Let $ABCD$ be a parallelogram and let $E$ be a point in the plane such that $AE\perp AB$ and $BC\perp EC$. Show that either $\angle AED=\angle BEC$ or $\angle AED+\angle BEC=180^\circ$.

The first of an infinite triangular spreadsheet the line contains one number, the second line contains two numbers, the third line contains three numbers, and so on. In doing so is in any $k$-th row ($k = 1, 2, 3,...$) in the first and last place the number $k$, each other the number in the table is found, however, than in the previous row the least common of the two numbers above it multiple (the adjacent figure shows the first five rows of this table). We choose any two numbers from the table that are not in their row in the first or last place. Prove that one of the selected numbers is divisible by another. [img]https://cdn.artofproblemsolving.com/attachments/3/7/107d8999d9f04777719a0f1b1df418dbe00023.png[/img]

The values of the polynomial $P(x) = 2x^3-30x^2+cx$ for any three consecutive integers are also three consecutive integers. Find these values.

It is known that triangle $ABC$ is not isosceles with altitudes of $AA_1, BB_1$, and $CC_1$. Suppose $B_A$ and $C_A$ respectively points on $BB_1$ and $CC_1$ so that $A_1B_A$ is perpendicular on $BB_1$ and $A_1C_A$ is perpendicular on $CC_1$. Lines $B_AC_A$ and $BC$ intersect at the point $T_A$ . Define in the same way the points $T_B$ and $T_C$ . Prove that points $T_A, T_B$, and $T_C$ are collinear.

Find all pairs $(a,b)$ of different positive integers that satisfy the equation $W(a)=W(b)$, where $W(x)=x^{4}-3x^{3}+5x^{2}-9x$.

Convex cyclic quadrilateral $ABCD$ satisfies $AC \perp BD$ and $AC$ intersects $BD$ at $H$. Let the line through $H$ perpendicular to $AD$ and the line through $H$ perpendicular to $AB$ intersect $CB$ and $CD$ at $P$ and $Q$, respectively. The circumcircle of $\triangle CPQ$ intersects line $AC$ again at $X \ne C$. Given that $AB=13$, $BD=14$, and $AD=15$, the length of $AX$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

Prove that for any prime $p,$ there exists a positive integer $n$ such that \[1^n+2^{n-1}+3^{n-2}+\cdots+n^1\equiv 2020\pmod{p}.\] [i]Robin Son[/i]

In triangle $ABC$, a point $M$ is the midpoint of $AB$, and a point $I$ is the incentre. Point $A_1$ is the reflection of $A$ in $BI$, and $B_1$ is the reflection of $B$ in $AI$. Let $N$ be the midpoint of $A_1B_1$. Prove that $IN > IM$.

Suppose $\, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions: (i) $\, m-n \,$ divides $\, q_{m}-q_{n}\,$ for $\, m > n \geq 0,$ (ii) there is a polynomial $\, P \,$ such that $\, |q_{n}| < P(n) \,$ for all $\, n$ Prove that there is a polynomial $\, Q \,$ such that $\, q_{n}= Q(n) \,$ for all $\, n$.

A $9\times9$ checkerboard is colored with 2 colors. If we choose any $3\times1$ region on the checkerboard we can paint all of the squares in that region with the color that is in the majority in that region. Show that with a finite number of these operations, we can paint the checkerboard all in the same color.

The cirumradius and the inradius of triangle $ABC$ are equal to $R$ and $r, O, I$ are the centers of respective circles. External bisector of angle $C$ intersect $AB$ in point $P$. Point $Q$ is the projection of $P$ to line $OI$. Find distance $OQ.$ (A.Zaslavsky, A.Akopjan)

The sequence $(x_n)_{n\in\mathbb N}$ is defined by $x_1=x_2=1$, $x_{n+2}=14x_{n+1}-x_n-4$ for each $n\in\mathbb N$. Prove that all terms of this sequence are perfect squares.

Let $ABC$ be an isosceles triangle with $\angle BAC = 108^o$. The angle bisector of the $\angle ABC$ intersects the circumcircle of a triangle $ABC$ at the point $D$. Let $E$ be a point on segment $CB$ such that $AB =BE$. Prove that the perpendicular bisector of $CD$ is tangent to circumcircle of triangle $ABE$ . (Bohdan Zheliabovskyi)

Consider triangle $ABC$ with side lengths $AB = 4$, $BC = 7$, and $AC = 8$. Let $M$ be the midpoint of segment $AB$, and let $N$ be the point on the interior of segment $AC$ that also lies on the circumcircle of triangle $MBC$. Compute $BN$.

Hexagon $ABCDEF$ has an inscribed circle $\Omega$ that is tangent to each of its sides. If $AB = 12$, $\angle FAB = 120^o$, and $\angle ABC = 150^o$, and if the radius of $\Omega$ can be written as $m +\sqrt{n}$ for positive integers $m, n$, find $m + n$.

Solve for real values of parameter $a$, the inequality : $$\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}$$

Let $n$ is positive integer, $D_n$ is a set of all positive divisor of $n$ and $f(n)=\sum_{d\in D_n}{\frac{1}{1+d}}$ Prove that for all positive integer $m$, $\sum_{i=1}^{m}{f(i)} <m$

Let $H$ be a complex Hilbert space. Let $T:H\to H$ be a bounded linear operator such that $|(Tx,x)|\le\lVert x\rVert^2$ for each $x\in H$. Assume that $\mu\in\mathbb C$, $|\mu|=1$, is an eigenvalue with the corresponding eigenspace $E=\{\phi\in H:T\phi=\mu\phi\}$. Prove that the orthogonal complement $E^\perp=\{x\in H:\forall\phi\in E:(x,\phi)=0\}$ of $E$ is $T$-invariant, i.e., $T(E^\perp)\subseteq E^\perp$.

Let $\vartriangle ABC$ be an acute triangle with incenter $I$. The line passing through $I$ parallel to $AC$ intersects $AB$ at $M$, and the line passing through $I$ parallel to $AB$ intersects $AC$ at $N$. Let the line $MN$ intersect the circumcircle of $\vartriangle ABC$ at $X$ and $Y$ . Let $Z$ be the midpoint of arc $BC$ (not containing $A$). Prove that $I$ is the orthocenter of $\vartriangle XY Z$

Show that the sequence $\{a_{n}\}_{n \ge 1}$ defined by $a_{n}=\lfloor n\sqrt{2}\rfloor$ contains an infinite number of integer powers of $2$.

Andriy and Olesya take turns (Andriy starts) in a $2 \times 1$ rectangle, drawing horizontal segments of length $2$ or vertical segments of length $1$, as shown in the figure below. [img]https://i.ibb.co/qWqWxgh/Kyiv-MO-2021-Round-1-7-2.png[/img] After each move, the value $P$ is calculated - the total perimeter of all small rectangles that are formed (i.e., those inside which no other segment passes). The winner is the one after whose move $P$ is divisible by $2021$ for the first time. Who has a winning strategy? [i]Proposed by Bogdan Rublov[/i]

Consider the set of solutions of the equation $$x^2+y^3=z^2.$$ in positive integers. Is it finite or infinite? (Folklore)

On side $AE$ of regular pentagon $ABCDE$ there is an equilateral triangle $AEF$, and on side $AB$ of the pentagon there is a square $ABHG$ as shown. Find the degree measure of angle $AFG$. [img]https://cdn.artofproblemsolving.com/attachments/7/7/0d689d2665e67c9f9afdf193fb0a2db6dddb3d.png[/img]

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]