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? $\textbf{(A) }64\qquad\textbf{(B) }128\qquad\textbf{(C) }160\qquad\textbf{(D) }320\qquad \textbf{(E) }640$

Let $ABC$ be an acute scalene triangle and let $P$ be a point in its interior. Let $A_1$, $B_1$, $C_1$ be projections of $P$ onto triangle sides $BC$, $CA$, $AB$, respectively. Find the locus of points $P$ such that $AA_1$, $BB_1$, $CC_1$ are concurrent and $\angle PAB + \angle PBC + \angle PCA = 90^{\circ}$.

Determine the remainder when $$2^6\cdot3^{10}\cdot5^{12}-75^4\left(26^2-1\right)^2+3^{10}-50^6+5^{12}$$ is divided by $1001$. [i]2016 CCA Math Bonanza Lightning #4.1[/i]

Given two positive number $a$, $b$ such that $a<b$. Let A.M. be their arithmetic mean and let G.M. be their positive geometric mean. Then A.M. minus G.M. is always less than: $\textbf{(A) }\dfrac{(b+a)^2}{ab}\qquad\textbf{(B) }\dfrac{(b+a)^2}{8b}\qquad\textbf{(C) }\dfrac{(b-a)^2}{ab}$ $\textbf{(D) }\dfrac{(b-a)^2}{8a}\qquad \textbf{(E) }\dfrac{(b-a)^2}{8b}$

Consider skew lines $a,b$ and a plane $\rho$ that intersect both of the lines (but does not contain any of them). Choose such points $X\in a,Y\in b$ that $XY\parallel\rho.$ Find the locus of midpoints $M$ of all segments $XY,$ when $X$ moves along line $a$.