Let $ABC$ be a triangle, and $D$, $E$, $F$ points on the segments $BC$, $CA$, and $AB$ respectively such that $$ \frac{BD}{DC} = \frac{CE}{EA} = \frac{AF}{FB}. $$ Show that if the centres of the circumscribed circles of the triangles $DEF$ and $ABC$ coincide, then $ABC$ is an equilateral triangle.

There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of number of girls to the number of boys in her class? $\textbf{(A) }3 : 4\qquad\textbf{(B) }4 : 3\qquad\textbf{(C) }3 : 2\qquad\textbf{(D) }7 : 4\qquad \textbf{(E) }2 : 1$

On the screen of a computer there is an $2^n\times 2^n$ board. On each cell of the main diagonal there is a file. At each step, we may select some files and move them to the left, on their respective rows, by the same distance. What is the minimum number of necessary moves in order to put all files on the first column? [i]Proposed by Vlad Spătaru[/i]

The function $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$.

Determine all positive integers $n$ such that for every positive devisor $ d $ of $n$, $d+1$ is devisor of $n+1$.

$A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles $B$ and $C$? $ \textbf{(A)} \ 55 \qquad \textbf{(B)} \ 56 \qquad \textbf{(C)} \ 57 \qquad \textbf{(D)} \ 58 \qquad \textbf{(E)} \ 59$

Dima wrote several natural numbers on the blackboard and underlined some of them. Misha wants to erase several numbers (but not all) such that a multiple of three underlined numbers remain and the total amount of the remaining numbers would be divisible by $2013$; but after trying for a while he realizes that it's impossible to do this. What is the largest number of the numbers on the board?

Let $ABCD$ be a parallelogram, let $X$ and $Y$ in the segments $AB$ and $CD$, respectively. The segments $AY$ and $DX$ intersects in $P$ and the segments $BY$ and $DX$ intersects in $Q$, show that the line $PQ$ passes in a fixed point(independent of the positions of the points $X$ and $Y$). I guess that the fixed point is the midpoint of $BD$.

Find all ordered pairs of positive integers $\left(a,b\right)$ such that $$\frac{1}{a}+\frac{a}{b}+\frac{1}{ab}=1.$$

In the figure, $ABCD$ is a rectangle and $EFGH$ is a parallelogram. Using the measurements given in the figure, what is the length $d$ of the segment that is perpendicular to $HE$ and $FG$? [asy] defaultpen(linewidth(0.8)); size(200); pair A=(0,8), B=(10,8), C=(10,0), D=origin; pair E=(4,8), F=(10,3), G=(6,0), H=(0,5); pair I=H+4*dir(H--E); pair J=foot(I, F, G); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); draw(I--J); draw(rightanglemark(H,I,J)); draw(rightanglemark(F,J,I)); label("$A$", A, dir((5,4)--A)); label("$B$", B, dir((5,4)--B)); label("$C$", C, dir((5,4)--C)); label("$D$", D, dir((5,4)--D)); label("$E$", E, dir((5,4)--E)); label("$F$", F, dir((5,4)--F)); label("$G$", G, dir((5,4)--G)); label("$H$", H, dir((5,4)--H)); label("$d$", I--J, SW); label("3", H--A, W); label("4", E--A, N); label("6", E--B, N); label("5", F--B, dir(1)); label("3", F--C, dir(1)); label("5", H--D, W); label("4", C--G, S); label("6", D--G, S); [/asy] $ \textbf{(A)}\ 6.8\qquad\textbf{(B)}\ 7.1\qquad\textbf{(C)}\ 7.6\qquad\textbf{(D)}\ 7.8\qquad\textbf{(E)}\ 8.1 $

Let $PAB$ and $PBC$ be two similar right-angled triangles (in the same plane) with $\angle PAB = \angle PBC = 90^\circ$ such that $A$ and $C$ lie on opposite sides of the line $PB$. If $PC = AC$, calculate the ratio $\frac{PA}{AB}$.

