How many positive integers $n<10^6$ are there such that $n$ is equal to twice of square of an integer and is equal to three times of cube of an integer? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of the above} $

On the sides of triangle $ABC$, isosceles right-angled triangles $AUB, CVB$, and $AWC$ are placed. These three triangles have their right angles at vertices $U, V$ , and $W$, respectively. Triangle $AUB$ lies completely inside triangle $ABC$ and triangles $CVB$ and $AWC$ lie completely outside $ABC$. See the figure. Prove that quadrilateral $UVCW$ is a parallelogram. [asy] import markers; unitsize(1.5 cm); pair A, B, C, U, V, W; A = (0,0); B = (2,0); C = (1.7,2.5); U = (B + rotate(90,A)*(B))/2; V = (B + rotate(90,C)*(B))/2; W = (C + rotate(90,A)*(C))/2; draw(A--B--C--cycle); draw(A--W, StickIntervalMarker(1,1,size=2mm)); draw(C--W, StickIntervalMarker(1,1,size=2mm)); draw(B--V, StickIntervalMarker(1,2,size=2mm)); draw(C--V, StickIntervalMarker(1,2,size=2mm)); draw(A--U, StickIntervalMarker(1,3,size=2mm)); draw(B--U, StickIntervalMarker(1,3,size=2mm)); draw(rightanglemark(A,U,B,5)); draw(rightanglemark(B,V,C,5)); draw(rightanglemark(A,W,C,5)); dot("$A$", A, S); dot("$B$", B, S); dot("$C$", C, N); dot("$U$", U, NE); dot("$V$", V, NE); dot("$W$", W, NW); [/asy]

A set $S$ of $n-1$ natural numbers is given ($n\ge 3$). There exist at least at least two elements in this set whose difference is not divisible by $n$. Prove that it is possible to choose a non-empty subset of $S$ so that the sum of its elements is divisible by $n$.

We have $n$ countries. Each country have $m$ persons who live in that country ($n>m>1$). We divide $m \cdot n$ persons into $n$ groups each with $m$ members such that there don't exist two persons in any groups who come from one country. Prove that one can choose $n$ people into one class such that they come from different groups and different countries.

In the fall of $1996$, a total of $800$ students participated in an annual school clean-up day. The organizers of the event expect that in each of the years $1997$, $1998$, and $1999$, participation will increase by $50 \%$ over the previous year. The number of participants the organizers will expect in the fall of $1999$ is $\text{(A)}\ 1200 \qquad \text{(B)}\ 1500 \qquad \text{(C)}\ 2000 \qquad \text{(D)}\ 2400 \qquad \text{(E)}\ 2700$

We have a collection of $1720$ balls, half of which are black and half of which are white, aligned in a straight line. Our task is to make the balls alternating in color along the line. The following greedy algorithm accomplishes that task for $2n$ balls: \begin{tabular}{l} 1: \textbf{FOR} $i$ \textbf{IN} $[2,3,\dots,2n]$ \\ 2: $\quad$ \textbf{IF} balls $i-1$ and $i$ have the same color: \\ 3: $\quad\quad$ $j\gets$ smallest index greater than $i$ for which balls $i-1$ and $j$ have different colors \\ 4: $\quad\quad$ swap balls $i$ and $j$ \end{tabular} Given a configuration $C$ of our $1720$ balls, let $\hat{\sigma}(C)$ denote the number of swaps the greedy algorithm takes, and let $\sigma(C)$ denote the minimum number of swaps actually necessary to perform the task. Find the maximum value over all configurations $C$ of $\hat{\sigma}(C)-\sigma(C)$.

If a,b,c positive numbers and such that $a+\sqrt{b+\sqrt{c}}=c+\sqrt{b+\sqrt{a}}$. Prove that if $a\neq c$ then $40ac<1$.

Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle? $ \textbf{(A)}\ \frac{1}{36} \qquad \textbf{(B)}\ \frac{1}{24} \qquad \textbf{(C)}\ \frac{1}{18} \qquad \textbf{(D)}\ \frac{1}{12} \qquad \textbf{(E)}\ \frac{1}{9}$

Suppose that the real numbers $a_{1},a_{2},...,a_{2002}$ satisfying $\frac{a_{1}}{2}+\frac{a_{2}}{3}+...+\frac{a_{2002}}{2003}=\frac{4}{3}$ $\frac{a_{1}}{3}+\frac{a_{2}}{4}+...+\frac{a_{2002}}{2004}=\frac{4}{5}$ $...$ $\frac{a_{1}}{2003}+\frac{a_{2}}{2004}+...+\frac{a_{2002}}{4004}=\frac{4}{4005}$ Evaluate the sum $\frac{a_{1}}{3}+\frac{a_{2}}{5}+...+\frac{a_{2002}}{4005}$.

