Is there a continuous function $f : \mathbb R \backslash \mathbb Q \to \mathbb R \backslash \mathbb Q$ for which the archetype of every irrational number has a positive Hausdorff dimension?

In two neigbouring cells(dimensions $1\times 1$) of square table $10\times 10$ there is hidden treasure. John needs to guess these cells. In one $\textit{move}$ he can choose some cell of the table and can get information whether there is treasure in it or not. Determine minimal number of $\textit{move}$'s, with properly strategy, that always allows John to find cells in which is treasure hidden.

Consider a $8\times 8$ chessboard where all $64$ unit squares are at the start white. Prove that, if any $12$ of the $64$ unit square get painted black, then we can find $4$ lines and $4$ rows that have all these $12$ unit squares.

If $z\in\mathbb{C},\arg{(z^2-4)}=\frac{5}{6}\pi,\arg{(z^2+4)}=\frac{\pi}{3}$, then the value of $z$ is________.

Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$ whenever $A$ and $B$ are two distinct subsets of $X$. [i] (Sergiu Novac)[/i]

The board used for playing a game consists of the left and right parts. In each part there are several fields and there’re several segments connecting two fields from different parts (all the fields are connected.) Initially, there is a violet counter on a field in the left part, and a purple counter on a field in the right part. Lyosha and Pasha alternatively play their turn, starting from Pasha, by moving their chip (Lyosha-violet, and Pasha-purple) over a segment to other field that has no chip. It’s prohibited to repeat a position twice, i.e. can’t move to position that already been occupied by some earlier turns in the game. A player losses if he can’t make a move. Is there a board and an initial positions of counters that Pasha has a winning strategy?

Let the vertices $A$ and $C$ of a right triangle $ABC$ be on the arc with center $B$ and radius $20$. A semicircle with diameter $[AB]$ is drawn to the inner region of the arc. The tangent from $C$ to the semicircle touches the semicircle at $D$ other than $B$. Let $CD$ intersect the arc at $F$. What is $|FD|$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac 52 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The tangent to $\omega$ through $B$ cuts the parallel to $BC$ through $A$ at $P$. The line $CP$ cuts the circumcircle of $\triangle ABP$ again in $Q$ and line $AQ$ cuts $\omega$ at $R$. Prove that $BQCR$ is parallelogram if and only if $AC=BC$.

A positive integer \( n \) satisfies the following conditions: [list] [*] The number \( n \) has exactly \( 60 \) divisors: \( 1 = a_1 < a_2 < \cdots < a_{60} = n \); [*] The number \( n+1 \) also has exactly \( 60 \) divisors: \( 1 = b_1 < b_2 < \cdots < b_{60} = n+1 \). [/list] Let \( k \) be the number of indices \( i \) such that \( a_i < b_i \). Find all possible values of \( k \). [i]Note: Such numbers exist, for example, the numbers \( 4388175 \) and \( 4388176 \) both have \( 60 \) divisors.[/i] [i]Proposed by Anton Trygub[/i]

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 $C$.