Given an integer $ m$, define the sequence $ \left\{a_{n}\right\}$ as follows: \[ a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1\] Find all values of $ m$ for which $ a_{2007}$ is the first integer appearing in the sequence. Note: For a real number $ x$, $ \left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to $ x$. For example, $ \left\lceil\pi\right\rceil\equal{}4$, $ \left\lceil 2007\right\rceil\equal{}2007$.

Prove that in a trapezoid with perpendicular diagonals, the product of the legs is at least as much as the product of the bases.

There are $n$ distinct lines in three-dimensional space such that no two lines are parallel and no three lines meet at one point. What is the maximal possible number of planes determined by these $n$ lines? We say that a plane is determined if it contains at least two of the lines.

The altitudes $AH_1,BH_2,CH_3$ of an acute-angled triangle $ABC$ meet at point $H$. Points $P$ and $Q$ are the reflections of $H_2$ and $H_3$ with respect to $H$. The circumcircle of triangle $PH_1Q$ meets for the second time $BH_2$ and $CH_3$ at points $R$ and $S$. Prove that $RS$ is a medial line of triangle $ABC$.

At each vertex of a regular $14$-gon, lies a coin. Initially $7$ coins are heads, and $7$ coins are tails. Determine the minimum number $t$ such that it’s always possible to turn over at most $t$ of the coins so that in the resulting $14$-gon, no two adjacent coins are both heads and no two adjacent coins are both tails.

Let $S_n = \{1, 2,\cdots, n\}$ and $f_n : S_n \to S_n$ be defined inductively as follows: $f_1(1) = 1, f_n(2j) = j \ (j = 1, 2, \cdots , [n/2])$ and [list] [*][b][i](i)[/i][/b] if $n = 2k \ (k \geq 1)$, then $f_n(2j - 1) = f_k(j) + k \ (j = 1, 2, \cdots, k);$ [*][b][i](ii)[/i][/b] if $n = 2k + 1 \ (k \geq 1)$, then $f_n(2k + 1) = k + f_{k+1}(1), f_n(2j - 1) = k + f_{k+1}(j + 1) \ (j = 1, 2,\cdots , k).$[/list] Prove that $f_n(x) = x$ if and only if $x$ is an integer of the form \[\frac{(2n + 1)(2^d - 1)}{2^{d+1} - 1}\] for some positive integer $d.$

Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following: There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements, $$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$ where $S(n)$ denotes sum of digits of decimal representation of $n$.

Nathan has discovered a new way to construct chocolate bars, but it’s expensive! He starts with a single $1\times1$ square of chocolate and then adds more rows and columns from there. If his current bar has dimensions $w\times h$ ($w$ columns and $h$ rows), then it costs $w^2$ dollars to add another row and $h^2$ dollars to add another column. What is the minimum cost to get his chocolate bar to size $20\times20$?

A number $n$ is satisfying the conditions below i) $n$ is a positive odd integer; ii) there are some odd integers such that their squares' sum is equal to $n^{4}$. Find all such numbers.

Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the trangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is $\text{(A)}\ \frac1{2m+1}\qquad\text{(B)}\ m \qquad\text{(C)}\ 1-m\qquad\text{(D)}\ \frac1{4m} \qquad\text{(E)}\ \frac1{8m^2}$

Let $a>1$ be a positive integer. Show that the set of integers \[\{a^2+a-1,a^3+a^2-1,\ldots ,a^{n+1}+a^n-1,\ldots\}\] contains an infinite subset of pairwise coprime integers. [i]Mircea Becheanu[/i]

For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? $\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$

Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $\$4$ per gallon. How many miles can Margie drive on $\$20$ worth of gas? $\textbf{(A) }64\qquad\textbf{(B) }128\qquad\textbf{(C) }160\qquad\textbf{(D) }320\qquad \textbf{(E) }640$