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]

Let $f$ be a strictly increasing function defined on the set of real numbers. For $x$ real and $t$ positive, set\[g(x,t)=\frac{f(x+t)-f(x)}{f(x) - f(x - t)}.\] Assume that the inequalities\[2^{-1} < g(x, t) < 2\] hold for all positive t if $x = 0$, and for all $t \leq |x|$ otherwise. Show that\[ 14^{-1} < g(x, t) < 14\] for all real $x$ and positive $t.$

Suppose $P(x) = x^{2016} + a_{2015}x^{2015} + ...+ a_1x + a_0$ satisfies $P(x)P(2x + 1) = P(-x)P(-2x - 1)$ for all $x \in R$. Find the sum of all possible values of $a_{2015}$.

Evaluate \[\lim_{a\rightarrow \frac{\pi}{2}-0}\ \int_0^a\ (\cos x)\ln (\cos x)\ dx\ \left(0\leqq a <\frac{\pi}{2}\right)\]

Let $ABC$ be a triangle with a right angle at $C$. Let $I$ be the incentre of triangle $ABC$, and let $D$ be the foot of the altitude from $C$ to $AB$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $A_1$, $B_1$, and $C_1$, respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$, respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$, respectively. Prove that the circumcircles of triangles $A_1EI$, $B_1FI$, and $C_1KL$ have a common point.

On the plane of triangle $ABC$ with $\angle BAC = 90^\circ$ we raise perpendicular lines in $A$ and $B$, on the same side of the plane. On these two perpendicular lines we consider the points $M$ and $N$ respectively such that $BN < AM$. Knowing that $AC = 2a$, $AB = a\sqrt 3$, $AM=a$ and that the plane $MNC$ makes an angle of $30^\circ$ with the plane $ABC$ find a) the area of the triangle $MNC$; b) the distance from $B$ to the plane $MNC$.

Let $m \ge n$ be positive integers. Frieder is given $mn$ posters of Linus with different integer dimensions of $k \times l$ with $1 \ge k \ge m$ and $1 \ge l \ge n$. He must put them all up one by one on his bedroom wall without rotating them. Every time he puts up a poster, he can either put it on an empty spot on the wall or on a spot where it entirely covers a single visible poster and does not overlap any other visible poster. Determine the minimal area of the wall that will be covered by posters.

In the Archery with an especial gun, the probability of goal is $90 \%.$ If we continue our work until we goal. [b]i)[/b] What is the probability which exactly $3$ balls consumed. [b]ii)[/b] What is the probability which at least $3$ balls consumed.

The friends Alex, Ben, and Charles prepared a lot of labels and wrote one of the numbers $2,3,4,5,6,7,8$ on each label. Then Mary joined them and glued one label onto the forehead of each friend. Of course, each of the friends can see the labels on the others’ foreheads, but not the one on his own forehead. Mary told them: ”The numbers on your foreheads are not all distinct, and their product is a perfect square.” Can any of the friends find out the number on his forehead?

Prove that the sum of dihedral angles in an arbitrary tetrahedron is greater than $2\pi$

Kevin colors a ninja star on a piece of graph paper where each small square has area $1$ square inch. Find the area of the region colored, in square inches. [img]https://cdn.artofproblemsolving.com/attachments/3/3/86f0ae7465e99d3e4bd3a816201383b98dc429.png[/img]

Let $ABC$ be a triangle with $AB=AC$. Let $D$ be the midpoint of $BC$, $M$ the midpoint of $AD$ and $N$ the foot of the perpendicular from $D$ to $BM$. Prove that $\angle ANC = 90^\circ$.