The $ABC$ triangle is given, point $I_a$ is the center of an exscribed circle touching the side $BC$ , the point $M$ is the midpoint of the side $BC$, the point $W$ is the intersection point of the bisector of the angle $A$ of the triangle $ABC$ with the circumscribed circle around him. Prove that the area of the triangle $I_aBC$ is calculated by the formula $S_{ (I_aBC)} = MW \cdot p$, where $p$ is the semiperimeter of the triangle $ABC$. (Mykola Moroz)

If $ABCDEF$ is a convex cyclic hexagon, then its diagonals $AD$, $BE$, $CF$ are concurrent if and only if $\frac{AB}{BC}\cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$. [i]Alternative version.[/i] Let $ABCDEF$ be a hexagon inscribed in a circle. Then, the lines $AD$, $BE$, $CF$ are concurrent if and only if $AB\cdot CD\cdot EF=BC\cdot DE\cdot FA$.

Let $a$ be a permutation on $\{0,1,\ldots ,2015\}$ and $b,c$ are also permutations on $\{1,2,\ldots ,2015\}$. For all $x\in \{1,2,\ldots ,2015\}$, the following conditions are satisfied: (i) $a(x)-a(x-1)\neq 1$,\\ (ii) if $b(x)\neq x$, then $c(x)=x$,\\ Prove that the number of $a$'s is equal to the number of ordered pairs of $(b,c)$.

Let $x$ and $y$ be relatively prime integers. Show that $x^2+xy+y^2$ and $x^2+3xy+y^2$ are relatively prime.

Let $ABCD$ be a parallelogram. The interior angle bisector of $\angle ADC$ intersects the line $BC$ in $E$, and the perpendicular bisector of the side $AD$ intersects the line $DE$ in $M$. Let $F= AM \cap BC$. Prove that: a) $DE=AF$; b) $AD\cdot AB = DE\cdot DM$. [i]Daniela and Marius Lobaza, Timisoara[/i]

You are given some $12$ non-zero, not necessarily distinct real numbers. Find all positive integers $k$ from $1$ to $12$, such that among these numbers you can always choose $k$ numbers whose sum has the same sign as their product, that is, either both the sum and the product are positive, or both are negative. [i]Proposed by Anton Trygub[/i]

Beto has rectangular chessboard where the number of rows and columns are consecutive numbers (for example, $30$ and $31$). Ana has tiles of two colors and different sizes: the red tiles are $5 \times 7$ rectangles and the blue tiles are $3 \times 5$ rectangles. Ana realized that she can cover all the squares of Beto’s board using only red tiles, which can be rotated, but must not overlap or extend beyond the board. Then, she realized she can also do the same using only blue tiles. What is the minimum number of squares that Beto’s board can have?

Find all the non-zero polynomials $P(x),Q(x)$ with real coefficients and the minimum degree,such that for all $x \in \mathbb{R}$: \[ P(x^2)+Q(x)=P(x)+x^5Q(x) \]

Show that there does not exist a right triangle with all integer side lengths such that exactly one of the side lengths is odd.

A consecutive pythagorean triple is a pythagorean triple of the form $a^2 + (a+1)^2 = b^2$, $a$ and $b$ positive integers. Given that $a$, $a+1$, and $b$ form the third consecutive pythagorean triple, find $a$.

Let $ABCD$ be a trapezium inscribed in a circle $k$ with diameter $AB$. A circle with center $B$ and radius $BE$,where $E$ is the intersection point of the diagonals $AC$ and $BD$ meets $k$ at points $K$ and $L$. If the line ,perpendicular to $BD$ at $E$,intersects $CD$ at $M$,prove that $KM\perp DL$.

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. A group of haikus Some have one syllable less Sixteen in total. The group of haikus Some have one syllable more Eighteen in total. What is the largest Total count of syllables That the group can’t have? (For instance, a group Sixteen, seventeen, eighteen Fifty-one total.) (Also, you can have No sixteen, no eighteen Syllable haikus) [i]Proposed by Jeff Lin[/i]

For a positive integer $n$, $r(n)$ denote the sum of the remainders when $n$ is divided by $1, 2,..., n$ respectively. Prove that $r(k) = r(k -1)$ for infinitely many positive integers $k$.

Let $\sigma(n)$ denote the sum of positive divisors of a number $n$. A positive integer $N=2^r b$ is given, where $r$ and $b$ are positive integers and $b$ is odd. It is known that $\sigma(N)=2N-1$. Prove that $b$ and $\sigma(b)$ are coprime. (J. Antalan, J. Dris)

Find the largest positive integer that cannot be written in the form $a + bc$ for some positive integers $a, b, c$, satisfying $a < b < c$.

Let $ \mathcal G$ be the set of all finite groups with at least two elements. a) Prove that if $ G\in \mathcal G$, then the number of morphisms $ f: G\to G$ is at most $ \sqrt [p]{n^n}$, where $ p$ is the largest prime divisor of $ n$, and $ n$ is the number of elements in $ G$. b) Find all the groups in $ \mathcal G$ for which the inequality at point a) is an equality.