Let $M$ be the incenter of the tangential quadrilateral $A_1A_2A_3A_4$. Let line $g_1$ through $A_1$ be perpendicular to $A_1M$; define $g_2,g_3$ and $g_4$ similarly. The lines $g_1,g_2,g_3$ and $g_4$ define another quadrilateral $B_1B_2B_3B_4$ having $B_1$ be the intersection of $g_1$ and $g_2$; similarly $B_2,B_3$ and $B_4$ are intersections of $g_2$ and $g_3$, $g_3$ and $g_4$, resp. $g_4$ and $g_1$. Prove that the diagonals of quadrilateral $B_1B_2B_3B_4$ intersect in point $M$. [asy] import graph; size(15cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-9.773972777861085,xmax=12.231603726660566,ymin=-3.9255487671791487,ymax=7.37238601960895; pair M=(2.,2.), A_4=(-1.6391623316400197,1.2875505916864178), A_1=(3.068893183992864,-0.5728665455336459), A_2=(4.30385937824148,2.2922812065339455), A_3=(2.221541124684679,4.978916319940133), B_4=(-0.9482172571022687,-2.24176848577888), B_1=(4.5873184669543345,0.057960746374459436), B_2=(3.9796042717514277,4.848169684238838), B_3=(-2.4295496490492385,5.324816563638236); draw(circle(M,2.),linewidth(0.8)); draw(A_4--A_1,linewidth(0.8)); draw(A_1--A_2,linewidth(0.8)); draw(A_2--A_3,linewidth(0.8)); draw(A_3--A_4,linewidth(0.8)); draw(M--A_3,linewidth(0.8)+dotted); draw(M--A_2,linewidth(0.8)+dotted); draw(M--A_1,linewidth(0.8)+dotted); draw(M--A_4,linewidth(0.8)+dotted); draw((xmin,-0.07436970390935019*xmin+5.144131675605378)--(xmax,-0.07436970390935019*xmax+5.144131675605378),linewidth(0.8)); draw((xmin,-7.882338401302275*xmin+36.2167572574517)--(xmax,-7.882338401302275*xmax+36.2167572574517),linewidth(0.8)); draw((xmin,0.4154483588930812*xmin-1.847833182441644)--(xmax,0.4154483588930812*xmax-1.847833182441644),linewidth(0.8)); draw((xmin,-5.107958950031516*xmin-7.085223310768749)--(xmax,-5.107958950031516*xmax-7.085223310768749),linewidth(0.8)); dot(M,linewidth(3.pt)+ds); label("$M$",(2.0593440948136896,2.0872038897020024),NE*lsf); dot(A_4,linewidth(3.pt)+ds); label("$A_4$",(-2.6355449660387147,1.085078446888477),NE*lsf); dot(A_1,linewidth(3.pt)+ds); label("$A_1$",(3.1575637581709772,-1.2486383377457595),NE*lsf); dot(A_2,linewidth(3.pt)+ds); label("$A_2$",(4.502882845783654,2.30684782237346),NE*lsf); dot(A_3,linewidth(3.pt)+ds); label("$A_3$",(2.169166061149418,5.203402184478307),NE*lsf); label("$g_3$",(-9.691606303109287,5.354407388189934),NE*lsf); label("$g_2$",(3.0889250292111465,6.727181967386543),NE*lsf); label("$g_1$",(-4.763345563793459,-3.4725331560442676),NE*lsf); label("$g_4$",(-2.663000457622647,6.878187171098171),NE*lsf); dot(B_4,linewidth(3.pt)+ds); label("$B_4$",(-1.5647807942653595,-3.0332452907013523),NE*lsf); dot(B_1,linewidth(3.pt)+ds); label("$B_1$",(4.955898456918535,-0.6583452686912173),NE*lsf); dot(B_2,linewidth(3.pt)+ds); label("$B_2$",(4.104778217816637,5.0661247265586455),NE*lsf); dot(B_3,linewidth(3.pt)+ds); label("$B_3$",(-3.4454819677647146,5.656417795613188),NE*lsf); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

In a circle with a radius of $1$ is an inscribed hexagon (convex). Prove that if the multiple of all diagonals that connects vertices of neighboring sides is equal to $27$ then all angles of hexagon are equals. [i]V. Petkov, I. Tonov[/i]

In the triangle $ABC$, the medians $BB_1$ and $CC_1$, which intersect at the point $M$, are drawn. Prove that a circle can be inscribed in the quadrilateral $AC_1MB_1$ if and only if $AB = AC$.

