Let $PA_1A_2...A_{12} $ be the regular pyramid, $ A_1A_2...A_{12} $ is regular polygon, $S$ is area of the triangle $PA_1A_5$ and angle between of the planes $A_1A_2...A_{12} $ and $ PA_1A_5 $ is equal to $ \alpha $. Find the volume of the pyramid.

Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.

Let $a_1$, $a_2$, $ ...$ be a sequence of numbers, each either $1$ or $-1$. Show that if $$\frac{a_1}{3}+\frac{a_2}{3^2} + ... =\frac{p}{q}$$ for integers $p$ and $q$ such that $3$ does not divide $q$, then the sequence $a_1$, $a_2$, $ ...$ is periodic; that is, there is some positive integer $n$ such that $a_i = a_{n+i}$ for $i = 1$, $2$,$...$.

Let $A$ be a nonempty set with $n$ elements. Find the number of ways of choosing a pair of subsets $(B,C)$ of $A$ such that $B$ is a nonempty subset of $C$.

Let $\omega_1$ and $\omega_2$ be circles with radii $3$ and $12$ and externally tangent at point $P$. Let a common external tangent intersect $\omega_1$, $\omega_2$ at $S$, $T$ respectively and the common internal tangent at point $Q$. Define $X$ to be the point on $\overrightarrow{QP}$ such that $QX=10$. If $XS$, $XT$ intersect $\omega_1$, $\omega_2$ a second time at $A$ and $B$, determine $\tan\angle APB$. . [i]2015 CCA Math Bonanza Individual Round #15[/i]

А positive integer $n{}$ is given. For every $x{}$ consider the sum \[Q(x)=\sum_{k=1}^{10^n}\left\lfloor\frac{x}{k}\right\rfloor.\]Find the difference $Q(10^n)-Q(10^n-1)$. [i]Alexey Tolpygo[/i]

Prove that after deleting the perfect squares from the list of positive integers the number we find in the $n^{th}$ position is equal to $n+\{\sqrt{n}\},$ where $\{\sqrt{n}\}$ denotes the integer closest to $\sqrt{n}.$

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

Let $ABC$ be a triangle such that $M$ and $N$ are the midpoints of $AC$ and $BC$, respectively. Let $I$ be the incenter of $ABC$ and $E$ be the intersection of $MN$ with $Bl$. Let $P$ be a point such that $EP$ is perpendicular to $MN$ and $NP$ parallel to $IA$. Prove that $IP$ is perpendicular to $BC$.

We define a lattice point as a point whose coordinates are integers, zero admitted. Then the number of lattice points on the boundary and inside the region bounded by the $ x$-axis, the line $ x \equal{} 4$, and the parabola $ y \equal{} x^2$ is: $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 35\qquad \textbf{(C)}\ 34\qquad \textbf{(D)}\ 30\qquad \textbf{(E)}\ \text{not finite}$

Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had form a geometric progression. Alex eats 5 of his peanuts, Betty eats 9 of her peanuts, and Charlie eats 25 of his peanuts. Now the three numbers of peanuts that each person has form an arithmetic progression. Find the number of peanuts Alex had initially.

Given a triangle $ABC$, the internal and external bisectors of angle $A$ intersect $BC$ at points $D$ and $E$ respectively. Let $F$ be the point (different from $A$) where line $AC$ intersects the circle $w$ with diameter $DE$. Finally, draw the tangent at $A$ to the circumcircle of triangle $ABF$, and let it hit $w$ at $A$ and $G$. Prove that $AF=AG$.

Seongcheol has $3$ red shirts and $2$ green shirts, such that he cannot tell the difference between his three red shirts and he similarly cannot tell the difference between his two green shirts. In how many ways can he hang them in a row in his closet, given that he does not want the two green shirts next to each other?

