Find all pairs of integers $(a,b)$ such that $(b^2+7(a-b))^2=a^{3}b$.

Determine if there exist prime numbers $ p_{1},p_{2},...,p_{2007}$ such that $ p_{2}|p_{1}^{2}\minus{}1,p_{3}|p_{2}^{2}\minus{}1,...,p_{1}|p_{2007}^{2}\minus{}1$.

If you have $65$ points in a plane, we will make the lines that passes by any two points in this plane and we obtain exactly $2015$ distinct lines, prove that least $4$ points are collinears!!

Let $A\in\mathcal{M}_n(\mathbb{C})$ be an antisymmetric matrix, i.e. $A=-A^t.$[list=a] [*]Prove that if $A\in\mathcal{M}_n(\mathbb{R})$ and $A^2=O_n$ then $A=O_n.$ [*]Assume that $n{}$ is odd. Prove that if $A{}$ is the adjoint of another matrix $B\in\mathcal{M}_n(\mathbb{C})$ then $A^2=O_n.$ [/list]

Anders is solving a math problem, and he encounters the expression $\sqrt{15!}$. He attempts to simplify this radical as $a\sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible values of $ab$ can be expressed in the form $q\cdot 15!$ for some rational number $q$. Find $q$.

For what $k$ is it possible to construct a cube $k\times k\times k$ of the black and white cubes $1\times 1\times 1$ in such a way that every small cube has the same colour, that have exactly two his neighbours. (Two cubes are neighbours, if they have the common face.)

Faith has four different integer length segments. It turned out that any three of them can form a triangle. What is the smallest total length of this set of segments?

Let $A$ and $B$ be two finite disjoint sets of points in the plane such that no three distinct points in $A \cup B$ are collinear. Assume that at least one of the sets $A, B$ contains at least five points. Show that there exists a triangle all of whose vertices are contained in $A$ or in $B$ that does not contain in its interior any point from the other set.

Can you find three natural numbers $a, b, c$ whose greatest common divisor is $1$ and which satisfy the equality $$ab + bc + ac = (a + b -c)(b + c - a)(c + a - b) ?$$

Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $ \$$1 each, begonias $ \$$1.50 each, cannas $ \$$2 each, dahlias $ \$$2.50 each, and Easter lilies $ \$$3 each. What is the least possible cost, in dollars, for her garden? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((6,0)--(0,0)--(0,1)--(6,1)); draw((0,1)--(0,6)--(4,6)--(4,1)); draw((4,6)--(11,6)--(11,3)--(4,3)); draw((11,3)--(11,0)--(6,0)--(6,3)); label("1",(0,0.5),W); label("5",(0,3.5),W); label("3",(11,1.5),E); label("3",(11,4.5),E); label("4",(2,6),N); label("7",(7.5,6),N); label("6",(3,0),S); label("5",(8.5,0),S);[/asy]$ \textbf{(A)}\ 108 \qquad \textbf{(B)}\ 115 \qquad \textbf{(C)}\ 132 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 156$

Assume that $ \alpha$ and $ \beta$ are two roots of the equation: $ x^2\minus{}x\minus{}1\equal{}0$. Let $ a_n\equal{}\frac{\alpha^n\minus{}\beta^n}{\alpha\minus{}\beta}$, $ n\equal{}1, 2, \cdots$. (1) Prove that for any positive integer $ n$, we have $ a_{n\plus{}2}\equal{}a_{n\plus{}1}\plus{}a_n$. (2) Find all positive integers $ a$ and $ b$, $ a<b$, satisfying $ b \mid a_n\minus{}2na^n$ for any positive integer $ n$.

Determine the prime numbers $p$ for which the number $a = 7^p - p - 16$ is a perfect square. Lucian Petrescu

Let $A$ be a subset of the irrational numbers such that the sum of any two distinct elements of it be a rational number. Prove that $A$ has two elements at most.

Let $p$ be a prime number. For integers $r, s$ such that $rs(r^2 - s^2)$ is not divisible by $p$, let $f(r, s)$ denote the number of integers $n \in \{1, 2, \ldots, p - 1\}$ such that $\{rn/p\}$ and $\{sn/p\}$ are either both less than $1/2$ or both greater than $1/2$. Prove that there exists $N > 0$ such that for $p \geq N$ and all $r, s$, \[ \left\lceil \frac{p-1}{3} \right\rceil \le f(r, s) \le \left\lfloor \frac{2(p-1)}{3} \right\rfloor. \]

Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger? $ \textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{10} \qquad\textbf{(D)}\ \frac{\sqrt{5}}{6} \qquad\textbf{(E)}\ \frac{\sqrt{10}}{5} $

$[AB]$ and $[CD]$ are not parallel in the convex quadrilateral $ABCD$. Let $E$ and $F$ be the midpoints of $[AD]$ and $[BC]$, respectively. If $|CD|=12$, $|AB|=22$, and $|EF|=x$, what is the sum of integer values of $x$? $ \textbf{(A)}\ 110 \qquad\textbf{(B)}\ 114 \qquad\textbf{(C)}\ 118 \qquad\textbf{(D)}\ 121 \qquad\textbf{(E)}\ \text{None of the above} $

Positive $x,y,z$ satisfy a system: $\begin{cases} x^2 + xy + y^2/3= 25 \\ y^2/ 3 + z^2 = 9 \\ z^2 + zx + x^2 = 16 \end{cases}$ Find the value of expression $xy + 2yz + 3zx$.

Show that a graph with 2n points and $n^2 + 1$ edges necessarily contains a 3-cycle, but that we can find a graph with 2n points and $n^2$ edges without a 3-cycle. please prove it without induction .

Let $A,B,P$ be three points on a circle. Prove that if $a,b$ are the distances from $P$ to the tangents at $A,B$ respectively, and $c$ is the distance from $P$ to the chord $AB$, then $c^2 =ab$.

Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$ intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.

For integral $m$, let $p(m)$ be the greatest prime divisor of $m.$ By convention, we set $p(\pm 1) = 1$ and $p(0) = \infty.$ Find all polynomials $f$ with integer coefficients such that the sequence \[ \{p \left( f \left( n^2 \right) \right) - 2n \}_{n \geq 0} \] is bounded above. (In particular, this requires $f \left (n^2 \right ) \neq 0$ for $n \geq 0.$)

Find the sum of the smallest and largest possible values for $x$ which satisfy the following equation. $$9^{x+1} + 2187 = 3^{6x-x^2}.$$

there are $n$ points on the plane,any two vertex are connected by an edge of red,yellow or green,and any triangle with vertex in the graph contains exactly $2$ colours.prove that $n<13$

Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.

Let the line $ z\equal{}x, \, y\equal{}0$ rotate at a constant speed about the $ z$-axis; let at the same time the point of intersection of this line with the $ z$-axis be displaced along the $ z$-axis at constant speed. (a) Determine that surface of rotation upon which the resulting helical surface can be developed (i.e. isometrically mapped). (b) Find those lines of the surface of rotation into which the axis and the generators of the helical surface will be mapped by this development.