Find all positive integers $m$ such that $$\{\sqrt{m}\} = \{\sqrt{m+ 2011}\}.$$

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$. Prove that\[ 6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8} \]

In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country: [i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]

A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of $8^\text{th}$-graders to $6^\text{th}$-graders is $5:3$, and the the ratio of $8^\text{th}$-graders to $7^\text{th}$-graders is $8:5$. What is the smallest number of students that could be participating in the project? $\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 79 \qquad \textbf{(E)}\ 89$

Let $A$ be a nonempty set of positive integers, and let $N(x)$ denote the number of elements of $A$ not exceeding $x$. Let $B$ denote the set of positive integers $b$ that can be written in the form $b=a-a^{\prime}$ with $a\in A$ and $a^{\prime}\in A$. Let $b_1<b_2<\cdots$ be the members of $B$, listed in increasing order. Show that if the sequence $b_{i+1}-b_i$ is unbounded, then $\lim_{x\to \infty}\frac{N(x)}{x}=0$.

Assuming that the positive integer $a$ is not a perfect square, prove that for any positive integer n, the sum ${S_{n}=\sum_{i=1}^{n}\{a^{\frac{1}{2}}\}^{i}}$ is irrational.

Given three circumferences of radii $r$ , $r'$ and $r''$ , each tangent externally to the other two, calculate the radius of the circle inscribed in the triangle whose vertices are their three centers.

Let $n$ be a positive integer and $k\in[0,n]$ be a fixed real constant. Find the maximum value of $$\left|\sum_{i=1}^n\sin(2x_i)\right|$$ where $x_1,\ldots,x_n$ are real numbers satisfying $$\sum_{i=1}^n\sin^2(x_i)=k.$$

Let $2\leq N\leq 2022$ be a positive integer. Find the sum of all possible values of $N$ such that the product of the distinct divisors of $N$ is $N^{\frac{21}{2}}$.

Every square on the infinite sheet of cross-lined paper contains some real number. Prove that some square contains a number that does not exceed at least four of eight neighbouring numbers.

Whether exist set $A$ that contain 2016 real numbers (some of them may be equal) not all of which equal 0 such that next statement holds. For arbitrary 1008-element subset of $A$ there is a monic polynomial of degree 1008 such that elements of this subset are roots of the polynomial and other 1008 elements of $A$ are coefficients of this polynomial's degrees from 0 to 1007.

The quadrilateral $ABCD$ is inscribed in the circle and the lengths of the sides $BC$ and $DC$ are equal, and the length of the side $AB$ is equal to the length of the diagonal $AC$. Let the point $P$ be the midpoint of the arc $CD$, which does not contain point $A$, and $Q$ is the point of intersection of diagonals $AC$ and $BD$. Prove that the lines $PQ$ and $AB$ are perpendicular.