$\vartriangle ABC$ is right with $\angle C = 90^o$. The internal angle bisectors of $\angle A$ and $\angle B$ meet at point $D$, while the external angle bisectors of $\angle A$ and $\angle B$ meet at point $E$. Suppose that $AD = 1$ and $BD = 2$. The value of $DE^2$ can be expressed as $x+y \sqrt{z}$ for integers $x$, $y$, and $z$, where $z$ is greater than $1$ and not divisible by the square of any prime. Compute $100x + 10y + z$. Note: For a generic triangle $\vartriangle PQR$, if we let $Q'$ be the reflection of $Q$ over $P$, then the external angle bisector of $\angle P$ is the line that contains the internal angle bisector of $\angle Q'PR$.

Consider the system of congruences $$\begin{cases} xy \equiv - 1 \,\, (mod z) \\ yz \equiv 1 \, \, (mod x) \\zx \equiv 1 \, \, (mod y)\end {cases}$$ Find the number of triples $(x,y, z) $ of distinct positive integers satisfying this system such that one of the numbers $x,y, z$ equals $19$.

Find all functions $f(x)$ from nonnegative reals to nonnegative reals such that $f(f(x))=x^4$ and $f(x)\leq Cx^2$ for some constant $C$.

Find all continuous functions $f: \mathbb R \to \mathbb R$ such that \[f(x^2-y^2)=f(x)^2 + f(y)^2, \quad \forall x,y \in \mathbb R.\]

The integer lattice in the plane is colored with 3 colors. Find the least positive real $S$ with the property: for any such coloring it is possible to find a monochromatic lattice points $A,B,C$ with $S_{\triangle ABC}=S$. [i]Proposed by Nikolay Beluhov[/i] EDIT: It was the problem 3 (not 2), corrected the source title.

Let $x\ne 1,y\ne 1$ and $x\ne y.$ Show that if \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y},\] then \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y}=x+y+z.\]

Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.

On the side $ CD $ of the square $ ABCD, $ consider $ E $ for which $ \angle ABE =60^{\circ } . $ On the line $ AB, $ take the point $ F $ distinct from $ B $ such that $ BE=BF $ and such that it is on the segment $ AB, $ or $ A $ is on $ BF. $ Moreover, $ M $ is the intersection of $ EF,AD. $ [b]a)[/b] Show that $ \angle BME =75^{\circ } . $ [b]b)[/b] If the bisector of $ \angle CBE $ intersects $ CD $ in $ N, $ show that $ BMN $ is equilateral.

Let $ABC$ be an equilateral triangle with the side length equals $a+ b+ c$. On the side $AB{}$ of the triangle $ABC$ points $C_1$ and $C_2$ are chosen, on the side $BC$ points $A_1$ and $A_2$, arc chosen, and on the side $CA$ points $B_1$ and $B_2$ are chosen such that $A_1A_2 = CB_1 = BC_2 = a, B_1B_2 = AC_1 = CA_2 = b, C_1C_2 = BA_1 = AB_2 = c$. Let the point $A^{’}$ be such that the triangle $A^{'} B_2C_1$ is equilateral, and the points $A$ and $A^{'}$ lie on different sides of the line $B_2C_1$. Similarly, the points $B^{’}$ and $C^{'}$ are constructed (the triangle $B^{'} C_2A_1$ is equilateral, and the points $B$ and $B^{’}$ lie on different sides of the line $C_2A_1$; the triangle $C^{'} A_2B_1$ is equilateral, and the points $C$ and $C^{'}$ lie on different sides of the line $A_2B_1$). Prove that the triangle $A^{'}B^{'}C^{'}$ is equilateral.

Determine all triples \( (a, b, c) \) of positive integers such that: \[ a + b + c = abc. \]

Positive reals $x, y, z$ satisfy $$\frac{xy+1}{x+1} = \frac{yz+1}{y+1} = \frac{zx+1}{z+1}$$ Do they all have to be equal? [i](Proposed by Oleksii Masalitin)[/i]

Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$.