence $a_0,a_1,\dots$ inductively by $a_0=1$, $a_1=\frac{1}{2}$, and \[a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n}, \quad \forall n \ge 1.\] Show that the series $\displaystyle \sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k}$ converges and determine its value. [i]Proposed by Christophe Debry, KU Leuven, Belgium.[/i]

Let $ABC$ be a triangle and let $D\in (BC)$ be the foot of the $A$- altitude. The circle $w$ with the diameter $[AD]$ meet again the lines $AB$ , $AC$ in the points $K\in (AB)$ , $L\in (AC)$ respectively. Denote the meetpoint $M$ of the tangents to the circle $w$ in the points $K$ , $L$ . Prove that the ray $[AM$ is the $A$-median in $\triangle ABC$ ([b][u]Serbia[/u][/b]).

16 students took part in a competition. All problems were multiple choice style. Each problem had four choices. It was said that any two students had at most one answer in common, find the maximum number of problems.

a) Prove that the values of $x$ for which $x=(x^2+1)/198$ lie between $1/198$ and $197.99494949\cdots$. b) Use the result of problem a) to prove that $\sqrt{2}<1.41\overline{421356}$. c) Is it true that $\sqrt{2}<1.41421356$?

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following equality for all $x,y\in\mathbb{R}$ \[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\][i]Proposed by Dániel Dobák, Budapest[/i]

There are $n\leq 99$ people around a circular table. At every moment everyone can either be truthful (always says the truth) or a liar (always lies). Initially some of people (possibly none) are truthful and the rest are liars. At every minute everyone answers at the same time the question "Is your left neighbour truthful or a liar?" and then becomes the same type of person as his answer. Determine the largest $n$ for which, no matter who are the truthful people in the beginning, at some point everyone will become truthful forever.

Suppose you are playing a game against Daniel. There are $2017$ chips on a table. During your turn, if you can write the number of chips on the table as a sum of two cubes of not necessarily distinct, nonnegative integers, then you win. Otherwise, you can take some number of chips between $1$ and $6$ inclusive off the table. (You may not leave fewer than $0$ chips on the table.) Daniel can also do the same on his turn. You make the first move, and you and Daniel always make the optimal move during turns. Who should win the game? Explain.

Let $K$ be the midpoint of the median $AM$ of a triangle $ABC$. Points $X, Y$ lie on $AB, AC$, respectively, such that $\angle KXM =\angle ACB$, $AX>BX$ and similarly $\angle KYM =\angle ABC$, $AY>CY$. Prove that $B, C, X, Y$ are concyclic. [i]Proposed by Mykhailo Shtandenko[/i]

Let $ABC$ be an isosceles triangle with base $AB$. Line $\ell$ touches its circumcircle at point $B$. Let $CD$ be a perpendicular from $C$ to $\ell$, and $AE$, $BF$ be the altitudes of $ABC$. Prove that $D$, $E$, and $F$ are collinear.

Find all integer pair $(m,n)$ such that $7^m=5^n+24$.

The midpoints of the sides of a regular hexagon $ABCDEF$ are joined to form a smaller hexagon. What fraction of the area of $ABCDEF$ is enclosed by the smaller hexagon? [asy] size(130); pair A, B, C, D, E, F, G, H, I, J, K, L; A = dir(120); B = dir(60); C = dir(0); D = dir(-60); E = dir(-120); F = dir(180); draw(A--B--C--D--E--F--cycle); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); G = midpoint(A--B); H = midpoint(B--C); I = midpoint(C--D); J = midpoint(D--E); K = midpoint(E--F); L = midpoint(F--A); draw(G--H--I--J--K--L--cycle); label("$A$", A, dir(120)); label("$B$", B, dir(60)); label("$C$", C, dir(0)); label("$D$", D, dir(-60)); label("$E$", E, dir(-120)); label("$F$", F, dir(180)); [/asy] $\textbf{(A)}\ \displaystyle \frac{1}{2} \qquad \textbf{(B)}\ \displaystyle \frac{\sqrt 3}{3} \qquad \textbf{(C)}\ \displaystyle \frac{2}{3} \qquad \textbf{(D)}\ \displaystyle \frac{3}{4} \qquad \textbf{(E)}\ \displaystyle \frac{\sqrt 3}{2}$

The function $f : \mathbb N \to \mathbb R$ satisfies $f(1) = 1, f(2) = 2$ and \[f (n+2) = f(n+2 - f(n+1) ) + f(n+1 - f(n) ).\] Show that $0 \leq f(n+1) - f(n) \leq 1$. Find all $n$ for which $f(n) = 1025$.