[b]p1.[/b] I open my $2010$-page dictionary, whose pages are numbered $ 1$ to $2010$ starting on page $ 1$ on the right side of the spine when opened, and ending with page $2010$ on the left. If I open to a random page, what is the probability that the two page numbers showing sum to a multiple of $6$? [b]p2.[/b] Let $A$ be the number of positive integer factors of $128$. Let $B$ be the sum of the distinct prime factors of $135$. Let $C$ be the units’ digit of $381$. Let $D$ be the number of zeroes at the end of $2^5\cdot 3^4 \cdot 5^3 \cdot 7^2\cdot 11^1$. Let $E$ be the largest prime factor of $999$. Compute $\sqrt[3]{\sqrt{A + B} +\sqrt[3]{D^C+E}}$. [b]p3. [/b] The root mean square of a set of real numbers is defined to be the square root of the average of the squares of the numbers in the set. Determine the root mean square of $17$ and $7$. [b]p4.[/b] A regular hexagon $ABCDEF$ has area $1$. The sides$ AB$, $CD$, and $EF$ are extended to form a larger polygon with $ABCDEF$ in the interior. Find the area of this larger polygon. [b]p5.[/b] For real numbers $x$, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. For example, $\lfloor 3\rfloor = 3$ and $\lfloor 5.2 \rfloor = 5$. Evaluate $\lfloor -2.5 \rfloor + \lfloor \sqrt 2 \rfloor + \lfloor -\sqrt 2 \rfloor + \lfloor 2.5 \rfloor$. [b]p6.[/b] The mean of five positive integers is $7$, the median is $8$, and the unique mode is $9$. How many possible sets of integers could this describe? [b]p7.[/b] How many three digit numbers x are there such that $x + 1$ is divisible by $11$? [b]p8.[/b] Rectangle $ABCD$ is such that $AD = 10$ and $AB > 10$. Semicircles are drawn with diameters $AD$ and $BC$ such that the semicircles lie completely inside rectangle $ABCD$. If the area of the region inside $ABCD$ but outside both semicircles is $100$, determine the shortest possible distance between a point $X$ on semicircle $AD$ and $Y$ on semicircle $BC$. [b]p9.[/b] $ 8$ distinct points are in the plane such that five of them lie on a line $\ell$, and the other three points lie off the line, in a way such that if some three of the eight points lie on a line, they lie on $\ell$. How many triangles can be formed using some three of the $ 8$ points? [b]p10.[/b] Carl has $10$ Art of Problem Solving books, all exactly the same size, but only $9$ spaces in his bookshelf. At the beginning, there are $9$ books in his bookshelf, ordered in the following way. $A - B - C - D - E - F - G - H - I$ He is holding the tenth book, $J$, in his hand. He takes the books out one-by-one, replacing each with the book currently in his hand. For example, he could take out $A$, put $J$ in its place, then take out $D$, put $A$ in its place, etc. He never takes the same book out twice, and stops once he has taken out the tenth book, which is $G$. At the end, he is holding G in his hand, and his bookshelf looks like this. $C - I - H - J - F - B - E - D - A$ Give the order (start to finish) in which Carl took out the books, expressed as a $9$-letter string (word). PS. You had better use hide for answers.

Given are three positive real numbers $ a,b,c$ satisfying $ abc \plus{} a \plus{} c \equal{} b$. Find the max value of the expression: \[ P \equal{} \frac {2}{a^2 \plus{} 1} \minus{} \frac {2}{b^2 \plus{} 1} \plus{} \frac {3}{c^2 \plus{} 1}.\]

Let $ABC$ be a right-angled triangle with right-angle at $A$. Let $X$ be the foot of the perpendicular from $A$ to $BC$, and $Y$ the mid-point of $XC$. Let $AB$ be extended to $D$ so that $|AB|=|BD|$. Prove that $DX$ is perpendicular to $AY$.

We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$. Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.

Determine all pairs of positive integers $(a,b)$ such that \[ \dfrac{a^2}{2ab^2-b^3+1} \] is a positive integer.

For nonnegative integers $p$, $q$, $r$, let \[ f(p, q, r) = (p!)^p (q!)^q (r!)^r. \]Compute the smallest positive integer $n$ such that for any triples $(a,b,c)$ and $(x,y,z)$ of nonnegative integers satisfying $a+b+c = 2020$ and $x+y+z = n$, $f(x,y,z)$ is divisible by $f(a,b,c)$. [i]Proposed by Brandon Wang[/i]

Let $A$, $B$, $C$, and $D$ be points in the coordinate plane on the hyperbola $\tfrac{x^{2}}{20}-\tfrac{y^{2}}{24}=1$ such that $ABCD$ is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than $BD^{2}$ for all such rhombi.

Let $ \alpha$, $ \beta$ be acute angles. Find the maximum value of \[ \frac{\left(1-\sqrt{\tan\alpha\tan\beta}\right)^{2}}{\cot\alpha+\cot\beta}\]

In a convex quadrilateral. eight segments are drawn, each of them connects a vertex with the midpoint of some opposite side. Seven of these segments have the same length $ a$. Prove that the eight one is also of length $ a$.

Let $M$ be the midpoint of the side $AC$ of acute-angled triangle $ABC$ with $AB>BC$. Let $\Omega $ be the circumcircle of $ ABC$. The tangents to $ \Omega $ at the points $A$ and $C$ meet at $P$, and $BP$ and $AC$ intersect at $S$. Let $AD$ be the altitude of the triangle $ABP$ and $\omega$ the circumcircle of the triangle $CSD$. Suppose $ \omega$ and $ \Omega $ intersect at $K\not= C$. Prove that $ \angle CKM=90^\circ $. [i]V. Shmarov[/i]