Let $a_0=1,a_1=2,$ and $a_n=4a_{n-1}-a_{n-2}$ for $n\ge 2.$ Find an odd prime factor of $a_{2015}.$

The area of the ring between two concentric circles is $12\tfrac12\pi$ square inches. The length of a chord of the larger circle tangent to the smaller circle, in inches, is: $\textbf{(A) }\dfrac5{\sqrt2}\qquad \textbf{(B) }5\qquad \textbf{(C) }5\sqrt2\qquad \textbf{(D) }10\qquad \textbf{(E) }10\sqrt2$

Given a board of $3 \times 3$ you want to write the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and a number in their boxes positive integer $M$, not necessarily different from the above. The goal is that the sum of the three numbers in each row be the same $a)$ Find all the values of $M$ for which this is possible. $b)$ For which of the values of $M$ found in $a)$ is it possible to arrange the numbers so that no only the three rows add the same but also the three columns add the same?

Graph $G$ is defined in 3d space. It has $e$ edges and every vertex are connected if the distance between them is $1.$ Given that there exists the Hamilton cycle, prove that for $e>1,$ we have $$\min d(v)\le 1+2\left(\frac{e}{2}\right)^{0.4}.$$

Let $RICE$ be a quadrilateral with an inscribed circle $O$ such that every side of $RICE$ is tangent to $O$. Given taht $RI=3$, $CE=8$, and $ER=7$, compute $IC$.

Does there exist a set $M$ in usual Euclidean space such that for every plane $\lambda$ the intersection $M \cap \lambda$ is finite and nonempty ? [i]Proposed by Hungary.[/i] [hide="Remark"]I'm not sure I'm posting this in a right Forum.[/hide]

Given a regular heptagon $A_1...A_7$. Prove that $$\frac{1}{|A_1A_5|} + \frac{1}{|A_1A_3| }= \frac{1}{|A_1A_7|}$$.

Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$. Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$. Let $\Gamma$ be the circle which is tangent to the line $l$, tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$. The diameter of $\Gamma$ through $D$ meets $l$ in $E$. Show that $AB=DE$.

Let $ K$ be the circumscribed circle of the trapezoid $ ABCD$ . In this trapezoid the diagonals $ AC$ and $ BD$ are perpendicular. The parallel sides $ AB\equal{}a$ and $ CD\equal{}c$ are diameters of the circles $ K_{a}$ and $ K_{b}$ respectively. Find the perimeter and the area of the part inside the circle $ K$, that is outside circles $ K_{a}$ and $ K_{b}$.

Suppose $a$, $b$ and $c$ are nonzero real numberss satisfying $abc=2$. Prove that among the three numbers $2a-\frac{1}{b}$, $2b-\frac{1}{c}$ and $2c-\frac{1}{a}$, at most two of them are greater than $2$.

In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. For what is the smallest amount of rubles you can rearrange chips in reverse order?

Let $\mathcal E$ be an ellipse with foci $F_1$ and $F_2$. Parabola $\mathcal P$, having vertex $F_1$ and focus $F_2$, intersects $\mathcal E$ at two points $X$ and $Y$. Suppose the tangents to $\mathcal E$ at $X$ and $Y$ intersect on the directrix of $\mathcal P$. Compute the eccentricity of $\mathcal E$. (A [i]parabola[/i] $\mathcal P$ is the set of points which are equidistant from a point, called the [i]focus[/i] of $\mathcal P$, and a line, called the [i]directrix[/i] of $\mathcal P$. An [i]ellipse[/i] $\mathcal E$ is the set of points $P$ such that the sum $PF_1 + PF_2$ is some constant $d$, where $F_1$ and $F_2$ are the [i]foci[/i] of $\mathcal E$. The [i]eccentricity[/i] of $\mathcal E$ is defined to be the ratio $F_1F_2/d$.)

Two circles with radii $R$ and $r$ intersect at $C$ and $D$ and are tangent to a line $\ell$ at $A$ and $B$. Prove that the circumradius of triangle $ABC$ does not depend on the length of segment $AB$.

Determine the largest integer $n$ such that there exist monic quadratic polynomials $p_1(x)$, $p_2(x)$, $p_3(x)$ with integer coefficients so that for all integers $ i \in [1, n]$ there exists some $j \in [1, 3]$ and $m \in Z$ such that $p_j (m) = i$.

Let $a,b,c$ be real numbers satisfying: \[ab-a=b+119\] \[bc-b=c+59\] \[ca-c=a+71\] Determine all possible values of $a+b+c$.

Prove that for all $x_1, x_2,\ldots , x_n \in \mathbb R$ the following inequality holds: \[\sum_{n \geq i >j \geq 1} \cos^2(x_i - x_j ) \geq \frac{n(n-2)}{4}\]

Colin has $900$ Choco Pies. He realizes that for some integer values of $n \le 900$, if he eats n pies a day, he will be able to eat the same number of pies every day until he runs out. How many possible values of $n$ are there?

Positive integers $a,b,c,x,y,z$ satisfy: $a^2+b^2=c^2$, $x^2+y^2=z^2$ and $|x-a| \leq 1$ , $|y-b| \leq 1$. Prove that sets $\{a,b\}$ and $\{x,y\}$ are equal.

Determine all possible values of $A^3+B^3+C^3-3ABC$ where $A$, $B$, and $C$ are nonnegative integers.

In the Cartesian plane $Oxy$ we consider the points ${A_1}\left( {40,1} \right), {A_2}\left( {40,2} \right), \ldots , {A_{40}}\left( {40,40} \right)$ as well as the segments $O{A_1},O{A_2},\ldots,O{A_{40}}$. A point of the Cartesian plane $Oxy$ is called "good", if its coordinates are integers and it is internal of one segment $O{A_i}, i=1,2,3,\ldots,40$. Additionally, one of the segments $O{A_1},O{A_2},\ldots,O{A_{40}}$ is called "good" if it contains a "good" point. Find the number of "good" segments and "good" points.

Prove that for $\forall$ $k\geq 2$, $k\in \mathbb{N}$ there exist a natural number that could be presented as a sum of two, three … $k$ cubes of natural numbers.

Let $ABC$ be a scalene triangle and $D$ be the feet of altitude from $A$ to $BC$. Let $I_1$, $I_2$ be incenters of triangles $ABD$ and $ACD$ respectively, and let $H_1$, $H_2$ be orthocenters of triangles $ABI_1$ and $ACI_2$ respectively. The circles $(AI_1H_1)$ and $(AI_2H_2)$ meet again at $X$. The lines $AH_1$ and $XI_1$ meet at $Y$, and the lines $AH_2$ and $XI_2$ meet at $Z$. Suppose the external common tangents of circles $(BI_1H_1)$ and $(CI_2H_2)$ meet at $U$. Prove that $UY=UZ$. [i]Proposed by Ivan Chan Kai Chin[/i]

Find all pairs $(p,q)$ of prime numbers which $p>q$ and $$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$ is an integer.

$ \lim_{n\to\infty } \left( (1+1/n)^{-n}\sum_{i=0}^n\frac{1}{i!} \right)^{2n} $

If $ AD$ is the altitude, $ BE$ the angle bisector, and $ CF$ the median of a triangle $ ABC$, prove that $ AD,BE,$ and $ CF$ are concurrent if and only if: $ a^2(a\minus{}c)\equal{}(b^2\minus{}c^2)(a\plus{}c),$ where $ a,b,c$ are the lengths of the sides $ BC,CA,AB$, respectively.